On the Complexity of
the st Connectivity Problem
by
Chung Keung Po on
A thesis submitted in conformity with the requirements
for the degree of Do ctor of Philosophy
Graduate Department of Computer Science
University of Toronto
c
Copyright by Chung Keung Po on
Abstract
On the complexity of the st connectivity problem
Chung Keung Po on
Do ctor of Philosophy
Department of Computer Science
University of Toronto
The directed st connectivity problem is fundamental to computer science There
are many applications which require algorithms to solve the problem in small space
and preferably in small time as well Furthermore its space and time space complex
ities are related to several long standing op en problems in complexity theory
Depth and breadth rst search are well known algorithms that solve the problem
in optimal i e O n m time while using O n log n space where n and m are the
numb er of no des and edges in the graph resp ectively It can also b e solved in O log n
O log n
space and time by Savitch s algorithm For space S b etween log n and
O log n log n S
n log n the b est running time is T mn due to Barnes et al
Establishing matching lower b ounds on the Turing machine mo del has b een one of
the ma jor challenges in complexity theory
In this thesis we intro duce the natural structured NNJAG No de named JAG
mo del which is a generalization of Co ok and Racko s JAG Jumping Automa
ton for Graphs mo del For the space complexity Co ok and Racko prove an
log n log log n lower b ound on the JAG mo del This is extended to a ran
domized JAG by Berman and Simon We further extend this to the probabilis
tic NNJAG mo del For the time space tradeo we prove an upp er b ound of T
O log n log n S O
n on a JAG via a simulation of the Barnes et al algorithm We
p
log n log n S
log log n
also prove a lower b ound of T nS log n on an NNJAG im
proving the previous b ounds of T n S on NNJAGs by Edmonds and
T n S log n on JAGs by Barnes and Edmonds Our result almost matches
the Barnes et al algorithm and seems to suggest that st connectivity cannot b e de
termined in p olynomial time and p olylogarithmic space simultaneously For example i
when the space is S n for any constant the NNJAG lower b ound implies
a sup er p olynomial running time This lower b ound on time space tradeo is joint
work with Edmonds ii
acknowledgement
I would like to thank my sup ervisor Allan Boro din for his guidance and supp ort
esp ecially during the last year of my PhD study
I would also like to thank the professors the o ce clerks and the graduate students
in the Department of Computer Science who together provided such an enjoyable
environment during my stay Among them I am esp ecially grateful to Je Edmonds
who constantly help ed me along and who is also a co author of the result presented
in Chapter
Finally but not the least I thank my parents and family for their supp ort and I
dedicated this b elated thesis to the memory of my father Wah Po on iii
Contents
Intro duction
The imp ortance of stcon
Previous results on random access machines
Previous results on structured computation
mo dels
Contributions and organization of this thesis
Mo dels of Computation
The branching program mo del
The JAG mo del
The non uniform mo del
The uniform mo del
The NNJAG mo del
Comparisons among the mo dels
Probabilistic NNJAGs
Nondeterministic NNJAGs
A Space Lower Bound
The deterministic case
The skinny trees
Blo cks of p ebbles
k computations
Extension to probabilistic NNJAG with sided error iv
Extension to probabilistic NNJAG with sided error
A Time space Upp er Bound
The Barnes et al algorithm
The mo di ed Savitch algorithm
The mo di ed breadth rst search
Simulation on an NNJAG
Simulation on a JAG
A Time space Lower Bound
Layered Graphs
Recursive Layered Graphs
De nition of Progress
Viewing NNJAGs as Branching Programs
Pro of Outline
Collapsing an NNJAG
Pro of of Main Lemma
Bounding SUM
The Branching Pro cess Game
The Edge Collision Game
Branching Pro cess Game vs Edge Collision Game
Bounding SUM
Probabilistic NNJAGs with sided error
Future Work
Variations of JAGs NNJAGs
Finite Mo del Theory v
Chapter
Intro duction
The imp ortance of stcon
Given a directed graph G and two distinguished no des s and t in G the directed st
connectivity problem stcon is to determine whether there is a directed path from
s to t in G It is a fundamental problem in computer science b ecause it is the natural
abstraction of many search problems In many applications e g in game searching
program veri cation and some database applications the underlying search graph is
exp onentially large compared with a machine s memory Hence we need to nd algo
rithms which require working space much less than the size of the graph Very often
reducing the space blows up the running time b ecause re computation is necessary
Therefore a more ambitious goal is to nd an algorithm that runs in small time and
space simultaneously These considerations lead to the following two questions
Q How much space is necessary for solving stcon and
Q Given a certain amount of space how fast can stcon b e solved
The theory of computational complexity addresses these questions in a more gen
eral setting Instead of studying individual problems we study classes of problems
Some commonly used notation for denoting complexity classes is listed b elow
DTIME f n the class of problems solvable in O f n time using a deterministic
O
Turing machine In particular the class DTIME n is denoted by P
DSPACE f n the class of problems solvable in O f n space using a determin
O
istic Turing machine The class DSPACE log n and DSPACE log n are
denoted by L and Polylogspace resp ectively
NSPACE f n the class of problems solvable in O f n space using a nondeter
ministic Turing machine The class NSPACE log n is denoted by NL
DTISP f n g n the class of problems solvable in O f n time and O g n space
O
simultaneously using a deterministic Turing machine The class DTISP n
O
log n is denoted by SC
With a prop er notion of problem reduction the complexity of a whole class of prob
lems b oils down to the complexity of a prototypical problem in that class Such a
problem is said to b e complete for the class under this typ e of reductions In this
resp ect stcon has an added imp ortance b ecause it is complete for the complexity
class NL under logspace reduction See Jon JLL for more problems complete
for NL and AHV for the relation b etween NL and relational database theory
Two ma jor op en problems concerning NL are
Q Is NL L and
Q Is N L S C
Understanding the space complexity and time space tradeo of stcon seems to b e
the key to answering the ab ove questions
For example proving that there is no deterministic O log n space algorithm for
stcon would separate the classes L and NL On the other hand devising such
an algorithm would show that NL L By a padding argument Sav such an
algorithm would in fact imply that DSPACE f n NSPACE f n for any space
constructible function f n log n
O
Similarly showing that there is no algorithm for stcon using n time and
O
log n space simultaneously would separate NL and SC while demonstrating such
an algorithm would put NL inside SC As is well known stcon is in P Polylogspace
Therefore proving that stcon is not in SC would also prove that SC ( P
Polylogspace which is another long standing conjecture in complexity theory Joh
Previous results on random access machines
On a determinstic random access machine depth and breadth rst search are well
known algorithms that solve stcon in O n m time using O n log n space where n
and m are the numb er of no des and edges resp ectively The running time is optimal
b ecause at least n m time is needed to access every edge and every no de in the
graph At the other extreme Savitch s algorithm Sav requires only O log n space
which is currently the smallest space achievable The main drawback is however
O log n
that it takes time Nevertheless these two results imply that stcon is in
P Polylogspace
Tompa Tom shows that a certain natural approach repeated squaring for com
puting stcon has no implementation which runs in p olynomial time and sub linear
space simultaneously However Barnes et al BBRS construct an algorithm which
p
log n
uses only n space which is sub linear while keeping p olynomial running
time This shows that the rep eated squaring metho d is to o restricted Futhermore
O log n log n S
their algorithm implies a general time space tradeo of T mn
for log n S n log n Note that when the space is n for any constant
O log n
the running time is which is sup er p olynomial Therefore they
still have not exhibited an SC typ e algorithm
For lower b ounds much less progress has b een made It is not known whether
Savitch s algorithm is space optimal and whether the Barnes et al algorithm is time
optimal for restricted space Even attempts to prove sup er logarithmic space lower
b ounds or sup er linear time lower b ounds for sub linear space have b een unsuccessful
Hence the questions Q and Q remain op en
Previous results on structured computation
mo dels
For most natural decision problems no non trivial lower b ound has b een proved
on any general mo del of computation such as branching programs Random Access
Machines or multitap e Turing machines In order to gain more insight into the prob
lems di erent structured mo dels are considered for di erent problems For example
comparison based branching programs have b een used to study the time space trade
o of the element distinctness problem by Boro din et al BFMadH and Yao
Yao See Boro din Bor for a recent survey on time space tradeo s for various
problems on various mo dels
For the st connectivity problem Co ok and Racko CR intro duce a natural
structured mo del called the Jumping Automaton for Graphs JAG Informally a
JAG has a set of p ebbles which have the dual purp ose of marking the no des in
the input graph and gathering information ab out the input The p ebbles can move
around in the graph by walking along the edges or jumping to other p ebbles A formal
de nition will b e given in the next chapter Although the mo del is restricted it is
p owerful enough to express all known deterministic algorithms for graph connectivity
problems with at most a p olynomial factor blow up in time and logarithmic factor
blow up in space For example depth and breadth rst search can b oth b e translated
directly to JAG algorithms Co ok and Racko CR show how to simulate Savitch s
algorithm using e ectively the same time and space resources
There has b een much more success in proving lower b ounds on this structured
mo del First Co ok and Racko CR prove a space lower b ound of log n log log n
which almost matches the upp er b ound of O log n This result is then extended
by Berman and Simon BS to a randomized JAG More precisely they show
c
log n
that any randomized JAG solving stcon with at most time requires at least
c log n log log n space where c is a constant dep ending on c
For the time space tradeo Barnes and Edmonds BE prove a lower b ound of
T n S log n on the JAG mo del On a more general mo del called the NNJAG
mo del No de Named JAG to b e de ned in the next chapter Edmonds Edm
proves a lower b ound of T n S However there is still a big gap b etween
the upp er and lower b ounds In particular the ab ove results do not yield any sup er
p olynomial lower b ound on time T no matter how small the space S b ecomes
There are also other approaches to answering question Q It has b een shown
that there is a signi cant relationship b etween complexity theory and nite mo del
theory also known as descriptive complexity Many complexity classes can b e char
acterized in terms of logic See Immerman Imm for example In particular
the st connectivity prop erty of a graph can b e expressed as a First Order logic FO
sentence containing a Transitive Closure TC op erator provided an ordering is im
p osed on the graph Conversely any prop erty expressible in FO TC over ordered
structures can b e checked in NL In the terminology of descriptive complexity NL
FO TC over ordered structures Similarly L FO DTC over ordered struc
tures where DTC stands for the Deterministic Transitive Closure op erator Thus sep
arating NL and L amounts to showing that stcon is not expressible in FO DTC
over ordered structures Gr adel and McColm GM show that stcon is not express
ible in FO DTC over unordered structures Recently Etessami and Immerman
EI have extended this result to one way local ly ordered structures
Contributions and organization of this thesis
In this thesis we take a step further in understanding the space and time space
complexities of stcon We have not answered the questions Q and Q but
have provided evidence that their answers are negative by proving NNJAG lower
b ounds on the space and time space tradeo
In Chapter we intro duce the NNJAG No de named JAG mo del which is a
generalization of the JAG mo del It follows from Sav that if we further allow an
NNJAG to p erform the strong jumping op erations then the resulting machine is as
p owerful as a branching program for solving graph theoretic problems Hence the
NNJAG mo del seems to b e a suitable intermediate mo del b etween the JAG mo del
and a general computation mo del for studying graph connectivity problems
We also give some evidence that a nondeterministic NNJAG is actually more p ow
stcon erful than a nondeterministic JAG On one hand we do not know how to solve
which is the complement problem of stcon in O log n space on a nondeterministic
JAG On the other hand we show that a nondeterministic NNJAG can solve stcon
in O log n space by simulating Immerman Szelep cs enyi s inductive counting tech
nique Imm Sze Using the same technique we show that a nondeterministic
NNJAG is essentially as p owerful as a nondeterministic branching program in solving
a large class of graph theoretic problems
In Chapter we extend the space lower b ound of Co ok and Racko CR on
the JAG mo del as well as that of Berman and Simon BS on the randomized JAG
mo del to the probabilistic NNJAG mo del Thus we b elieve we are a step closer
towards separating L from NL
O log n log n S O
In Chapter we show a time space upp er b ound of T n
on the JAG mo del This result further substantiates our claim that the JAG and
hence the NNJAG mo del is a reasonable structured mo del for graph connectivity
problems
In Chapter we give some evidence that stcon is not in SC and thus that
SC ( P Polylogspace In particular we improve the lower b ounds of Barnes and
p
log n log n S
log log n
Edmonds Edm BE to T nS log n on the probabilistic
NNJAG mo del This is the rst result proved on a probabilistic NNJAG which gives
a sup er p olynomial lower b ound on time when S is su ciently small For example
log n
log log n
This lower when S is at most n for any constant T is at least
b ound also closely matches the upp er b ound of Barnes et al BBRS which can
b e implemented on an NNJAG as implied by Chapter The result presented in
Chapter is joint work with Edmonds
Finally in Chapter some future directions for answering questions Q and
Q are discussed
Chapter
Mo dels of Computation
In this chapter we formally de ne the JAG mo del of Co ok and Racko CR the
Node named JAG or NNJAG mo del and the r way branching program BP mo del
of Boro din and Co ok BC as they will b e used in the following chapters We also
compare them with the well known mo dels of multitap e Turing machines TM and
Random Access Machines RAM We then de ne the probabilistic and nondetermin
istic NNJAG mo dels We conclude the chapter by showing that a nondeterministic
NNJAG is nearly as p owerful as a nondeterministic branching program for solving
many graph theoretic problems
The branching program mo del
An r way branching program BC is a general mo del of computation in fact the
most general one among the ve mo dels mentioned in this chapter
It is a non uniform mo del of computation That is there will b e a di erent
branching program for each input size and these branching programs need not have
any similarity among them For a xed value of input size the corresp onding branch
ing program is a directed acyclic graph with a designated source no de and a numb er
of sink no des Each sink no de in the graph is lab elled with either accept or reject
and each non sink no de is lab elled with an input variable Furthermore for each
p ossible value of the input variable that lab els a non sink no de there is a unique
out edge lab elled with that value emanating from that no de Hence the out degree of
the graph is at most r where r is the maximum numb er of di erent values p ossible
for an input variable
The no des in this graph represent the p ossible states of the machine s memory
In particular the source no de represents the initial memory state In each step the
machine queries an input variable dep ending on the current state of its memory and
changes its memory to another state based on the value returned Which variable to
query and which state to go to on each p ossible outcome are sp eci ed by the graph
It is easy to see that for every input there will b e a unique path in the graph from
the source no de to a sink no de We call such a path the computation path fol lowed
by the input We say that a branching program accepts an input if and only if the
computation path followed by the input leads to a sink no de lab elled with accept
For stcon an n no de input graph of out degree at most deg is sp eci ed by the
variables X for every u n and every deg Such a variable
h u i
X has value v if there is a directed edge from u to v with lab el or the value
hu i
unde ned if there is no such edge The height of a branching program is de ned
as the length of its longest path from the source to a sink The space S is de ned as
the logarithm of the numb er of no des in the branching program i e the numb er of
bits needed to sp ecify the current state of the machine s memory
The JAG mo del
We rst de ne the JAG mo del as a non uniform computation mo del for solving the
sp eci c decision problem of stcon Then we give the de nition of a uniform JAG
for solving more general graph problems
The non uniform mo del
As the mo del is non uniform there will b e a JAG for each size n of the input graph
For a xed value of n the corresp onding JAG J is an automaton consisting of a set
of p distinguishable p ebbles a set of q states and a transition function In general
the values of p and q will b e functions of n and the transition function will b e
sp eci cally designed to deal with n no de graphs
For stcon the input to J is a triple G s t where G is an n no de graph con
taining no des s and t For every no de in G its out edges are lab elled by consecutive
integers starting at The no des in G are also lab elled from up to n We de ne
the instantaneous description id of J as the pair Q where Q is the current state
and is a mapping of p ebbles to no des sp ecifying the current lo cation of each p ebble
in the graph When J is in id Q the transition function determines the next
move for J based on
the state Q and
which p ebbles are on the same no de and which are not according to the mapping
i e for each pair of p ebbles P and P whether P P
A move is either a walk or a jump A walk P i Q consists of moving p ebble P
along the edge lab elled i that comes out of the no de P and then assuming state
Q If there is no such edge the p ebble just remains on the same no de A jump
P P Q consists of moving p ebble P to the no de P and then assuming state
Q The JAG J is initialized to have state Q and with p ebbles P P on no de
p
s and p ebble P on no de t which makes no de t distinguishable from the rest It is
p
said to accept an input G s t if it enters an accepting state on this input It solves
stcon for n no de graphs if for every input G s t where G is an n no de directed
graph it accepts the input if and only if there is a directed path from s to t in G We
de ne the space used by the JAG as p log n log q i e the numb er of bits to sp ecify
an id The time used is the numb er of moves it has made
The uniform mo del
To motivate the ab ove de nition and to provide a mo del for easier description of
JAG algorithms we de ne a uniform JAG mo del called the RAM JAG mo del A
1
In CR the input graph is not lab elled Our de nition here is equivalent to theirs b ecause
the machine is only allowed to know the coincidence partition of the p ebbles
RAM JAG is a machine that contains a nite state control a set of ordinary registers
and a set of sp ecial registers called pebble registers Each register has log n bits where
n is the numb er of no des in the input graph The instruction set includes the usual
op erations allowed in a regular RAM but are restricted to op erate on the ordinary
registers only It also includes three sp ecial instructions namely walk jump and
compare The instruction w al k P j replaces the content which is u say of the
th
p ebble register P with the no de p ointed to by the j edge in the adjacency list
of no de u The instruction j ump P P copies the content of p ebble register P to
p ebble register P The instruction compar e P P R p erforms a comparison
on p ebble registers P and P and stores the result in the ordinary register R
For stcon the input graph is represented as an adjacency list and is accessible
only by the ab ove op erations on p ebble registers Two p ebble registers are designated
as the input registers containing the no des s and t For more general graph theoretic
problems such as computing f u u v v there will b e i input registers
i j
and j output registers for storing the input and output resp ectively Both the set of
input registers and output registers may consist of b oth regular and p ebble registers
The space is de ned as the numb er of distinct registers used times log n where n is
the numb er of no des in the graph
The NNJAG mo del
In the JAG mo del the machine can only detect the coincidence partition of p ebbles
i e p erform comparisons on the corresp onding p ebble registers We remove
this restriction in an NNJAG by allowing it to detect the names of the no des that
contain p ebbles In other words the machine is allowed to compute any arbitrary
function based on the whole content of its memory However it is still restricted in
the way it can change its p ebble registers
Formally a No de named JAG NNJAG is de ned in the same way as a JAG
except that the transition function determines the next move based on
the state Q and
the mapping
For simplicity we also make the following technical changes First we assume that
the lab els of no des s and t are always xed say as and n resp ectively Hence
s and t are not part of the input Second there is no need to mark no de t with a
sp ecial p ebble b ecause the p ebbles can see the lab el of the no des they are sitting
on Therefore all the p ebbles are on no de s initially
A RAM NNJAG can also b e de ned similarly to a RAM JAG It has the walk
and jump op erations that are present in a RAM JAG as well as the copy instruction
copy P R which copies the content of the p ebble register P to the ordinary register
R Note that the reverse op eration of copying an ordinary register to a p ebble register
is not allowed
There is evidence that the NNJAG mo del may b e strictly stronger than the JAG
mo del For example a more precise statement of a result in BE is that pT
n log n for a JAG that solves stcon with p p ebbles The result holds even if
the JAG has an in nite numb er of states and the input is restricted to a particular
class of graphs However an NNJAG with an in nite numb er of states can traverse
any graph in this class in linear time Before showing more evidence of this typ e and
extending the mo dels to probabilistic and nondeterministic ones let us rst have a
comparison of all the mo dels in the following section
Comparisons among the mo dels
The relative p ower in terms of generality in de nition among branching programs
RAMs multitap e Turing machines JAGs and NNJAGs are depicted in Figure
Branching programs RAMs and TMs are general computation mo dels while NN
JAGs and JAGs are structured mo dels for graph connectivity problems However in
terms of stcon RAMs and TMs are incomparable with NNJAGs and JAGs To ex
plain the diagram in more detail let us consider the main features of a machine mo del
A deterministic sequential machine mo del is usually de ned to consist of a control
unit a memory an input output unit and some communication channels b etween the BP
RAM NNJAG
RAM- NNJAG TM JAG
RAM-JAG
Figure Mo dels of Computations
control unit and the other comp onents The content of the accessed memory and the
status of b oth the control unit and communication channels constitute a con guration
i e a full description of the state of a machine The de nition of mo del must also
sp ecify the rules for transition from one con guration to another usually in the form
of a transition function The ab ove mo dels can b e compared in the following asp ects
the accessing of inputs memory and the transition function
Both a branching program and a RAM allow random access to the inputs and
memory In a TM the access of input is restricted according to a linear order imp osed
by an enco ding of the input An NNJAG JAG is also restricted in the order of
accessing of the input but in a di erent way than a TM In order to detect the existence
of an edge the machine has to place a p ebble on the source of the p otential edge
b eforehand In addition there may b e some parts of input which are not accessible
at all
Consider the second asp ect The transition function of a TM is based on and
a ects a constant numb er of tap e cells and the nite state control Thus the function
is only from a nite domain to a nite range A RAM is slightly less restricted in
that it can lo ok at a constant numb er of memory cells and compute some function
chosen from a reasonable instruction set In most de nitions of RAM mo dels a
memory cell has log n bits where n is the input size As well it changes a constant
numb er of memory cells Both the RAM and TM mo dels are uniform mo dels b ecause
the transition function has to b e de ned for all input sizes A branching program is
the most unrestricted among the ve mo dels In one step it computes an arbitrary
function based on the whole con guration and changes the whole con guration Thus
it is often unrealistically fast Therefore it is used primarily for studying space or
time space tradeo s esp ecially lower b ounds Similar to a BP the transition function
of an NNJAG is also based on the whole con guration which consists of the state space
and the p ebble space The p ebbles can b e viewed as a set of registers in a RAM or
a non constant numb er of tap e heads in a TM The transition function can change
the whole state space but only the contents of one register in the p ebble space A
JAG is slightly weaker than an NNJAG in the typ e of functions computable More
sp eci cally it can only p erform comparisons on the registers in the p ebble
space The BP NNJAG and JAG are all non uniform mo dels b ecause the transition
function need only work for one input size
Probabilistic NNJAGs
Following BS we intro duce randomness to an NNJAG J by allowing the machine
to access a numb er of fresh random bits at each step The transition function is
then extended so that it dep ends on the random bits as well The random bits R
t
obtained in step t will not b e available for the future steps unless the machine stores
them in its memory Also they cannot b e available b efore step t
For the purp ose of proving lower b ounds it is more convenient to view J as a
distribution of deterministic NNJAGs That is we think of the whole sequence of
random bits R R R as b eing chosen b efore the computation starts Then
for di erent R a di erent deterministic NNJAG is chosen to carry out the compu
tation However not every probabilistic machine M with space S can b e viewed as
a distribution of deterministic machines with the same space S The deterministic
machine somehow has to have a p ointer into the random string R in order to sp ecify
which random bits it is currently using This p ointer requires log T bits if M runs in
O S
time T If T each deterministic machine requires only O S space However
S
if T then each machine requires S space For example Boro din et al
BBR shows that a deterministic WAG a JAG whose p ebbles are not allowed
to jump with p p ebbles and q states satisfying pq O n requires in nite time to
traverse undirected cycles in the worst case Thus any distribution of deterministic
WAGs with pq O n will have in nite exp ected time for the worst case input How
ever a probabilistic WAG with p ebble and O states can traverse any such graph
in exp ected O n time AKL
Fortunately we shall only b e interested in time b ounded computations When a
probabilistic NNJAG J with p p ebbles and q states has execution time b ounded by T
steps its computation can b e simulated by a distribution of deterministic NNJAGs
J s with p p ebbles and q T states and with T time constructed as follows For
R
each state Q in J each deterministic NNJAG will have the states h Qi h Qi
hT Qi Supp ose R Q P P Q for some t T Then the transition
t
function of J is de ned so that ht Qi P P ht Q i The case is
R R R
similar when R Q P i Q From now on we view a probabilistic NNJAG
t
as a distribution of deterministic NNJAGs constructed as describ ed ab ove
A probabilistic NNJAG J is said to solve stcon resp ectively stcon with sided
error if for every input G s t the probability over R of J entering an accepting
state is at least when there is a path resp ectively there is not a path from no de
s to t and otherwise Similarly J is said to solve stcon with sided error if for
every input G s t the probability over R of J entering an accepting state is at
least when there is a path from no de s to t and at most otherwise
2
This is obviously the case for the study of time space tradeo s For proving space lower b ounds
results of Gill Gil show that a probabilistic machine with unb ounded time can actually have
nondeterministic p ower For example given unb ounded time a probabilistic JAG with p ebble
O (1)
and n states can solve directed stcon with high probability Thus we need to put a time
b ound on the machine b efore we can prove any interesting space lower b ounds
Nondeterministic NNJAGs
The extension of a deterministic NNJAG to a nondeterministic one can b e done quite
easily We simply allow the transition function to return more than one move A
stcon if for every nondeterministic NNJAG J is said to solve stcon resp ectively
input G there is a sequence of moves which results in J entering an accepting state
if and only if there is resp ectively there is not a directed path from s to t in G
Now we provide some evidence that a nondeterministic NNJAG is strictly stronger
than a nondeterministic JAG In particular we show that a nondeterministic NN
JAG can solve the st nonconnectivity problem stcon easily by simulating Immer
man Szelep cs enyi s inductive counting metho d However it is not known how to
solve stcon in O log n space on a nondeterministic JAG
Theorem For every n there is a nondeterministic NNJAG J which solves
O
stcon for al l n node graphs using O log n space and n time
Pro of Let N b e the set of no des reachable from no de s in at most k steps Let
k
N denote the cardinality of the set N The main idea in Immerman Szelep cs enyi s
inductive counting is to calculate N given N For each no de v increment a
k k
counter d if and only if v N To determine if v N do the following
k k
Guess N di erent no des u and verify that each is in N
k k
For each such u test if u v or u v is an edge in the graph If so v N
k
Otherwise v N
k
The detailed algorithm in Figure shows how Steps and can b e carried out
in a clever way so that the space is kept to b e O log n
Note that the ab ove algorithm requires the machine to generate all the p ossible
values for no des u and v Both a JAG and NNJAG can do so by using their state
space However the algorithm further requires the machine to check that u and v
function count k hy p
fcount k N returns N g
k k
b egin
d
for v to n do
c
for u to n do
if we guess u N then
k
verify that u N and increment c
k
f If u is not found to b e in N then halt g
k
if u v or u v is an edge then increment d
if c hy p then halt
return d
end
Figure The pro cedure to compute N given N
k k
satisfy certain prop erties which is b est done in the p ebble space More precisely it
wants to check whether there is a path of length at most k from s to u and if so
whether u v or u v is an edge An NNJAG can do so by nondeterministically
walking a p ebble P from s for at most k edges and see if it hits no de u That is it
compares the part in the p ebble space corresp onding to p ebble P with the part in
the state space storing the value u If so it checks whether P is actually on no de v
or P can reach no de v by walking one more edge Again this is done by comparing a
certain part of the p ebble space with a certain part of the state space However this
kind of comparison is not allowed in a JAG
In the next theorem we show that a nondeterministic NNJAG is nearly as p ow
erful as a nondeterministic Turing machine in solving graph theoretic problems The
inspirations were from Theorem of Boro din et al BBR and a discussion with
Allan Boro din and Stephen Co ok Note that any graph theoretic decision problem H
on a family of graphs G can b e simply de ned as a subset of G Here we are interested
in a family G of graphs called s reachable graphs A graph G is said to b e s reachable
if every no de in G is reachable from the no de s
Theorem Let G be a family of s reachable graphs For any graph theoretic prob
lem H G if H can be solved by a nondeterministic Turing machine in time T n
and space S n simultaneously then it can also be solved by a nondeterministic NN
JAG in time O nT n and space O S n log n simultaneously provided a pebble
is placed on node s initial ly
Pro of It is shown in Savitch Sav that a nondeterministic JAG with strong jump
ing can simulate an arbitrary nondeterministic Turing machine e ciently A strong
jump in a JAG is a move of the form jump p ebble P from its current no de u say
to no de u and assume state Q It su ces to show how to simulate a strong
O
jump P Q on a nondeterministic NNJAG with n extra states First we store
the value u using the extra states This takes only one step which changes the
state appropriately while keeping the p ebble lo cations intact Then we nondetermin
istically walk p ebble P for at most n steps using the extra states to count starting
from no de s If P is not on no de u we stop the computation Since every no de
in G is reachable from s P must b e on no de u for some computation path
Remark It can b e shown that an NOJAG a restricted NNJAG which can only
p erform comparisons on the no de names can also simulate the inductive
counting This seems to show that the relative ordering of the input structure is im
p ortant for solving the stcon problem Nisan and Ta Shma s SL algorithm NTS
for solving ustcon also seems to rely on the relative ordering of the input structure
For more ab out the ordering on input structures see Etessami s thesis Ete
Chapter
A Space Lower Bound
In this chapter we prove a space lower b ound for stcon on the probabilistic NNJAG
mo del We rst prove the result for the deterministic case and then extend it to a
probabilistic mo del
The deterministic case
Theorem Any NNJAG that solves stcon for every n node graph requires at
log n
least space
log log n
Pro of Let J b e an NNJAG that solves stcon for every n no de graph with p p ebbles
and q states Since an NNJAG with only one state is not very interesting we shall
assume that pq From Theorem to b e proved b elow there is a tree G with
p
p log pq no des such that there is a path from no de s to no de t and when
G is fed as input to J no de t is never visited by any p ebble That means none of
the in edges of t has b een traversed Let G b e the graph obtained by removing all
the in edges of t in G Then t is an isolated no de in G However J cannot tell
the di erence b etween G and G as none of the in edges of t in G has b een tried
Hence it must give the wrong answer for either G or G It follows that any NNJAG
J solving stcon for all n no de graphs with p p ebbles and q states must satisfy
p
n p log pq or p log n log p log pq If b oth p log n and
log q log n we can show that p log n log log n Therefore the space required
by J is p log n log q log n log log n
Theorem For every NNJAG J with p pebbles and q states such that pq
there is a graph G such that
G is a rooted tree with maximum out degree
p
G has at most p log pq nodes including two special nodes s and t
there is a directed path from s to t and
if J starts with al l pebbles on node s no pebble wil l ever visit node t during the
computation
Following the pro of technique of CR and BS we prove Theorem by rst
de ning a class of graphs called skinny trees from which the worst case graph for the
NNJAG will b e chosen Our graphs are recursively constructed as in CR BS
and are simpler than those in BS Then we characterize a sub computation of
an NNJAG on this typ e of graph according to the level of co op eration among the
p ebbles during this sub computation Finally we prove an inductive statement on sub
computations of various levels of p ebble co op eration from which Theorem follows
trivially The pro of uses a simpler counting argument than that used in Po o
The skinny trees
d
Let d b e an integer parameter to b e chosen later and let f g Then G is
a binary tree with vertex set V f v g f v v g f v v g and
s h i hd i h i hd i
edge set E E E where E f v v g f v v j i d g and
s
h i hi i i hi i
E f v v g f v v j i d g Edges in E have lab el and
s
h i hi i i hi i
edges in E have lab el Lab elling of no des will b e dealt with later The no de v
s
and v are called the start and goal of G resp ectively See Figure for an
hd d i
example of G
Vs = start
0 1
V1,0 V1,1
0 1
V2,0 V2,1 0 1
V3,0 V3,1 0 1
V4,0 V4,1 = goal
Figure G with d and
G is obtained by replacing each no de u in the graph G with a copy
k k
u
of G which will b e referred to as G Furthermore each
k k
u
edge u v in G is replaced by the edge u v where u goal of G
k k
v
and v start of G The lab el on u v is the same as that of u v
k
star t g oal
The start of G and the goal of G are chosen as
k k
the start and goal of G resp ectively See Figure for an example of
k
G
k
d
The family of skinny trees is the set of graphs f G j f g g
p p
p
Each graph in this family has d no des which are lab elled from up to
p
d in a way to confuse the NNJAG Supp ose we lab el the no des of a
p
tree by the depth rst ordering Then the tree can b e searched in O log d
space On the contrary lab eling the tree in the breadth rst ordering do es not seem
to help the searching In other words telling the NNJAG the depth of each p ebble
in the tree i e its distance from the start no de do es not seem to b e useful Thus
we shall lab el the graphs so that for every no de lab el l and every edge lab el j
L the distance of the no de with lab el l from start is the same in every skinny tree start
0 1 start
0 1 goal
0 1
0 1
goal
Figure Left a copy of G Right G with d and
k k
k
and
L the lab el on the destination no de of the edge with lab el j from the no de with
lab el l in G is xed for every skinny tree G provided such an edge exists in G
This can b e achieved by many di erent ways of lab elling but one simple way is the
consistent lab elling As a base case a set G of graphs G is said to b e consistently
label led with a set L of lab els if each no de in G has a unique lab el from L and
for all u f v g f v v g f v v g the lab el on no de u of
s h i hd i h i hd i
G is the same as that on no de u of G for all G G in G Recursively
a set G of G is said to b e consistently label led with a set of lab els L if
k
there is a partition of L into d sets L for u f v g f v v g
u s h i hd i
f v v g and for each u f v g f v v g f v v g
s
h i hd i h i hd i h i hd i
u
the set of graphs f G in G j G G g is lab elled
k k k
consistently with the set of lab els L It is easy to verify that a consistently lab elled
u
family of skinny trees satis es the ab ove lab elling prop erties L and L
For the deterministic case we shall set d p log pq and nd a di cult input
G s t for J where G is chosen from the family of skinny trees and s t are its start
and goal
Blo cks of p ebbles
To analyse the b ehaviour of an NNJAG on a skinny tree it is more convenient to
consider groups of p ebbles instead of individual p ebbles Since we shall study the
interaction among the p ebbles within an arbitrary sub computation the grouping of
p ebbles will b e chosen according to the particular sub computation in consideration
Consider a sub computation C which starts at time t A set of p ebbles is called
a block of the sub computation C if they lie on the same no de at time t and no other
p ebbles o ccupy that no de at the same time Note that the collection of all blo cks of a
sub computation forms a partition of the p ebbles so that two p ebbles are in the same
blo ck if and only if they are on the same no de b efore the sub computation starts
The continuation of a blo ck B at time t t denoted as B t is de ned recursively
as follows B t is just B itself For t t if the move from time t to t causes a
p ebble P of B t to hit either by a walk or jump some p ebbles in the continuation
of another blo ck B t then B t B t nf P g and B t B t f P g
All the other blo ck continuations are unchanged If the move do es not cause any
collision then B t B t for every blo ck B For convenience we shall use
blo ck to stand for continuation of blo ck when the context is clear
Note that the numb er of blo cks remaining at time t is non increasing as t increases
Blo cks are de ned at the b eginning of a sub computation New blo cks cannot b e
created even if p ebbles in the same blo ck move apart However old blo cks will
disapp ear when all of their p ebbles run into other blo cks If a p ebble in a blo ck B
hits a p ebble of another blo ck it no longer b elongs to blo ck B
An NNJAG may often want to jump the p ebbles of a blo ck to some other blo cks
For example it may want to use all the p ebbles of a blo ck that is trapp ed in some
subtree for traversing other parts of the graph Then the trapp ed blo ck disapp ears
We shall use the numb er of blo cks remaining at a certain time as a measure of the
level of co op eration among the p ebbles up to that moment The smaller the numb er
of blo cks is the higher is the co op eration This is one of the key ingredients in the
pro ofs of CR and BS
k computations
We now consider sub computations of an NNJAG on a skinny tree with various levels
of co op eration among the p ebbles A blo ck B of a sub computation C is said to
v
traverse a copy of G named G say if B lies on or ab ove
k k
v
the start of G at the b eginning of C and there is a step t such that some
k
v
p ebble of B t is on the goal of G A sub computation C on G
k p
is called a k computation if it stops as so on as
C some blo ck of C has traversed a copy of G or
k
C there are fewer than p k blo cks left or
C the NNJAG stops
If the NNJAG lo ops forever without satisfying C while maintaining at least p k
blo cks then the k computation do es not stop
The aim of a k computation is to traverse a copy of G while maintain
k
ing at least p k blo cks Hence it is said to b e successful if it ends with condition
C The lower b ound on the numb er of blo cks limits the p ower of the NNJAG in a
k computation by limiting the level of co op eration among the p ebbles When k
the co op eration is minimal All the p ebbles are isolated and no p ebble collisions are
allowed except p ossibly in the last step of an unsuccessful computation A blo ck
here is simply a p ebble which has to traverse a copy of G As k increases the
co op eration increases and a blo ck is something more than a p ebble However it also
has a harder task of traversing a copy of G At the extreme when k p
k
the NNJAG can work with its full p ower b ecause there is always at least one blo ck
in any computation In fact the whole computation of the NNJAG which starts
at the xed initial state Q and with all p ebbles on the start of G is a
p
p computation The task of that single blo ck on the start no de i e no de s is to
traverse the whole graph which is equivalent to sending some p ebble to the goal no de
i e no de t If we can nd a setting of such that any p computation fails
p
then Theorem is proved The following lemma shows that we can set the s
i
incrementally so that for any k p any k computation fails
d
Lemma For al l k p there exists f g such that for any
k
d
f g any k computation by J on G is not successful
k p p
We rst prove Theorem using Lemma Consider the p computation by J
which starts with all p ebbles on the start of G It follows from Lemma
p
that we can set so that it is not successful i e it do es not end with
p
condition C Since there is always at least one blo ck in any computation the
p computation must either end with condition C or never halt Hence the goal of
G is never traversed Finally note that the graph G has size
p p
p p
d p log pq Hence Theorem follows
We now prove Lemma by induction on k Let H k denote the statement of
Lemma for a xed value of k p
Base case H is true
Recall that no p ebble jump is allowed in a computation except p ossibly in the last
step of an unsuccessful computation Therefore each p ebble traces only one path
of a certain length The following lemma shows that not many di erent paths can b e
traced in computations
Lemma The sequence of id s a computation on G goes through can
p
be determined independent of by
p
A the initial id Q of the computation and
A for every pebble P the length of the path traced by P
Pro of Supp ose the computation starts at time t We shall prove by induction on
t t that the id at time t is uniquely determined
Base Case The id at time t is Q as sp eci ed by A
Induction Step Supp ose that the id at time t is known to b e Q and that
t t
the computation has not nished at time t If it has already nished then we need
not determine the id at time t at all Then Q determines the next move
t t
and hence the next state Q If the move is a jump then the new p ebble lo cations
t
are also xed Therefore let us assume the move is a walk P i Q Observe
t t
that and tell us the depth of P at time t and time t resp ectively b ecause of
t
the lab elling prop erty L Hence their di erence determines whether P has already
nished the distance sp eci ed in A If so P must b e a leaf and hence P will
t
remain on the same no de at time t Otherwise P must b e an internal no de
t
By lab elling prop erty L the edge lab el i and the no de lab el P determines
t
which no de P will go to Therefore in b oth cases can b e determined
t
p
p p
Note that there are d no des in the graph and hence q d
p
choices for A To b ound the numb er of choices for A observe that q d
the path traversed by a p ebble is longest when it starts from a child at the start of
a G In this case it has to traverse d edges to reach the goal of that G
one edge to the start of the next G and then d edges to reach the goal of that
new G In total it traverses as many as d edges b efore it completes traversing a
p
G Hence there are at most d choices for A Consequently there are at
p p
most q d di erent sequences of id s Of course some choices of A or some
combinations of A and A may not constitute a realistic computation but this
only decreases the numb er of p ossible computations In each computation at
p p
most p di erent paths can b e traced It follows that at most pq d s out of
d
the total p ossibilities will have a successful computation for some choice of A
and A When d p log pq there exists an such that the computation is
not successful for any choice of A and A Hence H follows
Induction step H k H k
Throughout this subsection assume H k is true and let b e chosen so
k
that no k computation on G is successful Let us call each copy of
p
G a sup erno de The structure within a sup erno de is xed by
k k
and how the sup erno des are connected in G is determined by
p k p
Note that blo cks cannot jump by de nition If a p ebble P in a blo ck B jumps to a
p ebble of another blo ck then P no longer b elongs to blo ck B Therefore each blo ck
can only traverse a numb er of adjacent sup erno des The following lemma shows that
when a blo ck has completed traversing a sup erno de the whole blo ck must b e within
that sup erno de
d
Lemma Suppose H k is true Then there exists f g such that
k
d
for any f g if some block B completed the traversal of a copy of
k p
u
G cal led G say at time t during the k computation
k k
u
on G started at time t then al l pebbles of B t lie within G
p k
at time t
Pro of Let us x such that for any any k computation is not
k k p
u
successful Let P b e the rst p ebble in B t that reaches the goal of G
k
u
Supp ose the path from the start to the goal of G is star t v v
k m
g oal We claim that for every i there exists an earliest time t such that if B is the
i i
set of p ebbles on no de v at time t then p ebble P is in B t the continuation of
i i i
B at time t B can b e viewed as a blo ck of the computation which starts at time
i i
t Obviously we can cho ose t t Supp ose there is a time t such that P is in
i m i
B t where B is the set of p ebbles on no de v at time t Then we can cho ose
i i i i
t to b e the smallest value such that no de v is continuously o ccupied b etween t
i i i
and t Since the only edge going into v is from v the move at time t must b e
i i i i
to walk some p ebble from no de v to v Hence no de v is o ccupied by p ebbles in
i i i
b b b
B t at any time t where t t t and B B t Thus B t B t
i i i i i i i i
and p ebble P is in B t It follows from the claim that p ebble P is in B t
i
Consider B t the continuation of B at t B is the set of p ebbles on one no de
at time t Therefore either B B t or B B t If B B t then
B t B t to o However p ebble P is in B t B t by the claim ab ove Hence
B B t
Let C and C b e the computations b eginning at time t and t resp ectively and
let b and b b e the numb ers of blo cks of C and C resp ectively remaining at time
t Then b p k by de nition of a k computation Moreover p k b
b ecause blo ck B traverses a copy of G during C and we x
k k
so that no k computation is successful So b b
For all blo cks B of C at time t if B B t and B B then B t i e
B disapp ears at time t Otherwise B t and B t are nonempty subsets of B t
and hence b b the set of p ebbles in B t are considered as one blo ck at time t in
C but are considered as more than one blo ck at time t in C a contradiction
u
Hence B t B t which lies completely inside G at time t
k
A corollary of Lemma is that a blo ck can only traverse a path of sup erno des
as opp osed to a tree of sup erno des during a k computation For example
u v
supp ose blo ck B is trying to traverse b oth G and G which
k k
w u
are the children of some sup erno de G If B has completed traversing G
k
u
rst then all its p ebbles are in G and it is now imp ossible for B to tra
k
v v
verse G Note that some no des but not the goal within G
k k
u
might have b een touched by p ebbles of B while it is traversing G Now
k
we can analyse the paths of sup erno des traced by blo cks in a k computation
very much like the way we analyse the paths traced by p ebbles in a computation
Lemma The sequence of id s a k computation on G goes through
p
can be speci ed independent of by
p
B the initial id Q of the k computation and
B for every block B of the computation the number of G s block B has
k
successful ly traversed and whether the goal of the last one is a leaf
Pro of Supp ose the k computation starts at time t We can verify by induction
on t t that the id at time t and the memb ers of B t for each blo ck B of the
computation can b e determined
Base Case The id at time t is Q and the blo cks are de ned by
Induction Step Supp ose the id at time t is known to b e Q and we also
t t
know the memb ers of B t for each blo ck B of the computation Supp ose the k
computation has not nished at time t and the next move is a walk P i Q Then
t
again the key is to determine if P is a leaf Once this is known the edge lab el
t
i and the no de lab el P determines which no de P will arrive at b ecause of the
t
lab elling prop erty L Then we can up date the memb ers of the blo cks involved
By lo oking at and which is xed we know whether P is the
t k t
u
goal of a G called G say Case P is not the goal
k k t
of a G Then determines if P is a leaf Case P is
k k t t
the goal of a G Let P b elong to the blo ck B t By Lemma every
k
u u
p ebble of B t must lie within G at time t If the goal of G
k k
i e P is a leaf blo ck B cannot traverse any more G s Therefore if
t k
the numb er of sup erno des traversed by B so far is strictly less than that sp eci ed in
B then P must not b e a leaf To know how many sup erno des each blo ck has
t
traversed we lo ok at and B B B t to infer the greatest depth
t
a blo ck has explored using the lab elling prop erty L On the other hand if the
numb er of sup erno des traversed by B is equal to that in B then blo ck B is on
the last copy of G and P is on its goal Hence B determines whether
k
P is a leaf
t
p
There are q d choices for B To b ound the numb er of choices for B
observe that each blo ck can traverse as many as d copies of G s b efore
k
it completes traversing a G That is there are d choices for the
k
numb er of copies of G traversed For each choice there are p ossible cases
k
dep ending on whether the goal of the last copy of G traversed is a leaf
k
p
Finally observe that the numb er of blo cks is at most p Therefore there are d
p p
choices for B Hence by Lemma there are at most q d d di erent
sequences of id s for a xed To traverse a G a blo ck has
k k
to traverse a path of d sup erno des which lie within that copy of G
k
and the sequence of lab els on the edges connecting these sup erno des has to match
p p
Hence at most pq d d s can have a successful k
k k
computation for some choice of B and B and a xed It follows
k
that for d p log pq there exists a choice of such that the k
k
computation is not successful for any choice of B and B Hence H k follows
This also completes the pro of of Lemma
Extension to probabilistic NNJAG with sided
error
For a probabilistic NNJAG that allows sided error we have the following result
Theorem For every c any probabilistic NNJAG that solves stcon with
c
log n
log n
sided error for al l n node graph within steps requires space at least
c log log n
Pro of Observe that if a probabilistic NNJAG J solves stcon with sided er
ror it cannot accept a graph G unless it has p ebbled no de t during the compu
tation Otherwise we can construct a graph G with isolated no de t and force
J to accept it Therefore the probability that J accepts G is upp er b ounded
by the probability that it has p ebbled no de t Now supp ose J has p p ebbles q
c
log n log n
states and time so that pq By Theorem to b e proved b elow
c
log n
log n p
n p log pq or p If b oth p log n and
c
log log p log log pq log n
log n
log q log n then p Hence Theorem follows
c log log n
Note that the result can b e re stated as a time lower b ound for space b ounded ma
log n
chines For every c any probabilistic NNJAG with at most space
c log log n
c
log n
requires at least time to solve stcon with sided error on n no de graphs This
complements the result obtained in Chapter
Theorem For every T time bounded probabilistic NNJAG J with p pebbles and
q states such that pq T there is a graph G such that
G is a rooted tree with maximum out degree
p
G has at most p log pq T nodes including two special nodes s and t
there is a directed path from s to t and
if J starts with al l pebbles on node s then the probability that node t is visited
by some pebble is less than
Pro of Again we shall cho ose a di cult graph for J from the set of skinny trees but
with d set to p log pq T As mentioned in Chapter we view J as a distribution
of deterministic NNJAGs each having p p ebbles and q T states Denote the deter
ministic NNJAG asso ciated with random string R by J Let M R b e
R k p
the event over random R that there exists t Q such that the k computation
by J on G from id t Q is successful Lemma b elow shows
R p
that Pr M R for any R assuming are chosen
p p p
p
d
randomly and uniformly from f g By Theorem in Yao Yao there exists
such that Pr M R Hence Theorem follows
p R p p
Lemma For al l R Pr M R k p
k p
p
Pro of By induction on k Let H k b e the statement of the lemma for a xed
value of k p
Base case Since J is deterministic we can apply Lemma with q replaced
R
p p d
by q T and have P r s t M R pq T d
p p
p when d p log pq T Hence P r M R p
p
p
Hence H is true
Induction step Assume H k is true and write P r as P r Observe that
p
P r M R
k p
h i
P r M R P r M R j M R
k p k p k p
By H k the rst probability is less than k p Hence we just need to show that
the second probability is less than p Since M R do es not happ en
k p
by Lemma each blo ck can only traverse a path of sup er no des during a k
computation By Lemma
Pr M R
k p
p
Pr s t M R
k p k p
k
p p d
pq T d d
p
when d p log pq T Hence H k is true and Lemma follows
Extension to probabilistic NNJAG with sided
error
The ab ove result can b e easily extended from sided error to sided error as follows
Theorem For every c any probabilistic NNJAG J that solves stcon with
c
log n
sided error for al l n node graphs within steps requires a storage of at least
log n
c log log n
Pro of By Theorem b elow we can construct a graph G so that the probability
of no de t b eing visited is less than Since G is st connected J has to accept with
probability at least Hence P r J has not p ebbled t but accepts G
Now construct G as b efore by removing all the in edges of no de t With the
same probability i e J has not p ebbled no de t and hence cannot tell the
di erence b etween G and G and will accept G
Theorem For every T time bounded probabilistic NNJAG J with p pebbles and
q states such that pq T there is a graph G such that
G is a rooted tree with maximum out degree
p
G has at most p log pq T nodes including two special nodes s and t
there is a directed path from s to t and
if J starts with al l pebbles on node s then the probability that node t is visited
by some pebble is less than
The pro of of Theorem is the same as that of Theorem except that we set
d p log pq T
Chapter
A Time space Upp er Bound
In this chapter we prove the following time space upp er b ound for stcon on the
JAG mo del
Theorem For al l S such that log n S n log n there is a JAG
O log n log n S O
algorithm that solves stcon for al l n node graphs in n time and
p
log n
while the O S space simultaneously In particular S can be as smal l as n
running time is stil l bounded by a polynomial
In the following section we describ e the algorithm that runs on a RAM implied by
Barnes et al BBRS In Sections and we describ e the simulations on the
NNJAG mo del and the JAG mo del resp ectively
The Barnes et al algorithm
We rst de ne a generalization of stcon as follows Let U and V b e sets of no des
We de ne
N U as the set of no des v such that there exists a no de u U and a path
l
from u to v of length i e the numb er of edges in the path at most l
N U N U N U
l l
l
stcon as the problem of computing the set V N U given input U V In
l l
particular stcon is the sp ecial case when l n U f s g and V f t g
The Barnes et al algorithm is constructed by combining a space e cient breadth rst
search and a time e cient Savitch s algorithm The latter algorithm solves instances
of stcon with l much smaller than n As p ointed out in BBRS such a problem
l
retains a lot of di culties that are present in stcon
We shall describ e the mo di ed Savitch s algorithm and breadth rst search in
Section and resp ectively In the presentation we assume the input graph
G is represented as an adjacency list and stored in the global memory Hence G do es
not app ear in the input parameters of the algorithms Also we let d dn log n S e
and V V V b e an arbitrary partition of the set of no des in the graph so that
d
each V has size at most dn de
i
The mo di ed Savitch algorithm
Savitch s algorithm determines whether t is within distance l from s by recursively
checking for each p ossible intermediate no de v whether v is within distance l from
s and whether t is within distance l from v The correctness follows from the fact
that
t N s v such that v N s and t N v
l
l l
Thus the algorithm solves the original instance of stcon by solving n instances of
l
O log l
stcon It runs in n time and O log n log l space We can compute V N U
l
l
by applying this algorithm on all pairs of no des u v U V However when
we have more space we can save some time by considering a set of intermediate
no des V instead of a single no de v and determining connectivity for the whole set
i
simultaneously More precisely we have
d
V N U V N V N U
l i
l l
i
Thus we solve an instance of stcon by solving d instances of stcon This metho d
l
l
n log n
O log l
requires approximately d time and O log l space
d
For our purp ose we are only interested in problem instances where V is of the
form f x y g for some x and y Hence V can b e represented by the pair of no des
x y and memb ership in V can b e tested in constant time Also we cho ose the
partition V s so that v V if and only if bv dc i Here we consider the no de
i i
name v as a numb er in the range n Then the subproblems will also satisfy
Savitch is shown in Figure this condition The pseudo co de for the algorithm mod
function mod Savitch l U V
fInput l is a p ositive integer U is an array of at most dn de no des V f x y g is
sp eci ed by x and y
Output no des in V N U stored as an array g
l
var k integer Ans X set of no des u v no des
b egin
Ans
if l then f Base Case g
for each u U do
if u V then Ans Ans f u g
for each edge u v G do
if v V then Ans Ans f v g
end for
else f Recursive Case g
for k to d do
X mod Savitch bl c U V
k
Ans Ans mod Savitch dl e X V
end for
end if
return Ans
end
Figure mod Savitch
Savitch l U V Let T l and S l b e the time and space used by a call to mod
when jU j jV j O n d For the base case there are O n d no des in U and O m
no des neighb ouring some no des in U We can check whether they are in V in O
time each Adding a no de to Ans can b e done in O time if we represent Ans by
a bit vector of length jV j We simply turn the corresp onding bit on Therefore
T O m n d O m assuming n m Also jV j O n d and hence
n log n
jAnsj O n d Therefore S O For the recursive case there are d
d
recursive calls to mod Savitch l and O n lo cal work to up date Ans Hence T l
d T l O n It requires S l space for the recursive calls and O jV j log n
n log n
lo cal space for the sets of no des X and Ans Hence S l S l O
d
n log n
O log l
Combined with the base case T l d m and S l O log l
d
The mo di ed breadth rst search
In the breadth rst search we construct the breadth rst search tree layer by layer
storing every single no de we have found so far In the mo di ed breadth rst search
th
we store only every d layer of no des That is we only store layers i d i b n
i dcd i for some i d Let us denote this set of no des by W
i
Observe that any no de v reachable from no de s must lie within distance d
from s or some no de in W Therefore to determine whether t is reachable from s we
i
just solve an instance of stcon with input W f s g f t g
d i
Obviously there is a set W of size jW j bn dc Storing this W requires only
i i i
n log n
O space To nd a small enough W we simply try from i Each
i
d
W will b e computed incrementally in b n i dc stages with W containing the
i
p ortion of W found so far If W ever contains more than bn dc no des we discard W
i
and try the next set W
i
We shall maintain the invariant that W contains all the no des in layers i d
i j d i just b efore stage j During stage j W will b e augmented with
no des in layer j d i and hence the invariant is maintained for the next stage The
no des in layer i can b e found by computing N s N s N s Augmenting W
i i
i
W f s g We with no des in layer j d i in stage j can b e done by computing N
d
have to include no de s and the no des in layers i d i j d i b ecause not
all no des at distance exactly d from layer j d i are on layer j d i Some may
lie on some layer k j d i
We construct an algorithm ExactNeighbour l U which computes N U using only
l
O n log n d space Moreover it will only return at most bn dc elements in N U
l
if it has size bigger than bn dc The idea is to compute V N U and V N U
l l
b oth of size O n d and then take the di erence This is rep eated with the sets
V V and so on until the set V is done or we have found more than bn dc no des
d
already The pseudo co de is shown in Figure It only uses O n log n d lo cal
Savitch l U V with space p erforms O n lo cal work and makes O d calls to mod
O log l O log l
jV j O n d Hence if jU j O n d also it will use d d m d m time
n log n n log n n log n
and O log l O O log l space
d d d
function ExactNeighbour l U
fInput l is a p ositive integer U is an array of no des
U j bn dc g U where bn dc x bn dc if jN U or any x no des of N Output N
l l l
var k integer Ans set of no des
b egin
if l then return U
Ans k
while jAnsj bn dc and k d do
Ans Ans mod Savitch l U V mod Savitch l U V
k k
k k
end while
return Ans
end
Figure ExactNeighbour
bfs s t that p erforms the mo di ed breadth rst search just The algorithm mod
describ ed is shown in Figure It makes O n calls to ExactNeighbour l U with
l d and jU j O n d p erforms O n lo cal work and uses O n log n d lo cal space
O log d O log n log n S
Recall that d dn log n S e Hence it runs in mn mn
O log n log n S O
n time and O n log n d O S space
Simulation on an NNJAG
The previous algorithms can b e simulated on a RAM NNJAG without much change
In a RAM NNJAG the registers are divided into two typ es regular registers and
p ebble registers The input graph can only b e accessed by applying walk or jump
op erations on p ebble registers only Furthermore we can only transfer data from
p ebble registers to regular registers but not the reverse Thus we need to b e careful
with the typ es of registers used to hold the input no des and output no des
For mod Savitch l U V we require that U b e stored in the p ebble registers and
function mod bfs s t
fTo determine if there is a directed path from s to t g
var i j integer D one b o olean W set of no des
b egin
i D one false
while i d and not D one do
fCompute W g
i
W ExactNeighbour i f s g
j
while jW j bn dc and j d i n do
fW contains layers i d i j d i g
j j
W W ExactNeighbour d W f s g
end while
if jW j bn dc then
D one true
else fTry the next group W g
i
i i
end while
if mod Savitch d W f s g f t g then
return not connected
else return connected
end
Figure mod bfs
the output returned in the p ebble registers Then when l we can apply the walk
op eration to the p ebble registers that store the no des in U to nd out their neighb ours
Savitch bl c U V are When l the output returned by the recursive call mod
k
stored in the p ebble registers and hence can b e used as input to the recursive call
mod Savitch dl e X V The input V can still b e represented by a pair of no des
x y stored in regular registers We can check whether a no de u stored in a p ebble
register is in V by copying it to a regular register and see if x u y
Similarly for ExactNeighbour l U we require that U b e stored in the p ebble
registers and the result returned in the p ebble registers This is b ecause U will b e
used as input to calls to mod Savitch and the result of ExactNeighbour will b e used
bfs as input to subsequent calls to ExactNeighbour by mod
For mod bfs s t we can assume that no de s and t are stored in p ebble registers
initially It is easy to see that all the previous three algorithms can run on a RAM
NNJAG using this input output convention The running time and space are the
same as b efore
Simulation on a JAG
Savitch l U V can The ma jor problem in the simulation is that the algorithm mod
not b e simulated directly This is b ecause the set V cannot b e sp eci ed as the
input Some no des in V may not b e reachable from no de s and hence V cannot
b e stored in the p ebble registers Furthermore the JAG has no way of comparing
the content of a p ebble register with that of a regular register Hence V cannot
b e stored in the regular registers to o Thus rather than sp ecifying V and asking
the pro cedure to compute V N U we sp ecify an integer k and ask the pro ce
l
th
dure to compute the k group of no des in N U More precisely we construct
l
Savitch l U k which returns a set of no des R N U such that a function mod
k l
jR j max min dn de jN U j k dn de and R R for all i j We
k l i j
rst give the construction of mod Savitch l U k Then we describ e how it can b e
bfs and ExactNeighbour to solve stcon used by mod
Savitch l U k will consist of recursive calls to The pro cedure for mod
Savitch l V i for suitably chosen V s and i s We rst explain how to compute mod
th
N U in small space and then how to select the k group of no des from N U
l l
Observe that N U N N U Therefore we can recursively compute
l
bl c dl e
X N U and then N X Let X b e computed by mod Savitch dl e U i
dl e bl c i
S S S
N X Let Y b e com X X Also observe that N Then X
i ij i i
bl c bl c
i i i
S
puted by mod Savitch bl c X j Then N X Y and hence N U
i i ij l
bl c
j
S
Y Note that each Y can b e computed by a call to mod Savitch dl e U i
ij ij
ij
which returns X and then mod Savitch bl c X j
i i
th
To select the k group of no des in N U we compute Y in lexicographic order
l ij
S
of i j and count j j To do this whenever we computed Y we Y
ij i j
i j i j
S
b
for all i i and all j and then Y Y re compute Y Then Y
ij ij i j i j
i i
b b b b
for all j j by the prop erty for all i i In addition Y Y Y Y
ij ij ij i j
S P
b
Savitch bl c X j Therefore j j Y j We can jY of mod
i i j i j
i j i j i j i j
st th
then select the k dn de to k dn de element in N U The pseudo co de
l
Savitch l U k is broken into three parts which are shown in Figure of mod
and
function mod Savitch l U k
th
f To compute the k group of no des in N U g
l
var c integer Ans array of p ebble registers
b egin
c f to count the numb er of elements generated so far g
Ans
if l then
Base Case
else
Recursive Case
end
Savitch Figure The outline of mod
for i to jU j do
c c
if k dn de c k dn de then
app end U i to Ans
end for
for i to jU j do
for each edge U i v G do
f Check if v has already b een discovered b efore g
f ound false
for j to jU j do
if U j v then f ound true
end for
for j to i do
for each edge U j w G do
if w v then f ound true
end for
end for
for each edge U i w G such that w v do
if w v then f ound true
end for
f Checking completed g
if not f ound then
c c
if k dn de c k dn de then
app end v to Ans
end for
end for
return Ans
Figure Base Case
c
rep eat
i i
th
Savitch bl c U i f Compute X the i group of N U g X mod
i
bl c
j
rep eat
j j
th
Y mod Savitch dl e X j f Compute Y the j group of N X g
ij i
dl e
f To check and remove from Y any no de that has b een found b efore g
i
rep eat
i i
g Savitch bl c U i f Compute X Z mod
i
if i i then
j
rep eat
j j
W mod g Savitch dl e Z j f Compute Y
i j
b
Y Y W f Compute Y g
ij
until W
until i i
f Checking completed g
for j to jY j do
c c
if k dn de c k dn de then
app end Y j to Ans
end for
until Y
until X
Figure Recursive Case
Let T l and S l b e the time and space used by mod Savitch l U k For the
n log n
For the recursive case there are base case T O n d and S O
d
four nested lo ops each lo oping for at most d times In the deep est lo op there is
log l
a recursive call to mod Savitch l Hence T l d T l d T
O log l
d n Also S l O n log n d S l O n log n log l d
The mo di cation of ExactNeighbour l U is rather easy For each group of no des
in N U which is computed and stored in V we compute all the no des in N U
l l
and remove from V any no de that app ears in N U Thus line in Fig
l
ure is replaced by the following The new ExactNeighbours makes O d calls to
O
mod Savitch p erforms n lo cal work and uses O n log n d lo cal space
Ans k
rep eat
k k
Savitch l U k V mod
Ans Ans V
k
rep eat
k k
V mod Savitch l U k
Ans Ans V
until V
until V or jAnsj bn dc
The mo di cation in mod bfs happ ens only at the end when we test whether t is
within distance d from W f s g More precisely line in Figure is replaced
i
by the following lines
c
rep eat
c c
Savitch d W f s g c T mod
if t T then return connected
until T
return not connected
n log n log d
O log d O
bfs requires d n time and O It is easy to see that the new mod
d
O log n log n S O
space or equivalently n time and O S space
Chapter
A Time space Lower Bound
In this chapter we prove a time space lower b ound for stcon on the probabilistic
stcon i e the complement problem NNJAG mo del We rst prove a lower b ound for
of stcon on the probabilistic NNJAG mo del with sided error This implies the
same lower b ound for stcon on a deterministic NNJAG Then we extend it to a
probabilistic NNJAG that allows sided errors
Theorem There is a positive integer n and positive constants and c
log n
n
such that for any n n and any S any probabilistic NNJAG that
log log n log n
solves stcon with sided error on n node graphs with out degree at most two using
q
c log n log n S
nS
log log n
walking steps S space requires more than
log n
The pro of of Theorem will require the following theorem
Theorem There is a positive integer n and positive constants and c
n
such that for any n n and any S there is a distribution on n node
log n
directed graphs G such that
G has maximum out degree
G is not st connected and
n log n
q
c log
S
nS
log log n
for any deterministic NNJAG J with S space and at most
log n
walking steps which rejects every st connected graph
Pr J accepts G
G
S
Note that a deterministic NNJAG with space S cannot run for more than steps
without going into in nite lo op Set S c log n log log n for a small enough con
n log n
q
c log
S
nS
S
log log n
Then by Theorem no deterministic stant c so that
log n
NNJAG with space at most S can accept every graph in Therefore any deter
ministic NNJAG that solves stcon and hence stcon requires space more than
S c log n log log n This is an alternative pro of of the deterministic case result
derived in Chapter However the same trick cannot b e applied to get the proba
bilistic case result in Chapter b ecause the machine there is allowed to run for more
S
than steps
stcon for Pro of of Theorem Let J b e a probabilistic NNJAG which solves
all n no de graphs with S space and at most T time where log n log log n S
p
c log n log n S
log log n
n log n and T nS log n By the construction in Chapter J
can b e simulated by a distribution J of deterministic NNJAGs which reject every
n no de st connected graph Moreover each NNJAG in J uses space at most S log T
and time at most T For the stated ranges on S and T
c log n log n S nS
S log T S log
log log n log n
c log n
S
log log n
S
Let c b e the constants mentioned in Theorem Set and cho ose a
n log n
c log
p
c log n log n S
S
log log n log log n
small enough constant c so that T nS log n
p
n S log n for all n n Then by Theorem the probability that such a
deterministic NNJAG J J accepts a graph G drawn from the distribution on
R
non st connected graphs is less than i e Pr J rejects G
G R
For all deterministic NNJAG J and graph G let r J G if and only if J
R R R
rejects G Theorem of Yao Yao implies that for any NNJAG distribution J and
any graph distribution G
X X
min Pr J r J G Pr G r J G max
J J R R G G R
R
J J G G
R
G J
R
or equivalently
min Pr J rejects G max Pr J rejects G
G G R J J R
R
J J G G
R
Hence for some worst case graph G J accepts G with probability less than
stcon for all n no de graphs with the This contradicts the assumption that J solves
stated time and space b ounds
In the rest of this chapter except Section we shall consider a variant of
deterministic NNJAGs that runs in at most S space and rejects every n no de st
connected graph for some S b etween log n and n log n and some n n In
Sections and we rst describ e some distributions of n no de graphs including
the distribution We then de ne a notion of progress that measures how much
an NNJAG has done towards solving stcon for graphs chosen from in Section
Section describ es this variant of NNJAGs and the related mo dels used in the
subsequent pro of Section outlines the pro of of Theorem Sections and
give the pro of of the supp orting lemmas used in Section Sections and
contains the pro of of the two required technical lemmas for proving the Main Lemma
Finally Section extends the result to probabilistic NNJAGs that have sided
error
Layered Graphs
A d x layered graph rst de ned in BE is a graph consisting of d layers each
containing x no des The layers are numb ered from to d The width x is the numb er
th
of no des p er layer The j no de in layer i is denoted by u Every no de has at
hi j i
most two out going edges to some not necessarily distinct no des in the next layer
Intuitively solving stcon for graphs containing a d x layered graph is di cult
As explained in BE if we connect no de s to every no de in the top layer of a
d
d x layered graph there may b e as many as x di erent paths from s to no des
in layer d The NNJAG must p otentially check each such path b efore it can b e sure
that s is not connected to t Although these paths will overlap in many places it is
di cult for the machine to rememb er with its limited space which sub paths have
b een traversed already Therefore many sub paths must b e traversed many times
b efore the NNJAG is sure they have all b een traversed In what follows we consider
a distribution of layered graphs on which we can quantitatively measure how much
work an NNJAG has done towards solving stcon
Let D b log n for some constant b to b e determined later The distribution
of base graphs with parameter x denoted as B x is a distribution on D x layered
graphs Each graph G in B x will have x hard paths to b e de ned shortly
of length D and is obtained as follows In each layer i except the top layer we
randomly pick without replacement a sequence of x no des in a way that all the
th
x x x p ossibilities are equally likely Let us denote the j no de picked
as v It is the no de u for some j These no des are called the hard nodes The
hi j i hi j i
remaining x no des in that layer are called the easy nodes For layer we cho ose
the sequence of no des u u u as the sequence of hard no des Next
h i h i h x i
we shall put in edges so that if an NNJAG moves a p ebble D steps down from
a hard no de in the top layer it is di cult for the p ebble to b e on a hard no de by
the time it reaches layer D In other words it is easy for the p ebble to reach an easy
no de
First the hard no des are connected by the edges v v with lab el l for
hi j i hi j i hi j i
each i D and each j x The path from v to v is called the
h j i hD j i
th th
j hard path The no des v and v are called the root and goal of the j hard
h j i hD j i
path resp ectively Thus there are x hard paths and the same numb er of ro ots and
goals in G The edge lab els l are chosen indep endently and uniformly at random
hi j i
from f g We denote the vector of edge lab els on the hard path ro oted at ro ot r
th
by If r is the j ro ot is the sequence of bits l l Clearly is
r r r
h j i hD j i
D
uniformly chosen from f g
For each layer i D each hard no de v will have one more out going edge
hi j i
lab elled l and each easy no de will have two out going edges The destinations
hi j i
of these edges are chosen indep endently with replacement at random from the set
of easy no des in layer i In this way the in degree of each hard no de in layer i
is kept to one
As we shall not directly use the distribution B x as an input distribution for the
NNJAG we shall not sp ecify the naming of no des in each graph in B x yet For the
time b eing it is su cient to say that for each i j D x the name of the
no de u is the same for every graph drawn from B x
hi j i
Figure shows a graph in B assuming D The no des in each layer are
rearranged so that all the hard no des are on the left hand side
hard easy to reach to reach
D
x
Figure A graph G in B x
A key observation of this distribution of graphs is that each no de in each hard
path has in degree one and out degree two If a p ebble is to reach a goal starting from
a ro ot it has to decide correctly which edge to take at each no de along the hard path
This is di cult even though an NNJAG can see the names of the no des b ecause
we randomly pick the edge lab els and the no des on the hard path We shall say that
progress is made if a p ebble successfully reaches a goal no de Lemma Main
Lemma shows that with high probability any short sub computation cannot make
much progress Then setting x n log n adding the no des s and t with the
appropriate edge connections and using a standard progress argument as intro duced
in Boro din et al BFK we can obtain a lower b ound of ST n log n To
obtain a larger lower b ound we need to consider layered graphs with height bigger
than log n and develop new techniques We consider distributions on layered
graphs recursively constructed from the distribution B x in the following section
Recursive Layered Graphs
p
nS log n and K blog n log D c Recall that D b log n De ne
log n log n S
One can verify that K when S n b log n and that
log log n
K O log n log log n since S log n We rst recursively construct K distributions
k k
H H H on layered graphs H is a distribution on D layered graphs
K k
k k
hard paths of length D Then G is formed by adding Each such graph has D
k
some more no des and edges to each graph in H Each G will b e an input distribution
k k
of n no de graphs and is more and more di cult for the NNJAG as k increases G
K
will b e chosen as the required distribution in Theorem
The distribution H is simply B For k a graph G in H is formed as
k
follows We cho ose a graph G from H and replace each layer i of G by a graph G
k i
k k k
chosen from B Note that each G has hard paths and each layer
i
th
of G has the same numb er of no des We identify the j hard path in G i e the
i
th
path from v to v of G with the j no de in layer i i e u of G Every
i
h j i hD j i hi j i
edge that go es into u of G will now go into v of G and every edge that go es
i
hi j i h j i
out of u of G will go out of v of G The easy no des in G are not connected
i i
hi j i hD j i
to any no de outside G
i
Since G is uniquely determined by G and G G k we often denote G as
D
a tuple hG G G k G i The graph G is called the col lapsed graph of G
D
denoted by C G The set of hard paths resp ectively ro ots and goals of G is the
union of all the sets of hard paths resp ectively ro ots and goals in G G G k
D
k k
Hence G has D hard paths and the same numb er of ro ots and goals in total
The edges within each of the G s are called the base edges of G The other edges i e
i
those connecting the G s are in corresp ondence with the edges in the collapsed
i
graph C G of G Hence they are called the col lapsed edges
Figure shows a graph G H on the right its collapsed graph C G H
k k
k
and the symb ol for a base graph in B on the left Again we rearranged the
no des so that all the hard no des in the G s app ear on the left half i
hard easy to reach to reach
G 1 G' D k-1
G 2
k G 3 D k-1 2 χ
G k -1 D D k k χ
2 χ 2
k
Figure Left top G H Left b ottom A graph in B Right G
k
hG G G k G i H
k
D
For each k K the distribution G is obtained by augmenting the following
k
auxiliary no des and edges to each graph in H
k
with w s and an edge from w to u of A a directed path w w w
k
j
h j i
k
G for each j
A the isolated no de t
A a sp ecial isolated no de referred to as the lost no de
A a numb er of isolated no des so that the total numb er of no des in the graph is
exactly n
See Figure These auxiliary no des and edges are xed for each graph G G k
k 2 χ s = w 1 w2 w 3
k D
lost t
Figure A graph in G
k
Hence G can still b e sp eci ed by a tuple h G G G k G i where G G k
D D
k
are in B and G is in H The col lapsed graph of G denoted by C G is the
k
graph G augmented with the auxiliary no des and edges needed to form a graph in
G from a graph in H Thus C G is in G Note that excluding the no des
k k k
k k
added in A each graph in G consists of a D layered graph a path with
k
k k k
no des the no de t and the lost no de These add up to a total of D
k
D n no des for k K by our choice of the parameters D and K Hence
we are not adding a negative numb er of no des in A
k k
Finally we lab el the no des so that for each i j D the name of
the no de u is xed for every G G Similarly the names of the auxiliary no des
hi j i k
added in the ab ove four steps are also xed Furthermore the names of the no des
are drawn from the range n
De nition of Progress
Our pro of will measure at various stages of the computation how much progress an
stcon on an input G For any sub computation NNJAG J has made towards solving
A and any input G G we de ne w G as the numb er of di erent goals in G that
k A
have b een p ebbled at some p oint during the sub computation A Intuitively w G
A
stcon is the amount of progress made The following lemma shows that if J solves
it must make progress on input G G
Lemma Let J be an NNJAG that rejects every st connected graph Then for
every graph G G accepted by J w G
J
Pro of Supp ose that G G Therefore it is not st connected and has goals
Supp ose there is a goal v which is never reached by any p ebble during the computa
tion Let G b e obtained from G by connecting no de v to no de t By de nition of
the NNJAG mo del the machine cannot learn whether there is an edge from v to t
Therefore the computations on G and G are the same Since G is st connected J
rejects G Hence J will also reject G
For k the computation on an input G G is more complicated Each
k
k
B such graph contains some base graphs G G connected by some
k
D
collapsed edges The NNJAG has to learn b oth the structures of the base graphs and
the collapsed edges The following lemma shows that w G is a go o d estimate on
A
the numb er of di erent collapsed edges traversed during a sub computation A
Lemma The number of di erent col lapsed edges of an input G G that can be
k
traversed during a sub computation A of an NNJAG is at most w G
A
Pro of An NNJAG can traverse an edge u v only if there is a p ebble on no de u
b efore the traversal If the edge is a collapsed edge u must b e a goal Since every goal
has out degree at most two p ebbling one such no de allows the NNJAG to traverse
at most two di erent collapsed edges
The next lemma characterizes the only signi cant di erence b etween the NNJAG
mo del and a general computation mo del
Lemma If at some step t a particular hard path does not contain any pebble
and at some later step t a pebble arrives at the goal of this path then each edge in
that path must be traversed by some pebble between step t and t
Pro of Observe that every no de in every hard path has in degree one and the NNJAG
mo del do es not allow a p ebble to arrive at a no de unless there is another p ebble to
jump to or it walks there
Viewing NNJAGs as Branching Programs
To facilitate the analysis we shall make some changes and assumptions on the ma
chine mo dels We rst mo dify the NNJAG mo del slightly and de ne a variant of
NNJAGs called pebble location redundant NNJAGs We shall see that it is convenient
to construct an NNJAG J for graphs in G from another NNJAG J for certain
k
graphs in G if they are b oth p ebble lo cation redundant We call this the collaps
k
ing of J to J We then construct a branching program from such an NNJAG The
resulting branching program will have structure imp osed on it by the structure of
NNJAGs Lemmas and together with the concept of p ebbles character
ize the required structure Finally we de ne a variant of branching programs called
sectioned branching programs The concept of sectioning a branching program will
b e useful for charging the time used by the branching program
First we allow the NNJAG to jump p ebbles to the lost no de the no de w and
j
k
u for any j Also we mo dify a step to b e taken from an id Q to
h j i
consist of the following substeps
Substep Based on Q it either walks a p ebble P along the edge with
lab el or jumps a p ebble P to the lost no de the no de w or u for some
j
h j i
k
j It can also cho ose not to move any p ebble Let sp ecify the new
p ebble lo cations
it p erforms a p ossibly empty sequence of Substep Based on Q and
p ebble jumps and then assume some state Q Here each p ebble jump will only
sp ecify a p ebble to jump to another p ebble rather than to a no de
Note that a sequence of moves in a normal NNJAG can b e group ed as a sequence
of macro steps each of which starts with a walk followed by a p ossibly empty
sequence of jumps Moreover each jump causes the normal NNJAG to enter a
unique next id Thus each macro step can b e p erformed in one step in this new
mo del It follows that a time lower b ound on this new mo del implies the same lower
b ound on the numb er of walking steps on the original mo del
This change in mo del also allows us to de ne the following variant of NNJAGs An
NNJAG is said to b e pebble location redundant if the current state always determines
b
the current lo cation of all the p ebbles In other words there is a function such
b
that if the NNJAG is in state Q then Q sp eci es the lo cations of all the p ebbles
For any arbitrary NNJAG J with p p ebbles q states and T time we can construct a
p ebble lo cation redundant NNJAG J so that for any p ossible id Q of J J will
have a state hQ i In this state J will p erform the same action as J do es on id
p
Q Thus J will have p p ebbles and q n states and hence at most times
1
Note that in general a normal NNJAG cannot b e made p ebble lo cation redundant b ecause if
the space of J Moreover it uses the same amount of time as J
Consider an arbitrary NNJAG J which takes inputs from a distribution of n no de
graphs with out degree We construct the corresp onding branching program A as
follows For every id Q of J there will b e a con guration vertex Q in A
Start accept and reject ids of J translate to start accept and reject con guration
vertices of A resp ectively Recall that an input graph for a branching program is
represented by the variables X where u n is a no de name and f g
hu i
is an edge lab el The variable X will have value v if there is an edge u v
hu i
lab elled with in the input graph and the value unde ned if there is no such
edge Thus if in id Q the rst substep of J walks a p ebble from no de u along
the edge with lab el the con guration vertex Q in A will b e lab elled with the
variable X Furthermore the vertex will have a directed edge lab elled with v to
hu i
con guration vertex Q if for some input graph X v i e the queried edge
hu i
has destination v and the subsequent jumps taken in the second substep by J bring
the machine to the id Q If J do es not walk any p ebble in the rst substep the
con guration vertex will not get any lab el and will have only one unlab elled out edge
p ointing to some con guration vertex Q dep ending on the second substep of
J For an NNJAG with p p ebbles q states and T time the corresp onding branching
program will have space p log n log q and time T
Finally we de ne a variant of branching program called sectioned branching
program A branching program is said to b e sectioned if its vertices are partitioned
into sections so that the out edges of a vertex in section i can only go to vertices in
section i or i Thus each computation path will go through each section at most
once A branching program A is said to b e properly sectioned for an input G if A
queries at most S log n di erent edges of G in each section where n and S are the
xed values mentioned at the b eginning of this chapter We de ne the s height of A
on input G denoted by h G as S log n times the numb er of sections A contains
A
the move taken from an id Q is a walk the new p ebble lo cation will dep end on the input
1 1 2
b
graph Hence the NNJAG cannot know which new state Q to move to so that Q In
2 2 2
contrast in the mo di ed NNJAG the p ebble lo cation after substep is uniquely determined
2
0
b
by Q and Hence it is p ossible for the NNJAG to cho ose a state Q so that Q
1 1 2 2 2
1
if A is prop erly sectioned for G and in nity otherwise Thus there may b e more than
one query to the same edge of the input graph within a section but they are only
charged as one query in the s height measure
Note that every branching program can b e levelled by at most doubling the space
and without increasing the time Also every levelled branching program of height T
can b e viewed as a sectioned branching program with T log n S sections each of
which queries at most S log n di erent input edges
In summary we can construct from a p ebble lo cation redundant NNJAG J with
S space and T time to a sectioned branching program A with at most S space and
S
O T sections where S p log n log q All our subsequent lemmas and claims
log n
are proved on those sectioned branching programs derived from some p ebble lo cation
redundant NNJAGs with at most S space Moreover if J takes T time on input G
A will have s height O T on G We shall also o ccasionally mention p ebbles in the
context of such a branching program
Pro of Outline
In the rest of this chapter we denote a directed edge from u to v with lab el by
the triple hu v i Also we denote by G O the induced distribution obtained by
selecting those graphs in G that satisfy O where O is some condition on the edge
connections on graphs chosen from a distribution G
We rst state two supp orting lemmas The rst one mainly concerns ab out the
traversing of base edges of graphs in G It b ounds the probability of a sub branching
k
program making a lot of progress within a short p erio d of time
Lemma Main Lemma For any k K and any sub branching program
A of a sectioned branching program derived from some NNJAG with at most S log n
pebbles
S
Pr w G S log n and h G
G G A A
k
The intuition is as follows When h G A queries at most di erent edges
A
in G Consider the probability of p ebbling the goal corresp onding to an arbitrary ro ot
r assuming that the hard path ro oted at r do es not contain any p ebble initially There
D
are p ossibilities for the vector of edge lab els on this hard path To remind
r
us of its dep endency on G let us use the symb ol G instead of for the rest
r r
of this chapter An NNJAG can move a p ebble down from r following some vector
D
f g of edge lab els hoping that G For G drawn from G this
r k
D
probability is The NNJAG can dynamically cho ose based on the names of
the no des on the path it has traced so far However this will not help much b ecause
the no des on the hard path are chosen randomly Recall that D b log n where b
p
D
and that nS log n Since S O n log n it follows that
Clearly by querying at most di erent edges in the input graph the NNJAG
cannot try many di erent s Hence the probability of having at least one of them
b eing successful is still small
On the other hand the NNJAG can eliminate some of the p ossibilities by detecting
collision of edges and hence increasing the probability of success For example when
it learns that two di erent edges have the same destination no de v it learns that this
no de v is not on the hard path since its in degree is bigger than one Hence any
path continuing from no de v need not b e traversed However within steps the
probability that an edge traversed by the NNJAG collided with some other traversed
edges can b e shown to b e at most By analysing a variant of branching pro cesses
AN we can show that the probability of eliminating a large numb er of vectors
D
f g in this way is small In other words with high probability the NNJAG
still has a lot of p ossible s to try out
The ab ove discussion considers only a single ro ot When there are many ro ots
we need to take care of the dep endencies among them b efore we can apply some
Cherno typ e b ounds The detailed pro of is delayed until Sections and
The second lemma concerns ab out the traversing of collapsed edges of graphs in
G It relates the computation of a p ebble lo cation redundant NNJAG J on inputs in
k
G E to that of a faster p ebble lo cation redundant NNJAG J on inputs in G
k k
k
where E is some complete setting of the base graphs G G k B and
D
G E represents the distribution of G conditional on these xed graphs For any
k k
k
complete setting E of base graphs G G k B de ne a function C from
E
D
no des in G G E to no des in C G G as follows
k k
k
C w w for all i
E i i
k k
C w l ost for all i
E i
C l ost l ost
E
C v l ost for all easy no de v and
E
th
C v u for all no de v on the j hard path in G
E i
hi j i
Note that the function is well de ned b ecause for input G G E whether a no de
k
u is a hard no de an easy no de or an auxiliary no de is xed
Lemma Collapsing Lemma For any k K any complete setting E of
k
and any pebble location redundant NNJAG the base graphs G G k B
D
J with p pebbles and q states there exists a pebble location redundant NNJAG J with
p pebbles and q states such that there is a correspondence between the states and
pebbles of J and J and for any G G E J is in state Q with a pebble on node u
k
in G in a step if and only if J is in the corresponding state with the corresponding
pebble on node C u in C G in the same step In particular J traverses a col lapsed
E
edge hu l v i in G if and only if J traverses the corresponding edge hC u l C v i
E E
in C G and J accepts G if and only if J accepts C G
The pro of is given in Section
With the two lemmas we can prove the following inductive statement
Lemma For any k K and any sectioned branching program A with at most
S
T sections derived from a pebble location redundant NNJAG J with at most
k
log n
S space that rejects every st connected graphs
S
Pr A accepts G and h G T k
G G A k
k
k
log n
where T and G is the distribution of graphs de ned in Section
k k
S
We shall prove the lemma by induction on k Let H k denote the statement for any
S
sections derived from a p ebble sectioned branching program A with at most T
k
log n
lo cation redundant NNJAG J with at most S space that rejects every st connected
S
A accepts G and h G T k graphs Pr Then H implies that for
A k G G
k
any deterministic NNJAG J with at most S space the probability that J accepts
log n
S
for graph G randomly drawn from G is less than Note that G in time O
S
S
for big enough n since S log n As each graph in G has at most D
no des excluding those added in A we could set n log n rather than
p
nS log n as mentioned in Section while keeping the size of the graph
at n Then we have ST n log n In itself this is a new result extending the
JAG result in BE to probabilistic NNJAGs
The statement H K however gives a stronger lower b ound It implies that
for any deterministic NNJAG J with space at most S the probability that J
S
accepts G in time O T for graph G randomly drawn from G is less than K
K K
p
log n log n S
log log n
nS log n Substituting and K as de ned in Section T
K
S
Also K for big enough n since K O log n log log n from Section
and S log n Hence Theorem follows for some suitably chosen constants c and
We now prove Lemma
Pro of of Lemma
Base Case
The pro of follows the same line of argument as intro duced in BFK and also
used in Boro din and Co ok BC Beame Bea Boro din et al BFMadH and
S
Yao Yao Let A b e a sectioned branching program with at most T sections
log n
derived from some p ebble lo cation redundant NNJAG with at most S space Consider
a graph G G such that A accepts G and h G T By Lemma w G
A A
log n
That is A makes progress on G Now break A into at most slices such that
S
log n
S
slice i consists of the p ortion of the branching program from section i T
log n S
log n
S
to section i T inclusively We count sections and slices from
log n S
log n
S log n progress Since There exists a slice i that makes at least
S
A is sectioned G passes through each slice at most once Consider the maximal
b
sub branching program A which lies within slice i and is ro oted at the no de through
which G rst enters slice i Then any sub branching program that G passes through
b b
in slice i is contained in A Therefore A has to make at least S log n progress on
b
G The fact that h G is nite means that A and hence A is prop erly sectioned
A
for G Therefore h G
A
Applying Lemma Main Lemma with k we have
S S
Pr w G S log n and h G Since there are at most
G G
A A
ro ots for such sub branching programs the probability that at least one of the sub
branching program makes at least S log n progress and at the same time has s
S S S
height at most on G is less than Hence the probability that A
log n
S
accepts G and h G T is less than
A
S
Induction Step
Assume H k is true Consider a sectioned branching program A with at most
S
T sections derived from some p ebble lo cation redundant NNJAG J with at most
k
log n
S space Supp ose by way of contradiction Pr A accepts G and h G T
G G A k
k
S
k We shall show that there exists some sectioned branching program A with at
S
sections corresp onding to some p ebble lo cation redundant NNJAG most T
k
log n
G T J with at most S space such that Pr A accepts G and h
k G G A
k
S
k This contradicts H k
S
We break A into at most T slices so that slice i consists of section i T T
k k k
log n
to section i T T inclusively Let F b e the set of G G such that
k k k
b
h G T and at least one sub branching program A which lies within a slice has
A k
b
w G S log n Since h G b eing nite implies A is prop erly sectioned for G
A
A
S
h G T T By Lemma Main Lemma and the fact that
k k
A
log n
S S
there are at most ro ots of sub branching programs in A Pr G F
G G
k
S
Therefore Pr A accepts G and h G T and G F k
G G A k
k
Let us cho ose a setting E of the base graphs G G k so that
D
S
Pr A accepts G and h G T and G F k By Lemma
G G E A k
k
we can construct from the p ebble lo cation redundant NNJAG J another p ebble
lo cation redundant NNJAG J that runs on G with the same numb er of p ebbles
k
and states as J
S
From J we can construct a sectioned branching program A with at most T
k
log n
sections The partitioning of vertices in A into sections is done as follows We put a
con guration vertex of A in section i if and only if the corresp onding con guration
vertex of A is in slice i In A out edges will only go out from vertices in slice i to
vertices in slice i or i Therefore in A out edges will only go out from vertices
in section i to vertices in section i or i Hence it is a legal way of partitioning the
vertices in A into sections Now consider an arbitrary graph G G E F At
k
most S log n progress is made in the unique maximal sub branching program that
G passes through in each slice of A By Lemma each such sub branching program
can query at most S log n di erent collapsed edges in G Hence each corresp onding
sub branching program in A queries at most S log n di erent edges in C G There
S
fore A is prop erly sectioned for C G for all G G E F Since A has T
k k
log n
C G T for all slices A has the same numb er of sections It follows that h
k A
G G E F
k
Since C G and the G s are chosen indep endently the distribution G E is iso
i k
morphic to the distribution G Therefore
k
C G T A accepts C G and h Pr
k A
C G G
k
Pr A accepts C G and G F
G G E
k
Pr A accepts G and G F
G G E
k
2 0
There is a corresp ondence b etween states of J and J It is not hard to see that there is also
0
a corresp ondence b etween con guration vertices of A and A
Pr A accepts G and h G T and G F
A k
G G E
k
S
k
Collapsing an NNJAG
Lemma For any k K any complete setting E of the base graphs G
k
G k B and any pebble location redundant NNJAG J with p pebbles and q
D
states there exists a pebble location redundant NNJAG J with p pebbles and q states
such that there is a correspondence between the states and pebbles of J and J
and for any G G E J is in state Q with a pebble on node u in G in a step if and
k
only if J is in the corresponding state with the corresponding pebble on node C u
E
in C G in the same step In particular J traverses a col lapsed edge hu l v i in G
if and only if J traverses the corresponding edge hC u l C v i in C G and J
E E
accepts G if and only if J accepts C G
b
Pro of J will have the same set of states as J Let b e the function that maps
b
states of J to its p ebble lo cations i e Q is the p ebble lo cations of J in state Q
Consider an arbitrary state Q The move of J in state Q will b e determined by
the action of J in state Q
In the rst substep of J there are three cases according to the rst substep of J
Case J do es nothing
Then J also do es nothing
Case J walks p ebble P along edge lab el l in the rst substep
Let P b e on no de u in state Q There are sub cases based on what u is
k
a If u is the lost no de or the no de w for some j then the
j
destination v P will reach is xed and C v is either the lost no de the
E
k
no de w or u for some j Hence J jumps p ebble P to C v
j E
h j i
b If u is a hard no de in layer D of some G i e u is a goal no de then the
i
out edges of u are collapsed edges In this case J walks p ebble P along
edge l
c If u is not a goal no de and not an auxiliary no de then the destination v
is xed for all G G E If C u C v then u must b e a hard no de
k E E
and v must b e an easy no de Hence J jumps p ebble P to C v which is
E
the lost no de If C u C v then J do es nothing
E E
Case J jumps p ebble P to no de v
k
Then v must b e either the lost no de or w or u for some j Hence
j
h j i
k
C v is either the lost no de w or u for some j J just jumps
E j
h j i
p ebble P to C v
E
Let b e the p ebble lo cations of J after the rst substep and let J assume state
Q in the second substep In the second substep J simulates the sequence of jumps
in the second substep of J That is if J jumps p ebble P to P J do es the same
Then it assumes state Q if and only if its p ebble lo cation after the rst substep is
where C denotes the p ebble lo cations so that C P C P for C
E E E E
any p ebble P and any p ebble lo cation
We claim that J has the prop erties stated in the lemma i e if J is in state Q
in step t on input G G E J is in the same state in step t on input C G G
k k
b
and C Q sp eci es the lo cations of its p ebbles in that step We shall prove the
E
claim by induction on the numb er of steps taken
b b
Initially J and J start with state Q Both Q and C Q sp ecify that
E
all p ebbles are on no de s
Supp ose at step t J is in state Q on input G and at the same step J is in state
b
Q on C G and its p ebble lo cations are sp eci ed by C Q In particular
E
p ebble P of J is on no de C u Let us check that in all the ab ove cases the p ebble
E
lo cations of J after the rst substep is C Since J only moves p ebble P from
E
no de u to v in the rst substep we just need to show that J moves P from no de
C u to C v in the rst substep This is obviously true in all the ab ove cases
E E
except b In Case b h u l v i is a collapsed edge in G By the de nitions of
G and G h C u l C v i is an edge in C G Hence p ebble P of J will b e on
k k E E
no de C v after the rst substep It follows that J will also assume state Q in
E
the second substep Moreover in the second substep J changes the p ebble lo cations
b
to Q by p ebble jumps By construction J will also change the p ebble from
b
lo cations from C to C Q
E E
Pro of of Main Lemma
Lemma For any k K and any sub branching program A of a sectioned
branching program derived from some NNJAG with at most S log n pebbles
S
Pr w G S log n and h G
G G A A
k
k
Pro of of Lemma Recall that every G G consists of D graphs G
k
k
a graph G chosen from H and G G k chosen indep endently from B
k
D
k
ro ots Therefore there are some xed auxiliary no des and edges Each G has
i
k
ro ots D
Our pro of will concentrate on the traversing of the base edges in G Thus we
assume that G is xed and known to A Hence the probability is only over the
k
graphs G G k B Let B b e the distribution
k
D
k
f h G G k i j G G k B g We allow the NNJAG to jump p ebbles
D D
to any no de in the top layer of any G Moreover each time a variable X is queried
i hu li
the followings are returned the value v of X whether v is a goal no de
hu li
and if so its corresp onding ro ot no de With these changes we assume that the
branching program do es not query any collapsed edge as there is no such need
k
Finally we assume that A is a ary decision tree i e it will not forget
any previous query A will not rep eat any previous query each computation
path in A queries at most di erent base edges since any graph G following a
computation path that queries more than di erent edges cannot satisfy h G
A
each in A discovers at most S log n di erent goal no des With all the
S
ab ove assumptions we just need to show that Pr w G S log n
G B A
k
Just b efore A starts each of the S log n p ebbles may already b e part way down a
hard path or even on a goal no de Therefore we assume that the goals of those hard
paths containing p ebbles initially are discovered by A There are at most S log n
such goals To p ebble the other goals we know by Lemma that the entire hard
path must b e traversed by the NNJAG In other words every edge in the hard path
has to b e queried by A Let w G b e the numb er of ro ots such that every edge on
A
its hard path in G has b een queried by A To prove the lemma it su ces to show
S
G S log n that Pr w
G B
A
k
Consider an arbitrary computation path in A It can b e sp eci ed by the sequence
of base edges E it has queried and the sequence of no de names R sp ecifying
whether a goal no de is discovered in each step and if applicable its corresp onding
ro ot For example supp ose queries the variable X which has the value v and
h u li
then the variable X which has the value v Supp ose v is not a goal but v is the
hu l i
goal no de of ro ot r then E hu l v i h u l v i and R r assuming no ro ot
has name
First let us understand what we can deduce ab out G the sequence of edge
r
lab els on the hard path in G ro oted at r given that G B E Note that G may
k
not actually follow We say that a no de v is a col lision node with resp ect to E
if E contains two distinct edges h u v i and h u v i with the same destination v
Since v has in degree at least two it is known to b e an easy no de This is useful
information As an extreme example supp ose that u is known to b e on the hard
path of r and hu w i is the other edge coming out from u then the NNJAG
learns that w is on the hard path of r to o This could b e rep eated D times learning
the entire hard path See Figure
D
In general we can classify f g according to and r as follows Supp ose
we trace out a path through the edges in E starting at the ro ot r and following
hard u u' w v
hard
Figure Using collisions to learn the hard path
the sequence of edge lab els until the next edge to b e taken is not contained in E
Then one of the following three p ossibilities will o ccur
The path passes through some collision no de with resp ect to E
The path reaches layer D without passing through any collision no de with
resp ect to E
The path stops b efore reaching layer D and do es not pass through any collision
no de with resp ect to E
D
We de ne Y and Z to contain the vectors f g such that when the
h r i h r i
ab ove pro cedure is applied the second and third outcomes will o ccur resp ectively
Claim For any computation path in A any input graph G B E and any
k
root r G Y Z
r h r i h r i
Pro of For any Y Z and any input G B E the path from the ro ot
k
h r i h r i
r lab elled with in G contains a collision no de Since collision no des have in degree
at least two in E they do not lie on the hard path Therefore G
r
For all G B E de ne Prog G as the random variable indicating that
k
h r i
all the edges in the hard path in G ro oted at r are mentioned in E Obviously
Prog G is true if G Y and false if G Z If G actually follows
r r
h r i h r i
h r i
and Prog G is true the goal of ro ot r is discovered Let y jY j and z
h r i h r i h r i
h r i
jZ j Intuitively the probability that Prog G is true given that G B E
k
h r i
h r i
y
h r i
is approximately
y z
h r i h r i
Let b b b e some constant to b e determined We say that ro ot r is a high col lision
b log n
root with resp ect to the computation path if y z Otherwise
h r i h r i
we say that it is a low col lision root with resp ect to We say that is a high
col lision computation if there are at least S log n high collision ro ots with resp ect to
Otherwise we say that it is a low col lision computation Let C b e the set of all
high collision computation paths Then
S
Pr w G
G B
k A
log n
X X
S
G w Pr G follows Pr and G follows
G B G B
A
k k
log n
C
C
SUM SUM
P
where SUM Pr G follows and
G B
k
C
P
SUM G S log n and G follows By Claim in Section Pr w
G B
A
k
C
S
SUM is at most
Consider SUM If b oth events w G S log n and happ en there
G
A
exists at least S log n ro ots r such that Prog G is true i e all the edges on
h r i
the hard path ro oted at r in G are in E For C at least S log n of these are
G low collision ro ots with resp ect to E For any and any G B E de ne w
k
as the numb er of ro ots r such that r is a low collision ro ot with resp ect to and
Prog G is true Then
h r i
X
S
and G follows w G Pr SUM
G B
k
log n
C
Since G follows implies G B E
k
X
S
Pr w G SUM and G B E
G B k
k
log n
C
X
Pr G B E max Pr w G S log n j G B E
G B k G B k
k k
C
C
Consider a xed graph G B Since every computation path in A queries at most
k
di erent base edges and discovers at most S log n di erent goal no des there
S log n S
S log n n di erent values of R Hence there are at are at most
S
most di erent computation paths such that G satis es E Otherwise there
and G satis es exists two di erent computation paths and such that R R
For and to b e di erent there must b e an edge hu l v i in E b oth E and E
such that v v Then G cannot satis es b oth E and and an edge in hu l v i in E
P
S
a contradiction Hence E Pr G B E and thus
G B k
k
C
S
w G S log n j G B E SUM max Pr
k G B
k
C
G S log n j G B E is at most w Claim of Section shows that Pr
k G B
k
S
for any in A In conclusion
S S S
SUM SUM
S
S
for big enough n Hence Lemma Main Lemma follows
Bounding SUM
This section b ounds the rst sum SUM at the end of the pro of for Lemma
Main Lemma
P
S
Claim Pr G fol lows
G B
k
C
Pro of We rst de ne two games called the branching process game and the edge
col lision game Let S and S b e the random variables indicating the success of the
br ed
P
corresp onding games Then we shall show that Pr G follows Pr S
G B ed
k
C
S
Pr S
br
The Branching Pro cess Game
Our branching process game is a variant of a standard branching pro cess We rst
give an alternative de nition of a branching pro cess For more standard de nitions
see Athreya and Ney AN for example
In the branching pro cess there is an in nite complete binary tree in which each
edge has probability of dying indep endent of the outcomes of the other edges A
no de u in the tree is said to b e alive if and only if all the edges along the unique path
from the ro ot to u are alive We are interested in the numb er of living no des in a
layer d We count the layers from onwards i e the ro ot is on layer
k
In our branching pro cess game there are D complete binary trees of height
D The ro ots are at layer and the leaves are at layer D For each tree there will b e
a particular path whose edges are alive Each of the other edges dies with probability
indep endent of the other edges A no de u in a tree ro oted at r is alive if and only
if all the edges on the path from r to u are alive A ro ot r is said to b e a high col lsion
b log n
ro ot if the subtree ro oted at r has at most living no des at layer D Let S
br
b e the random variable indicating the event that there are more than S log n high
collision ro ots
Let p b e the probability that r is a high collsion ro ot For i D let v b e
r i
the no de on layer i which is on the path whose edges are guaranteed to live and let
u b e the sibling of v If u is alive then the subtree of height D i ro oted at u
i i i i
can b e thought of as resulting from the rst D i layers of a standard branching
b e the probability pro cess Let I b e the numb er of live u s for i D and let p
i
r
b log n
that each subtree ro oted at an alive u has at most living no des in layer D
i
The exp ected conditional on the event that I D Then p Pr I D p
r
r
D
value of I is D D and hence Pr I D e by the Cherno
b ound
The value of p can b e estimated as follows Let N b e the numb er of living no des
d
r
at layer d in a standard branching pro cess Lemma of Pipp enger Pip says
u u d
Pr N O provided b oth u and d u tend to in nity
d
Recall that D b log n and that the subtree ro oted at u where i D has
i
b log n b log n log b log n
height D i D Let us re write as and
set b b so b log n D Then b oth D and D b log n tend
to in nity as n tends to in nity The probability that the subtree ro oted at u has
i
b log n D b log n
living no des in layer D is at most O There at most
D b log n
D b log n b b log n
fore we have p O O and
r
b log n
b log n b b log n b log n
consequently p e O for some
r
constant b dep ending on b
b log n
The exp ected numb er of high collision ro ots is n b ecause there are
at most n ro ots By cho osing big enough b can b e made arbitrarily small As each
tree is grown indep endent of the others we can apply the Cherno b ound and prove
that
Pr S
br
S
Pr numb er of high collsion ro ots
log n
S S
ln
log n log n
S
provided b is big enough
The Edge Collision Game
k
The edge col lision game is de ned as follows D graphs G G G k are
D
k
chosen randomly and indep endently from B The player is informed of the hard
path of each ro ot in each G He then cho oses at most S log n no des in the G s
i i
called the seeds After this initialization he queries edges of G one at a time When
i
the player queries an edge he sp eci es hu i where u is a no de and f g is
an edge lab el The destination no de v of the edge h u v i is then revealed to the
player Based on the result of the query he cho oses the next edge to query There
are however restrictions on the order in which the player is allowed to query the
edges For each query he can either query a no de at the top layer of any graph G
i
or one of the seeds or a no de that he has already learned to b e the destination of
another edge As well he is allowed to query at most edges in total
We shall measure the p erformance of the game player by two sets of parameters
y s and z s to b e de ned shortly The parameters are very similar to the y
r r h r i
z s de ned in Section Let E b e the set of base edges of the input graph G
h r i
that the player has queried during the game Recall that in Section a no de v
is called a col lision node with resp ect to E if it is the destination of more than one
edge in E Here each edge in E will b e assigned one of the two conditions alive or
dead An edge is said to b e dead if its destination no de is a collision no de or one of
the S log n seeds Otherwise it is alive We shall construct from E a collection of
binary trees by the following steps
Step Delete from G all the no des v together with all its in out edges if E do es
not contain a path from any ro ot r to v
Step Delete all no des together with all their in out edges which are prop er
descendents in E of the destinations of the dead edges The dead edges and
their destinations themselves are kept
The remaining edges in E that are alive are called the y edges and their desti
nation no des are called the y nodes Each such no de has a unique path from
a ro ot to it and the path contains no dead edges Hence the y edges form a
collection of disjoint binary trees ro oted at the ro ots r
Step Fill up the ab ove trees so that each no de has exactly two child edges More
precisely for each y no de that do es not have exactly two child edges counting
the dead edges add the missing edges and attach to each such edge a complete
binary tree down to layer D The no des and edges that are added in this way
are referred to as the z nodes and the z edges They do not corresp ond to actual
no des and edges in the input graph G Note that each z no de also has a unique
path from a ro ot to it and the path contains no dead edges
De ney andz as the numb er of y and z no des that are descendents of the ro ot no de
r r
b log n
r in layer D Again a ro ot r is said to b e a high col lision ro ot ify z The
r r
goal of the player is to make as many high collision ro ots as p ossible More precisely
the player wins if there are more than S log n high collision ro ots Let S b e the
e
indicator variable for this event
We now relate the edge collision game to a sub computation by an NNJAG Both
the edge collision game player and the NNJAG query edges in the G s The game
i
player is not told whether each no de v discovered is a goal and its corresp onding ro ot
but he can deduce this information b ecause he knows all the hard paths Moreover
he has more exibility in cho osing which edges of the input graph to query In the
NNJAG computation the machine starts computing from some initial id The game
player can cho ose the no des that contain p ebbles initially as the seeds in the game
In each step the NNJAG moves a p ebble from some no de u along an edge with some
lab el Note that no de u must contain a p ebble b efore the NNJAG can make this
move Hence a necessary but may b e not su cient condition for the walk from no de
u to b e a legal move by the NNJAG is either u contains some p ebbles b efore the
sub computation starts or u has b een p ebbled b efore the current move However
this condition is already su cient to allow the game player to query the edge h u l i
Therefore whenever the NNJAG queries an edge the edge collision game player can
always query the same edge provided he has queried all the edges queried by the
NNJAG so far Hence for all G the game player can always ensure that E E
Now consider the conditions for a ro ot to b e a high collsion one In the NNJAG
computation a ro ot r is a high collision ro ot with resp ect to E if y z
h r i h r i
b log n D
Note that a vector f g is not in Y Z if the path obtained by
h r i h r i
following from r contains a collision no de In the edge collision game if the set E of
edges queried by the game player includes E the same no de will b e a collision no de
with resp ect to E Hence the same ro ot r is also a high collsion ro ot by de nition of
the game It follows that Pr C Pr S
G ed
Branching Pro cess Game vs Edge Collision Game
This subsection proves the second inequality
Lemma The success of the edge col lision game is probabilistical ly dominated by
the success of the branching processes game i e Pr S Pr S
ed br
Pro of The edge collision game starts by randomly cho osing the graph G G G k
D
k k
in B For each i D the rst step in cho osing G according to the dis
i
k k k
easy no des at each layer into is to randomly partition the tribution B
k
no des and hard no des and to cho ose the hard path ro oted at each ro ot r This
information is revealed to the player The edges in these paths corresp ond to the
edges in the branching pro cess game that are guaranteed to live
k
is to cho ose The next step in cho osing G according to the distribution B
i
k
for each remaining edge its destination among the easy no des at the next
layer This only needs to b e done for those edges queried by the player For each
k
i D for each layer d D and for each de ne the variable
k
Supp ose that the v to uniformly and indep endently take on a value from
hi d i
player is querying hu i where u is a no de at layer d in the graph G and this is the
i
k
th
query to easy edges at this layer in this graph Then the variable v
hi d i
k
easy no des at the next layer this edge hu i will b e sp eci es which of the
connected to
Because the player is able to cho ose dynamically which edges hu i to query we
do not know in advance which edges h u i will b e queried However the random
variables v do tell us the resulting destinations of all the edges queried Hence
hi d i
without knowing which edges h u i the player will cho ose we do know which of the
th
queried edges will die Sp eci cally the edge queried at layer d in G will die if
i
either there is another query at this layer with the same destination
i e v v for some or the destination is one of the chosen seeds
hi d i hi d i
In order to compare the success probability of the edge collision game and the
branching pro cess game let us de ne random variables that will indicate which edges
die in the branching pro cess game For each ro ot r in each of the graphs G there is
i
a corresp onding ro ot in the branching pro cess game Consider the complete binary
tree of height D ro oted at such a ro ot r A sp eci c edge in this tree can b e sp eci ed
by a string f g For each such edge let x f g b e the random variable
hr i
indicating whether this edge dies If the edge is one of the edges that are guaranteed
to live in the branching pro cess game i e the xed hard path in the input graph then
Pr x Otherwise Pr x indep endent of the other x variables
hr i hr i
Now consider a xed algorithm for the edge collision game For each intermediate
th
time step t we de ne the t game as follows The game starts with t
time steps of the xed algorithm for the edge collision game Let E b e the resulting
t
th
base edges queried We want these edges to die in the t game if and only if they
die in the edge collision game As previously mentioned the edge asso ciated with
with v is dead if and only if there is another query at this layer
hi d i
the same destination i e v v or the destination is one of the S log n
hi d i hi d i
seeds The query asso ciated with v can o ccur either b efore or after time step t
hi d i
Either way we consider the edge asso ciated with v dead Given which edges in
hi d i
E have died the set E can b e transformed as describ ed in Steps to into a
t t
collection of binary trees made up of y no des and y edges the living ones z no des
th
and z edges as well as some collision no des and dead edges The t game is completed
by completing the branching pro cess on the z edges Namely each such edge will live
or die according to the corresp onding random variable x f g A no de u in
hr i
the resulting collection of trees is said to b e alive if all the edges on the path from
the ro ot of the tree to u are alive A ro ot is said to b e a high col lision ro ot if it has
b log n th
at most living no des in layer D The t game succeeds if there are more than
S log n high collision ro ots Let S b e the random variable indicating the success of
t
the game
th
Observe that the game is simply the branching pro cess game and hence S
br
th
S The game di ers from the edge collision game only in that in the edge
collision game all the z no des edges added in Step are treated as alive while in
th
the game some of the z edges may die according to the x variables
hr i
The additional children at layer D only hurt the edge collision game player There
fore Pr S Pr S What remains to prove is that for every t
ed
Pr S Pr S
t t
st
Let V sp ecify a p ossible computation up to and including the t query It
t
will sp ecify the values of t of the v variables Which of them are sp eci ed will
hi d i
dep end dynamically on the computation The computation V will also sp ecify the
t
set of queried edges in the graph E and the next query hu i made by the player
t
th
Let the no de u b e on layer d of G and the query b e the one at this layer in this
i
graph
By the rules of the edge collision game no de u can only b e either a ro ot a seed
or a no de that he has already learned to b e the destination of another edge There
are three cases In the rst case E do es not contain a unique path with no dead
t
edges from a ro ot to u In this case the descendant no des and edges of no de u will b e
st th
deleted from the y no de tree b oth in the t game and in the t game Hence
whether this edge dies has no e ect on either game In the second case hu i is on a
hard path and for b oth games the edge is guaranteed to live
In the third case E contains a unique path with no dead edges from a ro ot
t
to u and hu i is not on a hard path Let r and sp ecify the ro ot and the lab els
st
in this path In the t game whether the edge from h u i dies is sp eci ed by
th
the variable x f g In the t game the destination of the edge from h u i
hr i
is sp eci ed by the variable v Consider one setting V of all the v variables
hi d i t
other than those set by V and other than the variable v Consider as well one
t hi d i
setting X of all the x variables other than x
t hr i
i i h h
V V X In b oth V V X and Pr S Compare Pr S
t t
t t t t t t
cases the probability is only over the values of v and x Everything else is
hi d i hr i
xed by V V and X For every value of v and x which edges die
t t t hi d i hr i
st
b efore time step t and which die after time step t is the same for b oth the t and
th
the t game The only change in the game is whether or not the edge from hu i dies
i h
st
In the t game this edge dies with probability Pr x V V X
hr i t t t
th
In the t game this edge dies if there exists a for which
v v or v hits one of the S log n seeds V and V x at most
hi d i hi d i hi d i t t
di erent values of the variables v and the S log n seeds x at most another
hi d i
p
k
nS log n S log n S log n S log n di erent values in Since
k
Therefore the probability that The value for v is chosen uniformly from
hi d i
S
v collides with one of these values given V V X is at most
hi d i t t t
log n
th
Having a smaller probability of this edge dying can only hurt the t game player We
can conclude that
i i h h
V V X V V X Pr S Pr S
t t
t t t t t t
and hence
Pr S
t
i h i h
X
V V X V V X Pr Pr S
t
t t t t t t
V V X
t t t
i h i h
X
V V X V V X Pr Pr S
t
t t t t t t
V V X
t t t
Pr S
t
Bounding SUM
This section b ounds the third sum at the end of the pro of of Lemma
Claim For any computation path in A
S
Pr G S log n j G B E w
G B k
k
Pro of Recall that w G is the numb er of ro ots r in G such that r is a low collision
ro ot with resp ect to and Prog G is true By Claim b elow the exp ected value
h r i
of w G is
X
Pr Prog G j G B E
G B k
h r i
k
low collision ro ots r
X
y
h r i
y z
h r i h r i
low collision ro ot r
P
y
h r i
low collision ro ot r
b log n
b log n
P
b ecause y At most di erent edges are queried by By cho osing
h r i
r
big enough b which implies cho osing big enough b can b e made arbitrarily small
By Lemma in Edmonds Edm
S
Pr w G j G B E
G B k
k
log n
S S
ln
log n log n
S
provided b is big enough
Claim For any computation path in A any root r and any subset O of roots
indicating for which roots r other than r Prog G is true
h r i
y
h r i
Prog G j G B E and O Pr
k G B
h r i
k
y z
h r i h r i
Pro of Let us consider a xed and r Recall that G is the random variable
r
indicating the vector of edge lab els on the hard path ro oted at r in graph G drawn
from B and that G Y Z Recall as well that Prog G is true
k r
h r i h r i
h r i
if and only if G Y We shall drop the subscripts in Y Z y
r h r i h r i h r i h r i
z G Prog G and E when there is no confusion We shall also write
r
h r i
h r i
Pr j G B E as Pr j E Note that
G B k
k
Pr Prog G j E and O
Pr Prog G j E and O
Pr Prog G j E and O Pr Prog G j E and O
P
Pr G j E and O
Y
P P
Pr G j E and O Pr G j E and O
Z Y
Consider the denominator Let b e the vector in Y that maximizes
y
Pr G j E and O over Y and let b e the vector in Z that minimizes
z
Pr G j E and O over Z Then the ab ove probability is at most
y Pr G j E and O
y
y Pr G j E and O z Pr G j E and O
y z
y
Pr G jE and O
z
y z
Pr G jE and O
y
What remains to b e proven is that
Pr G jE and O
z
Pr G jE and O
y
Let N G and N G b e resp ectively the set of edges on the path with lab el and
y z y
from ro ot r in G Let H G b e the random variable sp ecifying the hard path ro oted
z
at r in G i e b oth the no des and the lab els G The fact that Y means that
y
the path following the edge lab els in is totally contained in E Therefore N G
y y
is equal to some xed value N determined by E Then the statements G
y y
and G are equivalent to H G N and H G N G resp ectively The
z y z
p ossible values N for the random variable N G i e the path in G ro oted at r with
z z
edge lab els can b e divided into two sets Let N A if and only if some edge
z z z
in N collides with some edge in E In this case N cannot b e the hard path That
z z
is for N A Pr H G N j E and O Now consider an N A Given
z z z z z
that G contains E N and satis es O it is equally likely for H G to b e N or
z y
N b ecause b oth xed paths N and N are contained in E N and neither have
z y z z
collisions with other edges in E N That is
z
Pr H G N j N G N and E and O
y z z
Pr H G N j N G N and E and O
z z z
Note as well that H G N implies N G N Therefore
z z z
Pr H G N and N G N j E and O
y z z
Pr H G N and N G N j E and O
z z z
Pr H G N j E and O
z
The ab ove ratio then b ecomes
Pr G j E and O
z
Pr G j E and O
y
P
Pr H G N j E and O
z
N A
z z
Pr H G N j E and O
y
P
Pr H G N and N G N j E and O
y z z
N A
z z
Pr H G N j E and O
y
Pr N G A j H G N and E and O
z z y
The input distribution B rst cho oses the hard paths Then every other edges is
k
added indep endently at random If N G is not a hard path at each level i D
z
k
its no de is chosen from the easy no des at this level For N G not to b e in
z
A its edges must not collide with the edges in E Let h b e the numb er of no des
z i
mentioned in E at level i that N G must avoid It follows that
z
Pr N G A j H G N and E and O
z z y
h
i
i D
k
P
h
i
i D
k k
P
b ecause h as E contains at most di erent edges This completes
i
i D
the pro of of the claim
Probabilistic NNJAGs with sided error
The ab ove can b e extended to sided error but not as straight forward as in Chapter
Here we use a similar technique as used in Edmonds Edm
Theorem There is a positive integer n and positive constants and c
o
such that for any n n and any S n log n there is a distribution on n node
directed graphs G such that
G has maximum out degree
n log n
q
c log
S
nS
log log n
for any deterministic NNJAG J with S space and at most
log n
time
J is correct for G Pr
G
Pro of We shall construct for each k the distribution G from G as follows First
k
k
we de ne the concept of sup er hard path and sup er goal for graphs in H and
k
th
G For k we de ne the j sup er hard path of a graph G H simply as the
k
th th
j hard path in G The no de of the j sup er hard path in layer D is called the
th
j sup er goal There are sup er hard paths and the same numb er of sup er goals
k
in every graph in H For k each graph G H is obtained by picking D
k
k th
graphs G G k from B and a graph G from H Recall that the j
k
D
th
hard path of G is identi ed with u of G Supp ose the j sup er hard path of G
i
hi j i
th
Then the j sup er hard path of G is obtained by is u u u
k
h i h i
D
i h
k
D
th th
replacing each no de u with the hard path of G The j sup er goal of G is
hi i i i
i
th k
the no de of the j sup er hard path of G in layer D Thus each graph in H also has
k
only sup er hard paths and the same numb er of sup er goals The sup er hard paths
and sup er goals of the graphs in G are de ned similarly Now the graph G G is
k
k
formed by picking a graph G G and cho osing with probability whether G is
k
st connected If so cho ose one of sup er goals of G to b e connected to t uniformly
at random The distribution is then G For each graph G which is not
K
st connected denote by G the graph formed by adding an edge from the sup er goal
v
v to t in G For each G and any sub computation A we de ne w G as the
A
numb er of sup er goals of G p ebbled during the sub computation A We divide the
domain of the p ossible graphs in into ve disjoint sub domains Let
A f G j G is not st connected and w G g
J
B f G j G is not st connected and w G g
J
b
A f G j G A g
v
b
B f G j G B and v is visited during the computation by J on G g
v
b
B f G j G B and v is not visited during the computation by J on G g
v
Then
J is correct for G Pr
G
h i i h
b b
Pr G A A Pr G B
G G
h i
b
Pr J is correct for G and G B B
G
Consider the rst probability
h i
b
Pr G A A
G
G A Pr
G
w G and G is not st connected Pr
J G
w G j G is not st connected Pr
J G
By Lemma b elow we can prove that this probability is at most for big
enough n
i h
b
G B Pr For the second probability Pr G B
G G
h i
b
For the third probability Pr J is correct for G and G B B
G
h i
b b
Pr J is correct for G j G B B For G B the computations of J on G
G v v
and on G which is in B are the same b ecause no de v is not visited Therefore J must
give the same answer to G and G If it accepts it is correct for G but wrong for G
v v
If it rejects it is correct for G but wrong for G Since for each G B some of the
v
i h
b b
the b est b et G B G B Pr corresp onding G s may b e in B Pr
G v G
h i
b
of J is to reject Hence Pr G B J is correct for G j G B B Pr
G G
J is correct for G Hence Pr
G
The following lemma is similar to Lemma
Lemma There exists a constant b such that for any k K and any sec
S
tioned branching program A with at most T sections derived from some pebble
k
log n
location redundant NNJAG with at most S space
S
Pr w G j G is not st connected k
G G
A
k
log n
k
where T
k
S
Pro of Note that when conditioned on graphs not st connected the distribution G
k
is the same as G for all k K Thus we can replace the probability stated in the
k
lemma by the probability Pr w G
G G
k A
The only di erence b etween the two lemmas is that in Lemma the machine has
to p ebble all the sup er goals while the machine here has to p ebble only sup er
log n log n
k k
goals Thus we change the time b ound from T to T
k
k
S S
Base Case
The pro of is similar to that of the base case of Lemma We cut the sectioned
log n
slices such that slice i consists of the p ortion of the branching program A into
S
log n log n
S S
branching program from section i T to section i T
S log n S log n
inclusively Then we can argue as b efore that for each graph G G such that
b
w G and h G T there is a sub branching program A that lies within
A
A
a slice such that w G S log n and h G Then the rest of the pro of is
A
A
the same
Induction Step
S
slices so that Supp ose H k is true As in Lemma we cut A into T
k
log n
slice i consists of section i T T to i T T inclusively We argue
k k k k
S
that if Pr w G and h G T k then there exists a sectioned
G G A
k A k
S
branching program A with at most T sections derived from some p ebble lo ca
k
log n
G tion redundant NNJAG J with at most S space so that Pr w
G G
k A
S
k This contradicts H k A key observation in the construction of J is
that its computation on graph C G completely mirrors that of J on G Thus J
visits a sup er goal v in G if and only if J visits the corresp onding sup er goal C v
E
in C G
Chapter
Future Work
Although we have proved almost matching space and time space lower b ounds the
b ounds are not exactly tight It would b e interesting to close these gaps On the
other hand it is more interesting in my opinion to prove similar lower b ounds on a
general mo del of computation To achieve that one p ossible approach is to start with
a JAG NNJAG like mo del and add more and more p ower pushing our way towards
the ultimate mo del of the branching program The other approach is to work on the
nite mo del theory In the following sections we describ e some new thoughts in these
two directions
Variations of JAGs NNJAGs
The main weakness of a JAG or NNJAG is its restricted access to the inputs As
p ointed out in EI the space lower b ounds of CR BS and that in Chapter
are proved on a tree However it is easy for a RAM to solve stcon on trees in
O log n space All it needs to do is to walk a p ebble from no de t backward and
see if it hits no de s
However if we give the machine the ability to access any no de it wants proving
lower b ounds b ecomes to o di cult with our current techniques For example we
de ne a strong jump NNJAG as an NNJAG with the extra strong jump move P v Q
which consists of jumping p ebble P to no de v and assuming state Q This mo del is
essentially as p owerful as a branching program Whenever a branching program asks
whether there is an edge from no de u to v the strong jumping NNJAG can jump
a p ebble to no de u and try out all the out edges of u to see if any of these edges is
p ointing at v This requires only O n time and O log n space to keep track of which
out edges have b een tried In other words a strong jumping NNJAG can simulate
each step of a branching program with O n time and O log n space overhead
Similarly we de ne a strong jump JAG as a JAG that can jump a p ebble P from
its current no de say no de i to the no de i mod n Note that the JAG do es not
know the lab el on the no des However it is easy for the JAG to assign names to the
no des itself Supp ose it always has a p ebble on no de s Then it can consider the no de
reached by jumping a p ebble to no de s and followed by i n strong jumps as
th
the i no de Using the same argument as for strong jump NNJAGs a strong jump
JAG can simulate each step of a branching program with O n time and O log n
space overhead
Another interesting mo del is the way NNJAG In this mo del the in edges of the
input graph are also lab elled like the out edges and the machine can walk a p ebble
backward along an edge Again this mo del is as p owerful as a branching program
The machine can view the graph as an undirected one and access whatever no de it
wants by means of a universal traversal sequence Once it is able to access an arbitrary
no de i it can simulate an arbitrary step of a branching program as describ ed b efore
Since p olynomial length universal traversal sequences are known the simulation takes
O
O log n space and n time overhead
There is however a mo del called the Stack NNJAG that can solve stcon for
trees in O log n space and yet on this mo del we can still prove the same space lower
b ound as in Chapter and time space lower b ound as in Chapter In this mo del
there is a constant numb er of stack p ebbles in addition to those regular p ebbles Each
stack p ebble has a stack which can rememb er the path that it has traversed since its
last jump More precisely all the p ebbles whether the regular or the stack ones are
initially on no de s The stack of each stack p ebble is empty initially Whenever a stack
p ebble walks along an edge u v the no de u is pushed onto the stack Whenever a
stack p ebble jumps to another p ebble P it empties its stack If P is also a stack
p ebble then P copies the stack of P to its own stack A stack p ebble can also
backtrack along the path i e to move to the no de v if v is the top of the stack and
then p op the stack Note that the p ebble is not allowed to visit any arbitrary no de
Any no de reachable by a stack p ebble must b e reachable from s by a directed path
The space for storing the stacks is given for free
To prove the lower b ound in Chapter observe that the height of the graph
p p
n S log n If n S log n S log n each stack can only store at most is O
O S log n no des Since a Stack NNJAG has a constant numb er of Stack p ebbles it
can b e simulated by a normal NNJAG with at most S log n extra p ebbles The
extra p ebbles simply jump to and remain on each no de that a stack p ebble reaches
p
This increases the space used by the algorithm by at most S If n S log n
S log n then S n log n The b ound we have for a normal NNJAG is T
log n
log log n
Now observe that the height of each stack is at most the height of the
p
n log n Hence any Stack NNJAG with space S can b e graph i e at most O
p p
simulated by a normal NNJAG with space O S n log n O n log n Hence
the same lower b ound applies
At last we suggest another mo del called random jump JAG In this mo del the
machine is allowed to jump a p ebble to a random no de The no de is chosen uniformly
from the set of all no des and the machine do es not know which no de the p ebble has
landed on Although the machine has access to all parts of the input and can solve
O
stcon for trees in n time and O log n space with high probability it do es not
seem p ossible for it to simulate a branching program e ciently Proving lower b ounds
on this mo del may still b e p ossible
Finite Mo del Theory
As mentioned in Chapter another approach to separating NL and L is to show that
stcon is not expressible in FO DTC over ordered structure Following the recent
work by EI which extended the result of GM by adding way lo cal ordering to
an unordered structure one can think of ways to add more to a way lo cally ordered
structure and prove a similar result For more details please see Etessami Ete
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