Symmetric Logspace Is Closed Under Complement Basic Research in Computer Science

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BRICS ISSN 0909-0878 September 1994 BRICS Report Series RS-94-31 Noam Nisan Amnon Ta-Shma Symmetric Logspace is Closed Under Complement Basic Research in Computer Science

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Symmetric Logspace is Closed Under Complement

Noam Nisan Amnon TaShma

noamcshujiacil amcshujiacil

September

Abstract

We present a Logspace manyone reduction from the undirected stconnectivity prob

lem to its complement This shows that SL co SL

Intro duction

This pap er deals with the complexity class symmetric Logspace SL dened by Lewis and

Papadimitriou in LP This class can b e dened in several equivalentways

Languages which can b e recognised by symmetric nondeterministic Turing Machines

that run within logarithmic space See LP

Languages that can b e accepted by a uniform family of p olynomial size contact schemes

also sometimes called switching networks See Raz

Languages which can b e reduced in Logspace via a manyone reduction to USTCON

the undirected stconnectivity problem

A ma jor reason for the interest in this class is that it captures the complexityofUSTCON

The input to USTCON is an undirected graph G and twovertices in it s t and the input

should b e accepted if s and t are connected via a path in G The similar problem ST CON

where the graph G is allowed to b e directed is complete for NL nondeterministic Logspace

wn to b e in SL or co SL eg colourabilityis Several combinatorial problems are kno

complete in co SL Rei

The following facts are known regarding SL relative to other complexity classes in the

vicinity

L SL RL NL

Here L is the class deterministic Logspace and RL is the class of problems that can b e

accepted with onesided error by a randomized Logspace machine running in p olynomial

This work was supp orted byBSFgrant and byaWolfeson award administered by the Israeli

Academy of Sciences The work was revised while visiting BRICS Basic Research in Computer Science Centre

of the Danish National ResearchFoundation

time The containment SL RL is the only nontrivial one in the line ab ove and follows

directly from the randomized Logspace algorithm for USTCON of AKL It is also

known that SL SC Nis SL L KWandSL DS P AC E log n NSW

After the surprising pro ofs that NL is closed under complementwere found Imm

Sze Boro din et al BCD asked whether the same is true for SL They could prove

only the weaker statement namely that SL co RL and left SL co SL as an op en

problem In this pap er wesolve the problem in the armativeby exhibiting a Logspace

manyone reduction from USTCON to its complement Quite surprisingly the pro of of our

theorem do es not use inductive counting as do the pro ofs of NL co NL and is in fact

even simpler than them however it uses the AKS sorting networks

Theorem SL co SL

It should b e noted that the monotone analogues see GS of SL and co SL are

known to b e dierent KW

SL SL SL

As a direct corollary of our theorem we get that L SL SL where L is the

class of languages accepted by Log space oracle Turing machines with oracle from SL and

SL is dened similarly b eing careful with the waywe allow queries see RST

SL SL

Corollary L SL SL

This also shows that the symmetric Logspace hierarchy dened in Rei collapses to

Pro of of Theorem

Overview of pro of

We show that we can upp er and lower b ound the numb er of connected comp onents of a

graph using connectivity problems We upp er b ound this numb er using a transitiveclosure

metho d which can b e easily done since we are allowed to freely use connectivity problems

However trying to lowerb ound the numb er of connected comp onents this way requires nega

tion The heart of the pro of lies in lowerb ounding the numb er of connected comp onents

and weachieve this in a surprisingly easy wayby computing a spanning forest

In subsection weshowhowtocombine many connectivity problems to one single con

nectivity problem In subsection we showhow to nd a spanning forest using connectivity

problems In subsection weshowhow to use this spanning forest to nd the number of

connected comp onents of a graph and howwe solvethest nonconnectivity problem with

Pro jections to USTCON

In this pap er we will use only the simplest kind of reductions ie Log S pace uniform pro jec

tion reductions SV Moreover we will b e interested only in reductions to USTCONIn

this subsection we dene this kind of reduction and weshow some of its basic prop erties

Notation Given f f g f g denote by f f g f g the restriction

of f to inputs of length n Denote by f the k th bit function of f ieiff f g

nk n n

k n

f g then f f f

n n

nk n

Notation We represent an nnode undirectedgraph G using variables x fx g

ij ij n

st x is i i j E GIff x operates on graphswewillwrite f G meaning that

the input to f is a binary vector of length representing G

Definition We say that f f g f g reduces to USTCONm m mn if

there is a uniform family of S pacelog n functions f g st for al l n and k

is a projection ie is a mapping from fi j g to f x x g

nk nk ij m i i in

Given x dene G to bethegraph G fmgE where

x x

g E fi j j i j or i j x and x or i j x and x

nk nk i i nk i i

It should hold that f x thereisapath from to m in G

If is restricted to the set f x g we say that f monotonically reduces to USTCONm

i in

Lemma If f has uniform monotone formulae of size sn then f is monotonical ly re

ducible to USTCONO sn

Pro of Given a formula recursively build G s t as follows

If x then build a graph with twovertices s and t and one edge b et ween them

lab elled with x

If and G s t the graphs for i then identify s with t and

i i i i

dene s s t t

If and G s t the graphs for i then identify s with t and

i i i i

s with t and dene s s t and t s t

Using the AK S sorting networks AKS which b elong to NC we get

Corollary Sort f g f g which given a binary vector sorts it is monotoni

cal ly reducible to USTCONpol y

Lemma If f monotonical ly reduces to USTCONm and g reduces to USTCONm

then f g reduces to USTCONm m where is the standard function composition

operator

Pro of f monotonically reduces to a graph with m vertices where each edge is lab elled

with one of f x g In the comp osition function f g each x is replaced by x g y

i i i i

which can b e reduced to a connectivity problem of size m Replace each edge lab elled x

with its corresp onding connectivity problem

Finding a spanning forest

In this section we showhow to build a spanning forest using USTCON This construction

was also noticed by Reif and indep endentl y by Co ok Rei

Given a graph G index the edges from to mWe can view the indices as weights to the

edges and as no two edges have the same weight weknow that there is a unique minimal

spanning forest F In our case where the edges are indexed this minimal forest is the

lexicographically rst spanning forest

It is well known that the greedy algorithm nds a minimal spanning forest Let us recall

how the greedy algorithm works in our case The algorithm builds a spanning forest F which

is at the b eginning empty F Then the algorithm checks the edges one by one according

to their order for each edge e if e do es not close a cycle in F then e is added to the forest

ie F F feg

At rst glance the algorithm lo oks sequential however claim shows that the greedy

algorithm is actually highly parallel Moreover all we need to check that an edge do es not

participate in the forest is one st connectivity problem over a simple to get graph

Definition For an undirectedgraph G denote by LF F G the lexicographical ly rst span

ning forest of GLet

SF G f g be

i j LF F G

SF G

otherwise

Lemma SF reduces to USTCONpol y

Pro of Let F b e the lexicographically rst spanning forest of GFor e E dene G to

b e the subgraph of G containing only the edges fe E j indexe indexeg

e E i is not connected to j in G Claim e i j F

Pro of Let e i j E Denote by F the forest which the greedy algorithm built at the

time it was checking eSoe F e do es not close a cycle in F

e F and therefore e do es not close a cycle in F but then e do es not close a cycle

in the transitive closure of F and in particular e do es not close a cycle in G

e e

e do es not close a cycle in G therefore e do es not close a cycle in F and e F

e e

Therefore SF G x i is connected to j in G

ij ij

Since x can b e viewed as the connectivity problem over the graph with twovertices

and one edge lab elled x it follows from lemmas that SF reduces to USTCON

Notice however that the reduction is not monotone

Putting it together

First wewant to build a function that takes one representative from each connected com

p onent We dene LI Gtobe ithevertex i has the largest index in its connected

comp onent

Definition LI G f g

i has the largest index in its connectedcomponent

LI G

otherwise

Lemma LI reduces to USTCONpol y

Pro of

LI G i is connected to j in G

j i

So LI is a simple monotone formula over connectivity problems and by lemmas

LI reduces to USTCON This is actually a monotone reduction

Using the spanning forest and the LI function we can exactly compute the number of

connected comp onents of G ie given G we can compute a function NCC whichisi

there are exactly i connected comp onents in G

Definition NCCG f g

thereare exactly i connectedcomponents in G

NCC G

otherwise

Lemma NCC reduces to USTCONpol y

Pro of

Let F b e a spanning forest of G It is easy to see that if G has k connected comp onents

then jF j n k

Dene

f G Sort LI G

g G Sort SF G

Then

f G ki

g G n ki kn i

and thus NCC G f G g G

i i ni

Therefore applying lemmas proves the lemma

Finally we can reduce the nonconnectivity problem to the connectivity problem thus

proving that SL co SL

Lemma USTCON reduces to USTCONpol y

Pro of

Given G s tdene G to b e the graph G fs tg

Denote byCCH the numb er of connected comp onents in the undirected graph H

s is not connected to t in G

CCG CCG

NCC G NCC G

i i

Therefore applying lemmas proves the lemma

Extensions

Denote by L the class of languages accepted by Log space oracle Turing machines with

oracle from SL An oracle Turing machine has a work tap e and a writeonly query tap e

with unlimited length which is initialised after every queryWe get

Corollary L SL

Pro of

SL SL

Let Lang b e a language in L solved by an oracle Turing machine M running in L

and x an input x to M

Lo ok at the conguration graph of M In this graph wehave query vertices with outgoing

edges lab elled connected and not connected Wewould like to replace the edges lab elled

connected with their corresp onding connectivity problems and the edges lab elled not

connected with the connectivity problems obtained using our theorem that SL co SL

However there is a technical problem here as the queries are determined by the edges

and not by the query vertices We can x this dicultyby splitting each query vertex to its

yes and no answers and splitting each edge entering a query vertex to connected and

not connected edges Nowwe can easily replace each edge with a connectivity problem

obtaining an undirected graph whichisst connected i x Lang and therefore Lang SL

As can easily b e seen the ab ove argument applies to any undirected graph with USTCON

query vertices thus if we carefully dene SL see RST we get that

Corollary SL SL

In particular the symmetric Logspace hierarchy dened in Rei collapses to SL

Acknowledgements

Wewould like to thank Amos Beimel Allan Boro din Rob ert Szelep csenyi Assaf Schuster

and Avi Wigderson for helpful discussions

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