Parallel and On line Graph Coloring
Magn s M Halld rsson
Science Institute
University of Iceland
IS Reykjavik Iceland
Abstract
We discover a surprising connection b etween graph coloring in two orthogonal paradigms
parallel and on line computing We present a randomized on line coloring algorithm with
p
a p erformance ratio of O n log n an improvement of log n factor over the previous b est
known algorithm of Vishwanathan Also from the same principles we construct a parallel
coloring algorithm with the same p erformance ratio for the rst such result As a byproduct
we obtain a parallel approximation for the indep endent set problem
Introduction
At the heart of most partition and allo cation problems is a graph coloring question The graph
coloring problem involves assigning values or colors to the vertices of a graph so that adjacent
vertices are assigned distinct colors with the ob jective of minimizing the number of colors used
We seek graph coloring algorithms that are b oth e ective and e cient E ciency means
among other things using computation time b ounded by a p olynomial in the size of the input
E ectiveness means that the quality of the coloring is never much worse than the optimal
coloring it is measured in terms of the performance ratio of the algorithm de ned as the ratio
of the number of colors used to the minimum number of colors needed maximized over all
inputs In this pap er we present a reasonably e ective and e cient graph coloring algorithm
that yields b oth parallel and on line algorithms
A parallel algorithm is said to b e e cient if it uses time p olylogarithmic and pro cessors
p olynomial in the size of the input The coloring problem is of central imp ortance in parallel and
distributed computation as a means of scheduling non con icting concurrent tasks Previous
parallel algorithm have attained the degree b ound of Bro oks s theorem coloring of planar
graphs and some other sp ecial cases but no non trivial p erformance guarantee on general
k k
graph was known We present a parallel algorithm that uses at most n colors on
k colorable graphs for a p erformance ratio of O n log n where n is the number of vertices
An algorithm is said to b e on line if the color assigned to vertex i is indep endent of ver
tices with index greater than i This corresp onds to various dynamic situations such as the
assignments of frequencies to mobile users The on line graph coloring problem has b een widely
studied and for various other reasons as well In particular an algorithm that improves
the trivial p erformance ratio to O n log n was given by Lov sz Saks and Trotter and
p
p
n on colorable graph and O n log n ratio on a randomized algorithm with ratio of O
email mmh rhi hi is Research partly p erformed at Rutgers University Tokyo Institute of Technology and
Japan Advanced Institute of Science and Technology Hokuriku and supp orted there by a DIMACS Graduate
Fellowship a Toshiba Endowed Chair and a PFU Endowed Chair resp ectively
general graphs was given by Vishwanathan We improve here the randomized algorithm of
for a p erformance ratio of O n log n
We rst present in Section a sequential graph coloring algorithm with a p erformance ratio
of O n log n and use it to derive b oth a parallel algorithm in Section and a randomized on
line algorithm in Section with the same p erformance ratio The randomized algorithm is an
p
log n improvement of an algorithm of Vishwanathan and the p erformance ratio of O n
compares favorably with the lower b ound for randomized on line coloring of n log n
The p erformance ratio of the parallel algorithm is the rst of its kind It also compares favor
ably with the b est p erformance ratio known for sequential coloring of O n log log n log n
and the b est announced lower b ound known for that quantity assuming P NP of
n for each Finally using the parallel coloring algorithm we obtain in Section a
O n log log n log n p erformance ratio for parallel indep endent set problem This also com
pares favorably to the sequential upp er b ound of O n log n and the announced lower b ound
of n for each See Table for comparisons
On line Parallel Sequential
Deterministic Randomized
p
log n O n lg lg n log n Known upp er b ound O n log n O n
Here O n log n O n log n
Known lower b ounds n log n n log n n n
Table New and known results on approximate graph coloring
This connection b etween apparently unrelated algorithmic paradigms is quite surprising as
the two paradigms we study are seemingly orthogonal and incompatible Parallel computation
is inherently o line and on line computation inherently sequential Nevertheless there are
certain strong similarities in particular b oth require a lo ose coupling among subproblems
Parallel algorithms b ene t greatly from lo cality which can b e thought of as a case of limited
information similar to the on line regime On line algorithms know more ab out certain other
subproblems but with less control over what they learn
Approximate Coloring Algorithms
In this section we present and analyze the underlying sequential o line coloring algorithm that
forms the basis of the parallel and on line variants
Let us rst review what we have learned from previous metho ds First consider an algorithm
due to Wigderson Given a three colorable graph on n vertices we lo ok for a vertex of degree
p
n The neighborho o d of that vertex must b e two colorable so we color these neighbors at least
with two fresh new colors in linear time and remove them from the graph We rep eat this until
p p
n for at most n iterations The remaining graph can easily b e maximum degree falls b elow
p p
n colors for a combined total of n colors This principle colored using max degree
can b e generalized to graphs of larger chromatic number where the key observation is that the
neighborho o d of any no de in a k colorable graph must b e k colorable This allows us the
break the problem into a collection of somewhat easier subproblems
The problem with this approach is that it lls the color classes in sequence We need a
metho d that accumulates many color classes concurrently rather than searching for individual
indep endent sets We extract an idea from the on line algorithm of Vishwanathan
A simple greedy algorithm sometimes referred to as First Fit works as follows The vertices
are pro cessed in sequence according to some arbitrary order and a vertex is assigned to the
rst color to which none of the preceding neighbors of the vertex have b een assigned If we are
given a limited number of colors it may not b e p ossible to color all the vertices The resulting
coloring of a subset of the vertices is called a maximal partial coloring of the graph
A maximal partial coloring is an assignment of some of the no des into xed number of
greedy color classes so that each remaining no de is adjacent to some no de in each greedy
class The imp ortant p oint is that each color class partitions the remaining no des into subsets
of the neighborho o ds of the no des in the color class If we choose a small class the remaining
uncolored no des induce relatively few subproblems The total number of colors used is then the
number of greedy color classes plus the sum of the number of colors used in the subproblems
Such a maximal partial coloring is easy to achieve sequentially via the First Fit algorithm which
assigns a vertex to the rst compatible color class if one exists
K Color G k
b egin
if k BipartiteColor G
else if k log n Assign each vertex a di erent color
else
k k k
s s n k n k
ResidueNo des MaximalPartialColor G s
Find the smallest greedy color class and let w w b e its no des
p
Partition the ResidueNo des into R R such that no des in R are adjacent to w
p i i
for i to p
Color R k
i
end
Figure An approximation algorithm for k colorable graphs
We state the main algorithm in Figure The algorithm nds a maximal partial coloring
partitions the remaining no des around the smallest color class and recurses on those subprob
lems with the recursion b ottoming out at bipartite graphs When the chromatic number is to o
large for us to handle with resp ect to the size of the graph we settle on the trivial coloring of
one color p er vertex
Theorem The number of colors used by Color on k colorable graphs is at most f k
log k
k k
n where f k O
k
Proof Let A n k denote the maximum number of colors used by the algorithm to color any
k k k
k colorable graph on n vertices Let f k denote k k and let
k k
A n k denote f k n Our prop osition is that A n k A n k for all p ositive
integers n k k
The pro of pro ceeds by induction on k with the base case k established by an optimal
bipartite algorithm Hence assume from here on that A n k A n k for all p ositive
integers n
The number of colors used by our algorithm is the number of greedy colors s s n k
plus the number of colors used in the t subproblems each on m vertices To maintain sanity
j
and simplicity of presentation we ignore all ceilings Note that A n k k s n k
so our main task is to show that the number of colors used in the subproblems b e at most
k k
k s n k k n Intuitively we nd that the worst case o ccurs when the
subproblems are all of equal size and then derive the worst case number of subproblems
If g is the total number of vertices in the greedy color classes then the number of vertices
in the smallest class and thus the number of subproblems is at most g s By the inductive
assumption we can assume that the number of colors used in the subproblems is b ounded by
A Thus
g s
X
A m k A n k s
j
j
Since the approximation function is convex the sum is maximized when the subproblem sizes
are all equal
P
X
m
g
j
j
A n k s maxf m g A k j n g
j
g
s g s
j
Substitution and simpli cation gives
n g k
k k k k
A n k s maxfg g s
k k
g
g
k
n g
k k k k
attains a maximum value of k n k The function f g g
k
g
conveniently simplifying the formula By straightforward algebraic manipulation we nd
that
k k
A n k s n s A n k
Our algorithm assumes that the chromatic number of the graph is given as input To
circumvent that we can run the algorithm for all natural numbers less than n and retain the
b est result A more e cient binary search approach add only a log k O log log n factor to the
time complexity Start with the range n as the algorithm is guaranteed to fail for k and
succeed for k n and run the algorithm for the mid value of the interval If it succeeds run it
on the lower half if it fails run it on the upp er half At the end we have determined a value k
for which the algorithm succeeds while failing for k Then k must b e a lower b ound on
the chromatic number of the graph and our p erformance no worse than if this was given
Corollary The performance ratio of Color is O n log n
Proof The ratio of the size of the approximation to the chromatic number of the graph is
O n which is maximized when log n
Parallel Coloring Algorithm
In this section we discuss how to parallelize the steps of the preceding coloring algorithm while
maintaining an identical p erformance
The main algorithm can b e parallelized easily by executing the lo op iterations in parallel
The bipartite graphs can b e colored exactly by nding a forest of ro oted spanning trees using
the Euler tour technique see computing the depth of each no de using list ranking and
assigning no des of o dd depth one color and no des of even depth the other color
The hard part is nding a maximal partial coloring A construction of Luby reduces it
to the maximal indep endent set problem which has known NC solutions
0
We create a new graph G that consists of copies of the original graph with vertices in
di erent copies adjacent i they corresp ond to the same original vertex Consider a maximal
0
indep endent set I in G and let C b e the set of vertices in I b elonging to the i th copy of G
i
0
in G Note that no two vertices in I corresp ond to the same vertex in G thus I and C map
i
0 0 0
of V Moreover each vertex in V G I is adjacent to some one to one to subsets I and C
i
MaximalPartialColor G x
b egin
0
Construct graph G on n x vertices
j
0
V G fv v V G j ng
i
i
j
0 z
i s j z v v E G g E G f v v
i y
y
i
0
I MIS G
i
C I G i x
i
x
C ResidueNo des G
i
i
return fC g ResidueNo des
i
end
Figure Maximal partial coloring algorithm in parallel
0 0
form as otherwise its addition to I would still b e indep endent Hence the C vertex in each C
i i
a maximal partial coloring of G
The complexity of the maximal partial coloring algorithm is dominated by the complexity
of the maximal indep endent set algorithm used The faster algorithm known for the latter
problem runs in O log jV j time due to Luby A pro cessor e cient algorithm of Goldb erg
and Sp encer runs in O log jV j time with jV j jE j log jV j pro cessors The number of
0 k k
vertices of the comp osed graph G is s n where s s n k n k is the
s
number of greedy bins allo cated The number of edges is then sjE j n s jE j ns and
thus the pro cessor complexity of our algorithm would b e s jE j ns log n n
The complexity of the main algorithm is also dominated by k recursion levels of calls to the
maximal partial coloring subroutine adding a factor of k O log n to the time complexity
On line Coloring Algorithm
We show in this section how to convert Color into an on line algorithm
The on line coloring problem is de ned as follows An adversary selects a graph and an
ordering of the vertices It feeds us one no de at a time along with its edges to the no des already
received We are to irrevocably assign that no de a color consistent with the colors of the previous
no des We are allowed arbitrary amount of time but we are not given any further information
ab out the graph including its size and chromatic number
In order to deter the adversary from foreseeing our every move randomization frequently
proves useful An adversary is said to b e oblivious if it picks the graph and the ordering b efore
we make any choices i e without knowledge of the values of the random bits rather than
constructing the graph adaptively according to our choices We measure the exp ected number
of colors used by a randomized algorithm on a given graph where the exp ectation ranges over the
values of the random bits The p erformance ratio is then de ned as the ratio of this exp ectation
to the chromatic number maximized over all graphs
For some on line problems randomized algorithm against adaptive adversary form an inter
mediate mo del b etween the deterministic and the randomized oblivious cases For the graph
coloring problem however no results are known that are sp eci c to this mo del
In order to convert the o line algorithm into an on line one several changes need to b e
made and hurdles to b e overcome The greatest change is that we must pro cess the no des fully
in sequence rather than pro cessing the chromatic levels in sequence We must also overcome
three di culties the size of partitioning classes will not b e known b efore they must b e chosen
the size of the input will not b e known and the chromatic number will b e unknown
The rst hurdle is that a partitioning class must b e selected in advance b efore we know its
eventual size The concern is that if the size will b e large many subproblems will b e spanned
The solution is to use randomization pick a uniformly random class as a partitioning class
Since the adversary must choose the ordering of the graph in advance the exp ected number of
no des in the class will b e su ciently small
The second problem is estimating the size of the input and recursively the sizes of the
subproblems For this we apply a well known trick in the on line trade see e g the
doubling strategy which works as follows Assume to b egin with that the size of the input n
i
equals for i initially equal When the actual number of no des exceeds this estimate we
increment the estimate to the next p ower of retire the old color classes and start afresh with a
new set of color classes We continue this way and when the input nally ends our estimate of n
is never o by more than a factor of Since the function giving our approximation p erformance
is necessarily convex it follows that the overhead caused by this strategy is no more than a small
constant
It is imp ortant to notice that we can apply the same strategy recursively for the subproblems
as well From the p ersp ective of an on line algorithm the subgraph induced by the no des in
a residue class i e in a subproblem is structurally no di erent from the original problem
except for the fact that the chromatic number is one less We shall therefore allo cate sets of
color classes to this subproblem completely indep endent of the other subproblems or the parent
problem This strategy is what primarily distinguishes our algorithm from Vishwanathan s
The third problem not knowing the chromatic number in advance can b e handled simi
larly To b egin with we assume the graph is colorable and as our assumptions are shattered
usually by the bipartite algorithm stumbling up on a non bipartite subgraph we increment our
estimate by one and start from scratch with a fresh set of colors
Finally we need to nd on line versions of the subroutines called by algorithm K Color
As stated in there is a simple algorithm for on line coloring bipartite graph using no more
than log n colors And a First Fit algorithm that is restricted to using only the greedy color
classes actually available is p erfect for p erforming a maximal partial coloring
The resulting algorithm is shown in Figures
On line Color G
On line color a graph G
b egin
k
R oot New Coloring Tree k
foreach v V G in sequence do
R esul t K Color Online R oot v
if R esul t F ail ur e then
k k
R oot New Coloring Tree k
end
Figure On line Graph Coloring Algorithm
The on line algorithm colors the graph by constructing and maintaining a coloring tree This
tree contains each previous vertex in a greedy color class at some no de in the tree Each no de
in the tree contains the following static information n the capacity of the subtree k the
colorability of the subtree s s n k the number of greedy bins and par t the index of the
1
Static in the sense that the values are given at the creation of the subtree ro oted by the no de
K Color Online T v
Re entrant procedure
Assign a color to the vertex v using coloring tree T
Return success or failure
b egin
with T do
if k then
return BipartiteColor T v
if N n then
T New Coloring Tree n k
N N
for j to s do
if v is non adjacent to all vertices in P j then
Assign v the color P j or C O LO R v P j
return Success
v must now b e adjacent to some no de in each color class
Non deterministically cho ose a neighb or w of v in partitioning class P par t
if R w is uninitialized then
R w New Coloring Tree k
return K Color Online R w v
end
Figure A randomized on line coloring algorithm
New Coloring Tree n k
Create and return a new coloring tree with capacity n and chromatic upp er b ound k
b egin
with T do
n n
k k
c k k
s s n k n k
P s A new set of colors
par t Random integer b etween and s
N
return T
end
Figure Create and initialize a new coloring tree
partitioning bin Each no de also contains the following variables N the current number of
vertices in the subtree P s the greedy bins the bins themselves are static their contents
are dynamic and R a residue tree for each vertex assigned to the partitioning bin P par t
w
Finally each no de implicitly contains a host of cut o subtrees
Here c is a constant whose optimal value is determined in the analysis
When a vertex v arrives the algorithm will assign it to one of the static greedy bins at
the ro ot whenever p ossible If this fails we choose any neighbor w of v in the partitioning bin
P par t The vertex is the sent recursively to the subproblem corresp onding to the subtree R w
which is constructed if necessary This continues until the colorability index of the tree is at
most two in which case we know how to on line color the no de e ciently
Two invariants must b e maintained in each subtree of the coloring tree
N n That is the number of vertices in the subtree at any time cannot exceed the
capacity of the subtree
The colorability index for the subtree ro oted at no de v must b e at least one greater than
the colorability index for the subtree ro oted at some child of v This insures that the
colorability index stays an upp er b ound on the chromatic number of the graph
The former invariant is maintained by K Color Online by creating a new subtree with double the
capacity whenever the vertex count exceeds capacity The old greedy bins color classes are
then set aside and never used again and the vertices they contain do not factor into the coloring
assignment other than that their number is still a part of the content of the subtree
The latter invariant is maintained by On line Color again by cutting o the old tree and
creating a new tree with a colorability index one larger Such an action is triggered by the
failure of the on line bipartite coloring algorithm at some leaf when some leaf b ecomes three
chromatic and the e ect lters up the path to the ro ot
Analysis
The analysis of the p erformance of the on line algorithm resembles that of the parallel approxi
mation but is more involved The heart of the analysis is contained in the following result
k k k
Theorem K Color Online uses at most O n log n expected number of col
ors on k colorable graphs with n vertices
Proof Let us rst de ne some notation Let A n k denote the maximum exp ected number
of colors the algorithm uses to color any k colorable graph on n vertices when n is known in
advance Let B n k similarly b e the maximum exp ected number of colors when n is not known
in advance Let c denote a real valued constant to b e de ned later We shall need the following
functions and constants
c
k k
c
k
k
k k k
s n k c n log n
k
A n k k s n k
c k
B n k A n k
B n k is the upp er b ound that we shall prove for B n k and similarly A n k is the b ound
for A n k s n k denotes the number of greedy bins allo cated as previously de ned Default
logarithm is base
Given the ab ove notation the prop osition of the theorem can b e restated as
B n k B n k for all n k k
The pro of pro ceeds by induction on k simultaneously for A and B For the base case observe
that B n A n log n A n B n To prove the inductive step we rst show
that A n k A n k for any p ositive integer n
The number A n k of bins used during the time that the size estimate n remains unchanged
equals the number of greedy bins plus at most the average cost of the subproblems over the s
di erent choices of a partitioning bin The latter amounts to at most
n
s
i
X X
i
B m k
j
s
j i
i
where n is the number of elements in greedy bin i and m the size of the subgraph Let g
i
j
denote the total number of elements placed in greedy bins By the inductive assumption and
the convexity of B is at most
n g
maxfg B s k g
g
s g
We now plug in the value of B obtaining
n g
k k c k k
max fg g c k s log n
k
g
g
As in Theorem the maximum of the inner function simpli es the formula leading to
A n k s k s n k A n k
Hence A n k A n k for any n We now show that this implies that the same holds for
B and B from which the theorem follows
In general we do not know the number of no des in advance at any level in the recursion
The algorithm xes a value for n and when that count has b een exceeded it doubles its exp ec
tation for the total number of vertices We obtain d dlog ne subproblems coloring the no des
d d d
f g f g f g f ng resp ectively Let r n denote the total number
d
of no des colored by the end of the last doubling step i e r n
The total number of bins used is at most the sum of those used on each of the size b ounded
subproblems the residue bins used on the last r n no des and the greedy bins allo cated in
the last step That is
d
X
r n
A n k k A r n k A B n k
i
k
i
The rst term evaluates to
d
X
i k k k k
A r n k A r n k
i
Maximizing the rst two terms over the choices for r using standard calculus yields
k k k k
A n k B n k
k
Insp ection shows that this is at most
k
B n k A n k
Thus we set c and the theorem follows
Theorem Our on line algorithm uses at most O n log n colors without
a priori know ledge of either n or the chromatic number G
Proof We maintain a lower b ounded estimate k on the chromatic number of the graph This
estimate is incremented only when the current graph has b een proven to b e non k colorable
Consider the no des colored while the estimate was i If we denote their count by N we know
i
that On line Color used no more than B N i colors to color those no des This can b e b ounded
i
from ab ove by B N k which is a convex function Thus the sum of these values i k
i
is maximized when all N are equal for a total of
i
n
k k k k
k k B n k n log n k B
k
number of colors
We obtain the same p erformance ratio as in the o line case with a similar pro of
Corollary Randomized on line coloring is O n log n approximable
Indep endent Sets in Parallel
The parallel coloring algorithm can also b e used to nd cliques or indep endent sets in parallel
k k
via an observation due to Wigderson Recall that the algorithm used at most n
colors where the graph was provably not k colorable since a k chromatic subgraph had b een
found This subgraph is formed by the pivot no des w at each of the k recursive levels
i
along with a non bipartite subgraph i e an o dd cycle at the end of the recursion These pivot
no des are by de nition mutually adjacent as well as adjacent to the o dd cycle thus pro ducing
a k clique Since the clique number of the graph is at most the chromatic number which is at
k k
most the number of colors we used this yields a n k approximation of the maximum
clique for an O n log n p erformance ratio Finally recall that we can obtain an indep endent
set algorithm with the same p erformance by applying our algorithm to the complement graph
Notice that the algorithm runs in p olylogarithmic time for all interesting values of k i e
k log n
Another source of a parallel indep endent set algorithm can b e found in the rst step of
Berger and Romp el s coloring algorithm It essentially involves choosing p olynomial number
of random vertex subsets of size log n and testing for non adjacency This can b e made deter
k
ministic via the pigeonhole principle without destroying the indep endence of the samples and
hence can b e parallelized trivially This metho d yields a n log n approximation
k
Notice that the two metho ds complement each other since our Wigderson s metho d p erforms
b est for small values of k while the ab ove random sampling p erforms relatively b etter for larger
values of k If we run b oth metho ds and retain the b etter result we get an algorithm with a
p erformance ratio of
k k
n n
max min
k
k log n
k
which is minimized when k log n log log n as O n log log n log n
Graphs with large indep endent sets
We shall show how given a graph with an indep endence number at least n we can nd
p
n in p olylogarithmic time an indep endent set of size
The metho d starts by removing a maximal collection of disjoint triangles cliques of size
from the graph Such a collection can b e obtained as a maximal indep endent set in the
graph whose vertices are all the triangles in the original graph with edges b etween intersecting
triangles Since at most one no de in each triangle can b e a member of a given indep endent set
at least n n no des remain after the triangle removal
TriangleFreeIS G
b egin
I MaximalIndep endentSet G w w w
t
N V G
for i to t do
N N fv V G v w E G g
i i i
N fv V G v w E G g N
i i i
return max N fw g N fw w w g N fw w w g
t t t
end
Figure Algorithm for nding indep endent sets in triangle free graphs
Given a triangle free graph we apply the algorithm of Figure which is based on a sequential
algorithm of The algorithm nds a maximal indep endent set with an ordering of the vertices
N of any pre x of the vertices as well as in the set and nds the common non neighborho o ds
i
the complementary neighborho o ds N Each neighborho o d must b e indep endent since the graph
i
contains no triangles and is also contained in an earlier non neighborho o d The nal result is
the largest of these indep endent sets considered and its size must b e at least the smallest t such
p
t
that n or approximately n
N is a pre x computation that can b e p erformed Finding the recursive non neighborho o ds
i
in logarithmic time with linear number of pro cessors The complexity is therefore asymptotically
equivalent to that of the MIS algorithm used
k
The ab ove construction can b e generalized to nd indep endent sets of size n in
k clique free graphs providing a constructive parallel pro of of the upp er b ound on Ramsey
numbers by Erd s and Szekeres That also allows us to nd such indep endent sets in graphs
with indep endence number greater than n k for k but alas the pro cessor complexity of
k
removing the k cliques grows as fast as n
Acknowledgements
I would like to thank Ravi Boppana and Sundar Vishwanathan for helpful discussions and the
two anonymous referees for comments that improved the presentation I also thank Sundar for
re inventing Luby s construction for me
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