Static Frequency Assignment in Cellular
Networks
y
Lata Narayanan Sunil M Shende
August
Abstract
A cellular network is generally mo deled as a subgraph of the triangular lat
tice In the static frequency assignment problem each vertex of the graph is a
base station in the network and has asso ciated with it an integer weight that
represents the numb er of calls that must b e served at the vertex by assigning
distinct frequencies p er call The edges of the graph mo del interference con
straints for frequencies assigned to neighb oring stations The static frequency
assignment problem can b e abstracted as a graph multicoloring problem We
describ e an e cient algorithm to optimally multicolor any weighted even or o dd
length cycle representing a cellular network This result is further extended to
any outerplanar graph For the problem of multicoloring an arbitrary connected
subgraph of the triangular lattice we demonstrate an approximation algorithm
which guarantees that no more than = times the minimum numb er of required
colors are used Further we show that this algorithm can b e implemented in
a distributed manner where each station needs to have knowledge only of the
weights at a small neighb orho o d
Keywords Frequency assignment cellular networks approximation algo
rithms graph multicoloring distributed algorithms
Department of Computer Science Concordia University Montreal Queb ec Canada H G M
email lata cs concordia ca FAX Research supp orted by NSERC Canada
y
Department of Computer Science Rutgers University Camden NJ USA email
shende crab rutgers edu FAX This research was conducted at Concordia Uni
versity during the summer of with partial supp ort from NSERC Canada
Intro duction
Cellular data and communication networks can b e mo deled as graphs with each ver
tex representing a base station sometimes called a cel l in the network Cells can
communicate with their neighb ors in the graph via directional radio transceivers At
any given time a certain numb er of active connections or cal ls in cellular network
terminology are serviced by their nearest base station This service consists mainly
of assigning a frequency to each client call in a manner that minimizes or avoids radio
interference b etween two distinct calls in the network However cellular networks use
a xed sp ectrum of radio frequencies and the e cient shared utilization of the lim
ited available bandwidth is critical to the viability and e ciency of the network The
static frequency assignment problem therefore consists of designing an interference
free frequency allo cation proto col for a network where the numb er of calls p er cell is
known a priori This forms the motivation for the problems studied in this pap er
In particular cellular networks are usually mo deled as nite p ortions of the in
nite triangular grid emb edded in the plane Vertices representing cells are placed
at the ap exes of similar triangles and each vertex has at most six other neighb ors
surrounding it in the grid The reason for adopting this particular geometry stems
from the fact that cells are uniformly distributed in the geographic area of the net
work and an individual cell generally has six directional transceivers Hence the
Voronoi region around a cell or equivalently that cell s calling area can b e idealized
as a regular hexagon The triangular tiling representing the network is simply the
planar dual of the resulting Voronoi diagram We shall refer to the resulting graphs
1
as hexagon graphs
The frequency assignment problem incorp orating interference constraints can b e
abstracted as follows Let G V E denote an hexagon graph Each vertex v V
has an asso ciated integer weight w v A w coloring or multicoloring of G
is an assignment of sets of colors to the vertices such that each vertex v is assigned
w v distinct colors whereby for every edge u v E the set of colors assigned to
the endp oints u and v are disjoint In particular we are interested in a minimum
multicoloring or a w coloring of G that uses the least numb er of colors
1
We note here that this is the most commonly used mo del for frequency assignment problems in
the cellular network literature However in practice cellular systems tend to b e more complicated
and there have b een recent studies that attempt to mo del more general interference patterns and cost
functions see for example Bornd orfer et al Nevertheless the hexagon graph mo del continues
to b e of signi cance from the historical standp oint and as an abstraction that is su ciently close to
reality to provide useful insights
In the context of frequency assignment a multicoloring as de ned ab ove provides
a useful abstraction of the essential interference constraints each color represents a
distinct frequency and it is assumed that two calls may use the same frequency if and
only if they originate in distinct cells that are not neighb ors It should b e noted that
in practice the available cellular frequency sp ectrum is a contiguous linear subinterval
of the radio sp ectrum and frequency reuse is controlled by a sequence of non negative
integers c c with c called distance reuse constraints Two distinct calls
0 1 0
in cells that are a distance i apart in the underlying graph must b e assigned frequencies
that di er by c in the frequency sp ectrum Hale discusses many generalizations
i
and versions of the frequency assignment problem We formulate our problem under
the simplest constraints viz when c c and c i Under this
0 1 i
formulation the problem reduces to b eing able to compute a minimum multicoloring
to a given hexagon graph
In the sequel we assume that G V E w denotes a hexagon graph i e it is
a vertex weighted graph that is a nite induced subgraph of the in nite triangular
grid Thus the graph is planar and every vertex v V has degree at most six
and an asso ciated integer weight w v The weighted chromatic numb er of G
denoted G is the minimum numb er of colors required in a w coloring of G Even
for graphs with a regular structure such as those considered in the pap er the problem
of determining G is non trivial In fact it has b een established only recently that
the corresp onding decision problem is NP complete and hence it is unlikely that
a p olynomial time algorithm for computing G can b e devised Naturally it is of
interest to study approximation algorithms for the problem
It is easy to see that G must b e greater than the total numb er of colors required
at any set of mutually adjacent vertices Thus the maximum over the sum of weights
at vertices in any maximal clique in the graph is a trivial lower b ound on G Note
that for hexagon graphs edges and triangles are maximal cliques In the direction of
upp er b ounds while there is a vast literature on algorithms for frequency assignment
on graphs esp ecially hexagon graphs that claim to use few colors generally there
are no proven bounds on the p erformance of the prop osed algorithms in terms of the
numb er of colors used in relation to the weighted chromatic numb er We
note here two exceptions A well known algorithm sometimes referred to as Fixed
Al location uses the fact that the underlying graph can b e colored The algorithm
uses three xed sets of colors one for each base color A vertex that has base color
uses colors from the rst set and a vertex that base color or uses colors from the
second or third sets resp ectively It is easy to show that this algorithm could use as
many as times the numb er of required colors Janssen et al prop ose a di erent
algorithm called Fixed Preference Al location that is guaranteed to use no more than
times the minimum numb er of colors required
In the next section we formally de ne some basic terminology and problems In
Section we present optimal algorithms for multicoloring cycles and outerplanar
graphs In Section we address the question of multicoloring an arbitrary hexagon
graph Our main result is an e cient approximation algorithm with a p erformance
guarantee of within of the optimal Finally in Section we show how to imple
2
ment the ab ove algorithm in a distributed manner We conclude with a discussion
of future directions in Section
Preliminaries
Let G V E w b e a hexagon graph with a non negative integer weight vector w
de ned on the vertices of the graph where w v represents the numb er of calls to
b e served at vertex v We assume hereafter that G has a xed planar emb edding
with vertices and edges contained in the in nite triangular lattice tessellation of the
plane Thus any vertex v can b e connected to at most neighb ors and for a xed
edge incident on v any other edge incident on v is at an angle of or
from that edge Since the triangular lattice is colorable in the ordinary sense
i e when each vertex has unit weight the underlying graph corresp onding to unit
weights at vertices of G is also colorable
A w coloring or multicoloring of the graph G V E w consists of a set of colors
C the color palette and a function f that assigns to each v V a subset f v of
the palette C such that
v jf v j w v each vertex gets w v distinct colors and
u v E f u f v two neighb oring vertices get disjoint sets of colors
The span of a multicoloring is the cardinality of the set C The weighted chro
matic number of G denoted G is the smallest numb er m such that there exists
a multicoloring of G of span m Thus given a hexagon graph G our ob jective is to
nd a multicoloring for G whose span is as close to G as p ossible
2
We note that in addition to showing the NP hardness of this problem McDiarmid and Reed
have also describ ed a di erent = approximate algorithm for the problem Our result was indep en
dently derived and unlike the McDiarmid Reed algorithm has the advantage of b eing implementable
in a distributed setting
The only maximal cliques in G b eing edges and triangles we de ne the weight of
an edge triangle in G to b e the sum of the weights of its endp oints ap exes Note
[2]
that the weight of any maximal clique of G is a lower b ound on G Let D and
G
[3]
D denote the resp ective maxima over the weights of edges and triangles in G and
G
[2] [3]
de ne D maxfD D g Then if there exists a multicoloring of G with span
G
G G
it follows that
G D
G
We will assume without loss of generality that any palette of available colors can
b e suitably ordered or partitioned in particular we will often assume that vertices
are assigned colors from the circularly ordered interval M f M g where
M is a p ositive integer that dep ends on the particular graph under consideration
For instance when a vertex is assigned the subinterval of colors i j from the palette
it means that the vertex is colored with the set fi i j g in a cyclic manner
where color is assumed to follow the color M
Optimal Multicoloring of Cycles
Consider a hexagon graph G V E w with n vertices in the form of a simple
cycle lab eled u u u in clo ckwise order For simplicity let w i n
1 2 n i
denote the weight w u of vertex u We show that any such hexagon graph can b e
i i
optimal ly colored with exactly G colors There are two cases to consider dep ending
on whether n the numb er of vertices on the cycle is even or o dd
Supp ose that n m i e the graph consists of an even length cycle Then all
[2]
maximal cliques of G b eing edges D D is the maximum weight of an edge in the
G
G
cycle We observe that a very simple greedy strategy su ces to multicolor G with the
color palette D The idea is to assign for i m the colors w to the
G 2i 1
o dd numb ered vertex u and the colors D w D to the even numb ered
2i 1 G 2i G
vertex u Noting that for i m
2i
D w w and
G 2i 1 2i
D w w
G 2i 2i+1
with subscripts interpreted cyclically it follows that the given multicoloring is prop er
By construction G D and the simple parity based algorithm thus provides an
G
optimal multicoloring of G
We note that a very similar idea was already used in the cellular network literature
but it was only applied to networks consisting of simple paths it is easy to
see that this strategy works in general for any bipartite graph Unfortunately the
parity argument fails to multicolor o dd length cycles precisely b ecause the underlying
unweighted o dd length cycle needs at least three colors in any ordinary coloring For
instance if G is a cycle with weight on each vertex it is easy to see that G cannot
b e multicolored with D colors but needs colors instead
G
De nition Let G V E w be an odd length simple cycle with vertices labeled
u u u m in clockwise order We de ne
1 2 2m+1
P
2m+1
w
0
i
[2]
i=1
eg D maxfD d
G
G
m
Theorem Let G V E w be a cycle of odd length n m Then
0 0
G D and G can be optimal ly multicolored with exactly D colors Further the
G G
multicoloring can be obtained in time O n
2m+1
w
[2]
i
i=1
e Pro of It is clear that D is a lower b ound on G we establish that d
G
m
0
and hence D is a lower b ound on G Since the size of an indep endent set in G
G
is at most m any single color can b e used only at m vertices or fewer in the cycle
2m+1
The total numb er of colors needed at all vertices b eing w we conclude that
i
i=1
2m+1
0
w
i
i=1
G d e Thus D is indeed a lower b ound on G
G
m
0
Next we show that G can b e colored with D colors using a linear time algorithm
G
this completes the pro of of the theorem We rst observe that there must b e a smal lest
index k k m which satis es the inequality
0
2k +1
w k D
i
i=1 G
Note that this prop erty holds true for the index m from De nition and hence k
is well de ned and can b e found easily in linear time
The vertices of the cycle are now colored as follows
Vertices u through u are assigned contiguous colors in a cyclic manner from
1 2k
0
the palette D Sp eci cally for j k vertex u is assigned the colors
j
G
j j 1
X X
w w
i i
i=1 i=1
cyclically By construction this ensures that the path u u u is prop erly
1 2 2k
0
[2]
multicolored since D D
G G
Vertices u through u are colored based on their parity as in the even
2k +1 2m+1
cycle algorithm In particular for k i m the vertex u is assigned
i
0 0
the colors w if i is even or the colors D w D if i is o dd Again
i i
G G
this ensures that the path u u u is prop erly multicolored
2k +1 2k +2 2m+1
0 0
Since vertex u has the colors w and vertex u the colors D w D
1 1 2m+1 2m+1
G G
the edge u u is also prop erly multicolored All that remains is to verify that
1 2m+1
the edge u u is prop erly multicolored this is a consequence of the minimality
2k 2k +1
0
2k 1
of k for we know that w k D Hence no color assigned to u can b e
i 2k
i=1
G
0 0
among the colors D w D assigned to vertex u
2k +1 2k +1
G G
We illustrate the lab eling scheme using a cycle with weight on each vertex as
0
5
an example As D maxf g we use the palette Since w
i
G i=1
0 0
3
D but w D we color the rst four vertices in a cyclic manner
i
G i=1 G
always taking the next four available colors in the palette For the last ve vertices we
assign colors as in a bipartite graph from the two ends of the interval Finally
we note that our algorithm can actually multicolor any cycle and not just cycles that
are hexagon graphs i e emb edded in the triangular lattice
Theorem can b e used to derive an optimal multicoloring of any outerplanar
graph A graph is said to b e outerplanar if it can b e emb edded in the plane so that
every vertex of G lies on the b oundary of the exterior face It is straightforward to
see that the weighted chromatic numb er of any graph is the maximum taken over the
weighted chromatic numb ers of its biconnected comp onents Thus it su ces to con
sider a biconnected outerplanar graph any such graph is a cycle with non intersecting
chords A biconnected outerplanar graph G without chords is a simple cycle and can
b e multicolored optimally using the construction in the pro of of Theorem Oth
erwise let u v b e a chord and let G and G b e the two parts of G on the sides of
1 2
this chord each one including the edge u v Recursively color G and G Relab el
1 2
the color assignment of G so that the colors assigned to u and v in G agree with
2 2
those assigned in G The following corollary is immediate
1
Corollary Let G V E w be an arbitrary outerplanar graph Then G can be
colored optimal ly using G colors
A careful implementation of the algorithm sketched ab ove results in a linear time
multicoloring We remark that Corollary applies to any outerplanar graph and
not just outerplanar hexagon graphs
Approximate multicoloring of hexagon graphs
In this section we consider the problem of computing an approximate multicoloring
of an arbitrary hexagon graph Since a hexagon graph may contain an o dd cycle as an
[3]
induced subgraph it follows as a consequence of Theorem that D the maximum
G
weight taken over all triangles is not always a tight b ound For example consider
a cycle where every vertex is given the weight k for some integer k While
[3] [2]
D D k for the graph we know from Theorem that G d k e We
G G
cho ose a cycle b ecause it is the smallest o dd cycle that can b e an induced subgraph
of the triangular lattice This shows that any algorithm to color hexagon graphs must
[3] [3]
use at least d D e colors on some graphs G with triangle b ound D In fact we
G G
demonstrate an e cient approximation algorithm that can multicolor any hexagon
[3]
graph using at most dD e colors and hence at most d G e colors
G
Without loss of generality we assume that G is connected since disconnected com
p onents of G can b e indep endently colored without any color con icts For simplicity
[3]
we let M dD e and we cho ose the following color palette in our algorithm Start
G
with a base coloring of G so that every vertex gets base color red blue or green With
each base color we asso ciate a class of M hues identi ed with the interval M
In addition we have at our disp osal a class of auxiliary purple hues again identi ed
with the interval M The entire collection of M distinct hues forms our color
palette
The idea is to let each vertex v use as many hues from its base color class as
p ossible b efore trying to use hues either from the remaining two base classes or from
the auxiliary purple class We describ e the algorithm as pro ceeding in ve phases
we maintain the invariant that at the end of each phase the graph is partially but
correctly colored To facilitate reasoning ab out the correctness of the algorithm we
let G V E w denote the remaining graph after phase i i has b een
i i i i
completed We also assign an arti cial priority to vertices red vertices dominate over
blue ones which in turn dominate over green ones This priority scheme is used in
phases and to select in each case a suitable subset of vertices for partial coloring
We illustrate our algorithm with a running example shown in Figure a graph G
[3]
for which M D can easily b e veri ed to b e Hence the color palette consists
G
of colors equally divided among the red blue green and purple hues
A vertex v G is de ned to b e light if w v M and to b e heavy otherwise This
distinction is critical to each of the the ve phases b elow
Phase Every vertex v is assigned the rst w v hues from its base color class in
particular the hues minfw v M g All the light vertices thus get completely 7 7
7 1 7 2 10 Red
1 10 3 8 Blue
7 1 7 Green
3 8 8 3 14
72 7
Figure A hexagon graph with initial weights
colored and are deleted from the graph The weight of every remaining heavy
vertex v is decreased by M resulting in the graph G
1
It is easy to see that G has no maximal cliques of size greater than b ecause
1
every triangle in G must contain at least one light vertex that is eliminated in the
rst phase Let H denote the subgraph of G induced by the degree vertices in G
1 1
Note that if a vertex v G has three neighb ors say in clo ckwise order in the xed
1
emb edding in G then the incident edges to the neighb ors form successive angles of
1
radians in order furthermore the geometry implies that all three neighb ors have
the same base color It follows that each connected comp onent in H contains vertices
that b elong to at most two base color classes Thus every connected comp onent of H
consists of either an isolated vertex or contains only red and blue or red and green
or blue and green vertices
Call a vertex v H a priority vertex if and only if it has the highest priority
among its neighb ors if any in H recall that red dominates blue which dominates
green Clearly the priority vertices form an indep endent set in fact a dominating
set in H
Phase Without loss of generality let v b e a red priority vertex in H with three
blue neighb ors in H Let g v b e the maximum among the weights of the
three green neighb ors of v these vertices must have b een light vertices in G
Then v can b orrow from among the last M g v green hues these su ce to
color the remaining weight on v since all three blue neighb ors of v are heavy
vertices and hence w v M g v Accordingly v is assigned the green
1 r1-6 b1-6 1 1 1 1 c d b1-6 g1 r1-6 b1-2 g1-6 1 1 4 b g1 1 r1-6 b1-3 1 g1-6 4 a e b1-6 4 g1 r1-6 2 1 4 1 2 x g1-3 1 r1-6 b1-6 1 g1-3 r1-6 g5-6 u v t 12 2 1 8 2 8 y b1-6 g1-2 r1-6 1 1
(a) (b)
Figure Color assignment during a Phase and b Phase
hues M g v M and eliminated from further consideration Note that
the partial color assignment at the end of phase has no color con icts among
neighb ors and the remaining graph is designated G
2
Figure details the partial color assignment at the end of phase and phase
resp ectively Note that the six red hues are denoted as r and so forth in the
gure Since the subset of priority vertices eliminated in phase is a dominating set
of H the degree vertices of G every remaining vertex in G now has degree at
1 2
most Equivalently the connected comp onents of G consist of isolated vertices
2
cycles and paths in the triangular grid Note also that any edge of G has a residual
2
weight of at most M a consequence of the de nition of heavy vertices If the graph
G contains only even cycles or paths then we can color all the vertices using the M
2
purple colors However G may contain isolated vertices and o dd cycles In the next
2
phase we essentially eliminate al l p otential cycles in G
2
Call a vertex v G a corner vertex if and only if it has two neighb ors x y of the
2
same base color class in G such that the angle subtended at v by the incident edges
2
v x and v y is exactly radians Further a corner vertex v is a priority vertex
in G if and only if v has the highest priority among all its neighb ors if any that are
2
also corner vertices It is not di cult to see that the subset of priority vertices in G
2
forms an indep endent set in G Also every corner vertex is either itself a priority
2
vertex or is adjacent to a priority vertex hence the subset of priority vertices is
a dominating set of the subgraph induced by the corner vertices in G Finally by
2
de nition every cycle in G contains at least one corner vertex Thus coloring priority
2
vertices and eliminating them also breaks all cycles For example in Figure b a x p Red t u r Blue v Green
b y q
Figure Lo cal geometry around a blue priority vertex v in Phase
0 0
the vertices lab eled v v and x are corner vertices of G among them v and v are
2
priority vertices
Phase Without loss of generality let v b e a blue priority vertex in G with red
2
neighb ors x and y in G as shown in Figure Note that u denotes the third
2
neighb or of v of the same base color class as x and y It is also easy to see from
priority considerations that x and y must b e non corner vertices and hence the
blue vertices p and q must b e light vertices While u is absent in G there are
2
two p ossibilities
i u was eliminated in phase i e w u let g v b e the maximum
1
over the weights of v s green neighb ors i e vertices a b and t in Figure
in G Since u did not participate in phase v can b orrow from among the
last M g v green hues these su ce since two of v s red neighb ors are heavy
vertices and hence w v M g v Accordingly v is assigned the green hues
2
M w v M and eliminated
2
ii u was a priority vertex in phase consider a and b the common light
green neighb ors of u and v in G Let w b e the maximum among w a and w b
ab
and recall that u was assigned the last w u green hues M w u M
1 1
during Phase Since w v w u w M it app ears that v could b orrow
2 1 ab
the green hues w w w v from the midd le of the green sp ectrum
ab ab 2
However we must ensure that this assignment will not con ict with the green
hues assigned to t see Figure
If w t w then clearly the new assignment do es not con ict with prior
ab
color assignments However if w t w then we can recolor t as follows its
ab
original assignment of the rst w t green hues in phase is changed so that t
now uses the rst w and the last w t w green hues The new assignment
ab ab g2 1 1 1 p1 1 p1-4
1 g2-5 b4-5 4
1 1 p1 p1
g3-4 g6-2
1 1 8 p1 p1 p1-6,b5-6
(a) (b)
Figure Color assignment during a Phase and b Phase
to t cannot con ict with the green hues w w w v b orrowed by v
ab ab 2
since w t w v M
2
It remains to verify that recoloring t do es not cause a cascading con ict with any
other neighb or of t Clearly such a con ict could o ccur only if some other neighb or of
t also b orrows green hues in Phases or From the observations ab ove we conclude
that this would b e imp ossible for vertices x y they are not priority vertices in either
phase and p q they are light in G The only remaining p ossibility viz vertex r is
not a problem either r could not have b een a priority vertex in phase since it has
three consecutive light neighb ors p t and q and if it were a corner vertex in phase
it would have green neighb ors in G and would b orrow blue hues Figure a
2
depicts the color assignment during phase for the running example note that the
light vertex lab eled t in Figure b is recolored as describ ed ab ove
Thus at the end of phase we have a correct partial assignment of colors and
furthermore the remaining graph G consists only of isolated vertices or straight line
3
paths It is easy to see that any remaining isolated vertex may have a residual weight
b etween and M whereas the weight on every remaining edge is at most M
Phase To any isolated vertex v G we rst assign minfw v M g purple hues
3 3
If w v M then we still need to nd w v M M additional colors to
3 3
nish coloring v Without loss of generality we assume that v is a red vertex
and observe that all the neighb ors of v must have b een light vertices in G
Hence the blue green neighb ors of v must have had colors either assigned
to them in Phase or in Phase as a result of recoloring We claim that
v can still b orrow colors from either the blue or the green palettes without
con icting with any neighb oring assignment
From the description of phases and observe that the light neighb ors of v may
b e using colors from either end of their base color palettes Let b b b M
3 2 1
and g g g M b e the weights of v s blue and green neighb ors in G
3 2 1
resp ectively Clearly regardless of the manner in which the blue neighb ors are
assigned blue hues v has at least M b b hues available for its use from
1 2
the blue palette Likewise there are at least M g g green hues available
1 2
to g Since the green vertex with weight g forms a triangle in G with v and
1
either one of the blue vertices with weight b or with weight b In any event
1 2
M b g A similar argument shows that M g b It follows
2 1 2 1
that
maxfM b b M g g g
1 2 1 2
or in other words that v can obtain the remaining colors by b orrowing either
only blue hues or only green hues without any color con icts with assignments
prior to the phase
Figure b demonstrates the colors assigned in Phase to the remaining isolated
vertices in our running example At the end of phase the remaining graph consists
only of straight line paths i e paths in which any two consecutive edges subtend an
angle of degrees at the common vertex Further as noted ab ove every remaining
edge has a residual weight of at most M
Phase Since every remaining connected comp onent is a straight line path with a
weighted chromatic numb er of at most M it su ces to use the greedy parity
based strategy describ ed in Section to nish coloring the graph using the M
purple hues This cannot cause any con ict with previously colored vertices
since the purple hues were used only in Phase to color isolated vertices in P
the latter are disconnected from any remaining vertex in the current phase
Figure shows the entire color assignment constructed by our algorithm for the
running example The following result is immediate
[3]
Theorem An approximate multicoloring that uses no more than dD e colors
G
for any hexagon graph G V E w can be e ciently computed in linear time r1-6,g2 b1-6,p1
b1-6,p1 g1 r1-6,p6 b1-2 g1-6,p1-4
g1 r1-6,g2-5 b1-3 g1-6,b4-5
b1-6,p1 g1 r1-6,p1 g1-3 r1-6,g5-6 g6-2
b1-6,g3-4 r1-6,p1-6,b5-6
b1-6,p1 g1-2 r1-6,p1
Figure Complete color assignment in the example graph
A distributed implementation
The algorithm given in the previous section has the additional prop erty that it can b e
implemented in a completely distributed manner We consider the hexagon graph as
mo deling a network of pro cessors base stations with each pro cessor resp onsible for
a single vertex in the hexagon graph The network has the same spatial emb edding as
the graph and pro cessors at neighb oring base stations in the network can exchange
lo cal information e ciently For ease of description we will sometimes identify the
vertices of the hexagon graph with their pro cessors
In the xed planar emb edding of the in nite triangular grid we can select an
arbitrary vertex to b e the origin and three directional axes that intersect the origin
one designated the horizontal axis and the remaining two at angles and
from the horizontal axis It is easy to see that any path in the graph where the angles
subtended by all intermediate edges in the path are exactly is oriented along one
of the three directional axes For every vertex we wish to assign a parity with respect
to each directional axis This can easily b e done as follows The parity of a vertex v
along the horizontal axis is de ned to b e the parity of the length of the path oriented
along the horizontal axis from v to a vertex on the axis that intersects the origin
Similarly the parity of v along the axis axis is the parity of the length of
the path from v oriented along the axis axis to a vertex on the horizontal
axis intersecting the origin Thus given an arbitrary nite hexagon graph any path
in the graph that is oriented along one of the directional axes has a coloring that
can b e pre computed according to the parities of the vertices along the path
Our algorithm assumes that each pro cessor initially has access to the following
information
A base coloring for the graph is known each pro cessor knows whether its vertex
is red green or blue and also the color of each of its neighb ors
A coloring along each path oriented along a directional axis is known This
means each pro cessor knows three bits corresp onding to whether it has even or
o dd parity along each of the directional axes
[3]
The value of D is known this also implies that the division of base color
G
classes among pro cessors is known Additionally this implies that every vertex
has access to a known set of M purple hues if needed
The distributed algorithm consists of each pro cessor determining whether it should
participate in one or more of the ve phases of the approximation algorithm describ ed
in Section The algorithm starts with three rounds of information gathering after
which no more communication other than informing neighb ors of the current color
assignment is required essentially pro cessors can continue indep endently to compute
the hues to assign to themselves We describ e the communication rounds from the
p ersp ective of a xed pro cessor p
Round p sends its weight to each of its six neighb ors
Round Having received the weights of all its neighb ors p decides if it would b e a
degree vertex after Phase and sends this information a single bit to each
of its neighb ors This would b e the case if p is itself a heavy vertex and has
three neighb ors that are heavy vertices
Round The information received in the previous two rounds su ces for p to decide
if it will b e a priority vertex in Phase as well as if any of its neighb ors will
b e priority vertices in Phase For instance p will b e a priority vertex if it
is a blue vertex with degree after phase either with no neighb ors that will
also b e degree vertices after phase or with green neighb ors of degree after
phase see Section
Next p determines if it will b e a corner vertex in phase and sends this in
formation a single bit to each of its neighb ors p is a corner vertex if all the
following conditions are met
p is a heavy vertex
p will not b e a priority vertex in Phase
p has exactly two neighb ors of the same color class that are heavy vertices
but not priority vertices in Phase
Round The information derived from Round enables p to determine if it will b e
a priority vertex in Phase as describ ed in Section If p would b e a priority
vertex in Phase and would fall into case ii of Phase then it has a light
neighb or say q that would need to b e recolored in that phase In this case p
sends a message to q with the maximum weight of its remaining two neighb ors
of the same color as q so that p can color itself appropriately
Round A light vertex that got a message to recolor itself in Round informs
all its neighb ors of how many colors it will use from the end of its base color
sp ectrum This enables an isolated vertex in its neighb orho o d to color itself
appropriately in Phase
From the weights of its neighb oring pro cessors and its limited global knowledge
and the information collected in the communication rounds describ ed ab ove a pro
cessor can easily compute the colors it will use in each of the ve phases Without
loss of generality consider a pro cessor that corresp onds to a blue vertex v G The
pro cessor emulates the ve phases of the sequential algorithm as follows
Phase If w v M the pro cessor assigns to itself the appropriate blue hues
Otherwise it assigns to itself all the blue hues reduces its weight by M and
continues
Phase If the pro cessor is a priority vertex in Phase it simulates phase and
stops or else continues to phase Recall that the colors that a priority vertex
would b orrow from one of its neighb oring color class are the last colors from
that class thus no consultation is required with neighb ors to compute the colors
at a priority vertex
Phase If the pro cessor is a priority vertex in Phase it can determine the colors
it needs to b orrow from the appropriate neighb oring color class
If however the pro cessor is a light vertex that would have undergone recoloring
in phase of the sequential algorithm then it can emulate this b ehavior in the
distributed algorithm as well To do so it uses the information it received in
its neighb or in Round and recolors itself as describ ed in Section so that no
con ict app ears among the blue colors
Phase If a pro cessor is an isolated vertex in this phase it assigns itself any purple
hues that it needs Additional colors that it may need are b orrowed from one
of the neighb oring color classes Since any of its light neighb ors that may have
recolored itself in the last phase sent information in Round ab out the numb er
of colors it would use from the end of its base color sp ectrum the pro cessor can
determine all colors used by its light neighb ors and can b orrow colors without
any p ossibility of con ict
Phase Any vertex with unassigned colors at this stage lies along some straight line
path in the grid In particular it can detect its one or two incompletely assigned
neighb ors that lie along exactly one of the three directional axes The pro cessor
can easily compute the identity of the particular axis from the information
gathered after Round Dep ending on whether it is an even or o dd vertex
along this axis it assigns itself the necessary hues from the b eginning or end of
the set of purple hues as in the coloring for bipartite graphs
[3]
Theorem An approximate multicoloring that uses no more than dD e colors
G
for any hexagon graph G V E w can be e ciently computed in constant time in a
completely distributed manner after an initial constant time communication protocol
where each processor exchanges ve messages with each of its neighbors
Discussion
In this pap er we have cast the problem of frequency assignment in cellular net
works as a multicoloring problem for hexagon graphs For some particular induced
subgraphs of hexagon graphs i e cycles and outerplanar graphs we show e cient
algorithms for multicoloring them using an optimal numb er of colors In Section
[3] [3]
we describ e a multicoloring algorithm that uses at most dD e colors where D
G G
the maximum weight on any clique in G is a trivial lower b ound on the minimum
[3]
numb er of colors required We showed also a hexagon graph that requires d D e
G
colors Determining an exact b ound on G the weighted chromatic numb er of an
arbitrary hexagon graph is NP hard our results do establish that for all hexagon
[3]
graphs G G dD e Whether or not there is an approximation algorithm
G
[3]
for hexagon graphs which always uses at most d D e colors remains an intriguing
G
op en problem A useful feature of our algorithm is that it can b e implemented in a
distributed manner In contrast the algorithm of McDiarmid and Reed which
has the same p erformance ratio as ours seems inherently centralized and cannot b e
made distributed in any obvious way
An interesting avenue for future research is the generalized version of the problem
where the frequencies assigned at a particular vertex or at adjacent vertices are re
quired not merely to b e di erent but also to b e far enough apart Another recently
prop osed mo del considers arbitrary interference graphs with pre de ned costs on
the edges these costs re ect the interference p enalties when the same or adjacent
frequencies are assigned to neighb oring no des The ob jective here is to minimize the
total cost or equivalently the net interference of an assignment Finally the dynamic
version of the problem involves changing weights at vertices It would b e interesting
to see if the distributed algorithm we describ e in Section can b e adapted to work
in this setting and what b ounds can b e proved on its p erformance
Acknowledgments
We thank Jeannette Janssen for intro ducing us to the problem of channel assignment
and for comments that greatly improved the presentation of Section We are grateful
to the anonymous referees for their useful comments and for suggesting the current
form of Corollary
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