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 ecient 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 HG M email latacsconcordiaca FAX Research supp orted by NSERC Canada y Department of Computer Science Rutgers University Camden NJ USA email shendecrabrutgersedu 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 ecient shared utilization of the lim ited available bandwidth is critical to the viability and eciency 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 cells 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 Borndorfer et al Nevertheless the hexagon graph mo del continues to b e of signicance from the historical standp oint and as an abstraction that is suciently close to reality to provide useful insights In the context of frequency assignment a multicoloring as dened 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 nonnegative 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 dier 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 ie it is a vertex weighted graph that is a nite induced subgraph of the innite 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 nontrivial In fact it has b een established only recently that the corresp onding decision problem is NPcomplete 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 wellknown 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 dierent 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 dene 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 ecient 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 nonnegative integer weight vector w dened 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 innite 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 ie 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 NPhardness of this problem McDiarmid