Heuristics for Multiway Partitioning
in Hexagonal Cellular Systems
Kyungshik Lim and YannHang Lee
Computer and Information Sciences Department
University of Florida
Gainesville, FL 32611
Abstract
Given a hexagonal mesh of base stations in cellular systems we consider the problem of
finding a cover of disjoint clusters of base stations which generate multiple types of traffic
among themselves. The problem differs from general graph partitioning problems in that it
considers not only communication costs but also the underlying topology among base stations,
such that base stations in a cluster are connected in their physical topology. The objective is to
minimize the total communication cost for the entire system where intercluster communication
is more expensive than intracluster communication for each type of traffic. The problem is
transformed into the dual based on a topology matrix and a relative cost matrix. We develop
several heuristics for the dual. These heuristics produce optimal partitions with respect to
the initial partition, based on the techniques of moving or interchanging the boundary nodes
between adjacent clusters. The heuristics are compared and shown to behave quite well through
experimental tests and analysis.
1 Introduction
In wireless communications networks, whenever there is a need to establish communication with any
particular user, the network has first to find out which one of base stations can communicate with
the user. In the fullinformation strategy, each mobile user transmits location update messages
whenever it moves to a new base station and every base station maintains a complete location
information for every user. This makes a move operation to perform handoff very expensive but
makes a find operation to locate the current base station of a mobile user very cheap. In the
noinformation strategy, on the other hand, mobile users never send location update messages and
whenever there is a need to locate a particular user, the global search over the entire network is
performed. This makes a move operation very cheap but makes a find operation very expensive.
Based on the combination of these two extreme strategies, a number of efficient location tracking
strategies have been reported in the literature [1, 2, 3, 4]. A hierarchical location server structure
using the graphtheoretic concept of regional matching is constructed to give the upper bound of
the communication cost for a sequence of find and move operations in [1, 2]. A partial information
strategy based on the reporting center concept is presented in [3] and the issues of querying locations
in wireless environments are discussed in [4].
Let us assume that we have an efficient location tracking strategy which involves the two primi
tive operations. Given the frequencies of move and find operations among base stations, we consider
the problem of assigning location servers to base stations. Location servers are connected among
themselves and to base stations by fixed networks. Since base stations are used as the interface
between mobile users and fixed networks, they are regarded as traffic sources and destinations from
the prospective of location servers. The cost of tracking mobile users within the area administered
by a location server is usually much lower than that of between the areas administered by different
location servers. Thus, the location server assignment problem is concerned with optimal parti
tioning of base stations, so as to minimize the total communication cost of the entire system for a
sequence of move and find operations.
In addition to the frequencies of move and find operations among base stations, the problem
also considers the physical deployment of base stations so that base stations in a cluster are con
tiguously connected in their physical topology, not scattered. While the frequency is represented by
a complete directed graph among base stations for a possible communication between mobile hosts
through different base stations, the physical topology is represented by a linear graph in highway
cellular systems and by a hexagonal mesh graph in cellular systems. Hence, the location server
assignment problem is concerned with the two types of graphs, the topology graph for partitioning
base stations and the frequency graph for optimizing communication cost.
For general graphs, the problem of finding a cover of disjoint sets such that the sum of all edge
weights, whose two endpoints are in two different sets, is equal to or less than a given positive
integer is known NPcomplete for the arbitrary size of a set[5]. The kcut problem of finding a
partition of vertices into k nonempty clusters such that the total edge weight between clusters is
minimum is also NPcomplete for arbitrary k. The kcut problem with specified vertices which is an
extension of the 2cut problem solvable via repeated applications of a maxflow mincut algorithm
becomes NPhard even for k = 3[6]. Given an souterplanar embedding of an souterplanar graph,
an optimal solution for maximum independent set can be obtained in time O(8n) by a dynamic
programming technique, where n is the number of nodes[8]. If an nvertex planar graph G is given
with an souterplaneseparable planar embedding of G, then the optimal partition of two clusters
of fixed size is determined in time O(s2n323s) by a dynamic programming technique[9].
,.../, A ,,,
S0,( 20 2 4 6 8t1
(a) The (column row) indexing (b) The twodimensional labeling
Figure 1: The Labeling of the Physical Topology for H3
Given the frequencies of move and find operations among n base stations in highway cellular
systems, we have shown an efficient optimal partitioning algorithm of O(mn2) for an arbitrary
number of clusters mn by dynamic prograiimmin[12]. In this paper, we consider the location server
assignment problem in hexagonal cellular systems. Section 2 introduces a labeling scheme to for
malize the problem mathematically. With the labeling scheme, the problem is transformed into
the dual based on a topology matrix and a relative cost matrix. The topology matrix reflects the
underlying topology constraint on clustering, while the relative cost matrix reflects the communi
cation cost of multiple types of traffic to be optimized. Section 3 presents several heuristics for
the dual. These heuristics produce optimal partitions with respect to the initial partition, based
on the techniques of moving or interchanging the boundary nodes between adjacent clusters. The
heuristics are compared and shown to behave quite well through experimental tests and analysis
in Section 4. Finally we conclude in Section 5.
2 The Problem Formalization
2.1 The Labeling Scheme
The physical deployment of n base stations is represented by a planar graph of a hexagonal mesh
of n base stations, where the vertices represent base stations and the edges represent the adjacency
of base stations. The vertices on the exterior face of the graph are level 1 vertices and the vertices
on the exterior face of the subgraph induced by removing level 1 vertices are level 2 vertices, and
so on. A planar graph is souterplanar if it has no vertices of level greater than s. Denote H, as
an souterplanar graph of a hexagonal mesh. It is then known that the number of vertices in an
Ha is n = 3s2 3s 1 and the number of columns in each of three directions is d = 2s 1. Before
formalizing the problem, it is necessary to introduce a labeling scheme to describe the problem
mathematically.
Starting from the leftmost column of an Hs, each column is indexed from 0 through d 1 in
sequence. Then the bottom vertices of every column constitute row 0, the next vertices of every
column row 1, and so forth. Once the column index c and the row index r of a vertex is determined,
the vertex is labeled (i,j) such that i = 2c and j = 2r. In this twodimensional coordinate system
of an Hs, a point (i,j) for 0 < i,j < 2(d 1) can represent either a vertex vij if i and j are even or
an edge cij otherwise. Figure 1 illustrates the labeling of the physical topology for an H3 of n = 19
nodes.
If given a vertex vij, at most six edges, each leading to an adjacent vertex, are directly identified
by the labeling scheme as follows:
e(j+l), e(+l)(j+) e(i+l), ei(j _1), e(i l) and e(il)j if i < d 1;
(j+l), e(i+l)j, (i+l)(j), ci(j_1), (i l)_ ), and c(iej if i = d 1;
(j+l), (i+l)j, (i+l)jl), ci(j_1), e(il), and (i l)(j)if > d 1.
On the other hand, if given an edge ej, the two vertices connected by the edge are directly
identified by the labeling scheme as follows:
_i)j and [)j if i is odd and j is even;
Vi(j_) and Vi(j+l) if i is even and j is odd;
_i)(j_) and ', t)(j+l) if both i and j are odd and i < d;
'* t)(j_) and _i)(j+i) if both i and j are odd and i > d.
In the example of Figure l(b), a vertex v24 leads to the six edges e13,e14,e23,e25, e34, and e35,
which are connected to the neighboring vertices, v02,V04,V22,v26,v44, and v46, respectively.
2.2 The Topology Matrix
To handle the underlying topology constraint on clustering, we construct a topology matrix T =
(tij) for 0 < i,j < 2(d 1), where tij corresponds to either a vertex vij if i and j are even or an
edge cij otherwise in the labeling scheme. If vij currently belongs to cluster P,, the element tij is
set to x. If the two vertices connected by eij belong to clusters Px and Py, the element tij is set
to xy for x < y. Figure 2(b) shows the topology matrix T derived from an initial partition of 3
clusters in Figure 2(a).
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
cluster 1 0 2 2 1 0 2 22 2 12 1
.. 1 1 22 22 22 12 11 11
0 2 2 2 1 1 2 222 212 1 11 1
3 .3 22 22 22 12 1 11 1111 11
4 2 2 4 2 22 212 1 11 1 111
 *
523 23 23 U
S 6 3 3 3 3 6 3 33 3 33 3 33 3
S 7 33 33 33 33 33
cluster 2 er3 8 3 3 3 8 3 33 3 33 3
(a) A partition of 3 clusters (b) The topology matrix T
Figure 2: The Topology Matrix T for a Partition of H3
From the topology matrix T the indices of the boundary edges between a pair of adjacent
clusters P, and Py can be represented by a list of edge elements Ly, = {efijtij = xy, x < y}. Using
the list Lxy, the boundary vertices between them can be directly identified by the labeling scheme
itself. Denote V~ and Vx, as the indices of the boundary vertices for P, and Py, respectively. In the
example of Figure 2, L12 = {03, 613, e23, C33, 643}, V2 = {v04, v24, v44}, and V" = {v02, V22, V42}.
2.3 The Relative Cost Matrix
The communication among base stations is considered as a full mesh of pointtopoint logical
network to represent a possible communication between mobile hosts through different base stations.
The communication network is described by a complete directed graph G = (V, E), where IV = n.
The vertices of the graph represent base stations and the edges represent directional communication
links between base stations. Each edge is assigned a move frequency by a function fm : V X V R+
and a find frequency by ff : V x V R+. Denote .,1 and ? as the weight of a move operation
within a cluster and between clusters, respectively, and w1 and wf as that of a find operation within
a cluster and between clusters, respectively. Then we define a relative cost function c : V x V R+
for all (vii, vkl) E V V as
c(vij, vkl) = fm (ij, Vkl)( )+ ff(vij, Vkl)(w w}), (1)
where (., ? ., ) is the relative weight of a move operation and (w w1) is the relative weight
of a find operation. The cost c(vij, vkl) represents the total relative cost of fm(vii, Vkl) move and
ff(vij, vkl) find operations from vij to vkl if they are in different clusters. Denote c(vij, vkl) as Cij;kl
The relative cost matrix C = (Cij;kl) for 0 < i,j, k, I < 2(d 1) is derived by the equation (1) from
the move and find frequencies among n vertices.
2.4 The Dual
Let II = {PI,..., Pm} be a partition of m clusters of contiguous vertices such that P, n Py = 0
and UP, = V for x,y = 1,...,m and x 5 y. The intracluster communication cost for all
(vii, Vk1) E V x V is
Costintra = (f.m(vij, Vkl 1 + ff(vii, Vkl ,2 )
and the relative intercluster communication cost of II is
ml
RcostinterT(l) = (Cij;kl + Ckl;ij).
X=1 v3 EP ,Vkl EPy,
The total communication cost of II is then
COSttotal(I) = Costintra + ROStinter (II).
Because Costintr, is constant independent of how to partition, it holds that II minimizes
Costtotal iff I minimizes Rcostintr. Let us define the relative intracluster communication cost
of II as
m
Rcostintra,(I) = (Cij;,k + Ckl;ij).
X=l v,2 ,vklIEP
Because the sum of Rcostintra and Rcostifter is constant, our task of finding an optimal partition II
which minimizes Rcost;iter is equivalent to that of finding an optimal partition II which maximizes
RCOStintra
Hence, the dual problem is: given the topology matrix T = (tij) and the relative cost matrix
C = (cij;kl) for a system of n hexagonal cells, find a cover of m disjoint clusters of contiguous base
stations II = {PI,..., Pm}, so as to maximize Rcostintra(lI).
3 Heuristics
We consider heuristics for the problem: starting with an arbitrary partition II of m clusters, we
try to increase the initial relative intracommunication cost Rcosti,,a(II) by repeated applications
of a twoway optimization procedure to pairs of adjacent clusters. In the twoway optimization
procedure, we try to increase the sum of the initial relative intracommunication cost of each
cluster by moving or interchanging boundary nodes until no further improvement is possible. The
resulting pair of adjacent clusters are then pairwise optimal with respect to the initial partition.
Because the twoway optimality for a pair of adjacent clusters may affect that for another pair of
adjacent clusters, more than one pass through all pairs of adjacent clusters may be required. This
process can be repeated with another initial partition II of m clusters, and so on, so as to obtain
as many locally maximum partitions as we desire. Then, one of the resulting partitions has a fairly
high probability of being a globally maximum partition.
3.1 Interchanging or Moving Boundary Nodes
Consider a pair of adjacent clusters P, and Py in the twoway optimization procedure. The inter
change of a pair of boundary nodes, vij E V, and vkl E VV, is said to be feasible if the resulting
clusters preserve the underlying topology constraint. In other words, all nodes in P, (Py) adjacent
to Vij (vkl) must be connected in their physical topology before the interchange and vij (vkl) must
be connected to at least one of the nodes in Py (P,) after the interchange.
To determine feasible pairs of interchanging nodes, we use the topology matrix T where an
element tij corresponds to a node vij or an edge eyj. Given a node vij E V,, at most six edges
adjacent to the node can be directly identified by the labeling scheme itself. Denote Adj(vij) as
an ordered list of those edges which are sequentially arranged in a circular fashion. Then it is said
that the node vij can be moved into Py for the interchange when the following two conditions are
satisfied:
1. There is only one continuous subsequence of edges which are equal to xx in Adj(vij).
2. There is at least one edge in Adj(vij), which is equal to xy and does not lead to the node
Vkl E Vxy.
Assume that initially all nodes of each cluster are connected in their physical topology. Con
dition 1 implies that the remaining nodes of P, after removing vij still preserve the connectivity
in their physical topology. Condition 2 implies that the removed node vij is also connected to its
adjacent cluster Py whose nodes are already connected in their physical topology. In the same way,
the node vkl must also satisfy the above two conditions so that it can be moved into PX for the
interchange. Therefore, the interchange of a pair of boundary nodes vij E V, and vkl E VX is
feasible iff both vij and vkl satisfy Condition 1 before the interchange and Condition 2 after the
interchange.
In the example of Figure 2(b), we consider interchanging v22 E P2 and v44 E P1. For node v22,
since the only one continuous subsequence {e12, e11, e21, e32} in Adj(v22) = {23, e12, e11, e21, e32, e33}
0 1 2 3 4 5 6 7 8
2 22 2 12 1
22 12 12 12 11 11
2 12 1 11 1 11 1
22 22 12 12 12 11 11 11
2 22 2 22 2 12 1 11 1
23 23 23 23 23 13 13 13
3 33 3 33 3 33 3
33 33 33 33 33 33
3 33 3 33 3
(a) after the interchange of (2,2) and (4,4.)
0 1 2 3 4 5 6 7 8
2 22 2 12 1
22 22 22 12 11 11
2 22 2 12 1 11 1
22 12 12 22 12 11 11 11
2 12 1 12 2 12 1 11 1
23 13 13 23 23 13 13 13
3 33 3 33 3 33 3
33 33 33 33 33 33
3 33 3 33 3
(b) after the interchange of (4,2) and (4,4)
Figure 3: Interchanging The Boundary Node Elements
is equal to 22, Condition 1 holds. In addition, since 623 = 12 and it does not lead to v44, Condition 2
also holds. Thus, v22 can be moved to P1 for the interchange. At the same time, for node v44, since
the only one continuous subsequence {e45, 634} in Adj(v44) = {34, 33, 643, 653, 654, 45} is equal to
11, Condition 1 holds. In addition, since e43 = 12 and it does not lead to v22, Condition 2 also
holds. Thus, v44 can be moved to P2 for the interchange. Hence, v22 and v44 are a feasible pair of
interchanging nodes between P1 and P2. Figure 3(a) shows the resulting topology matrix T after
the interchange. On the other hand, the nodes v42 E P2 and v44 E P1 cannot be interchanged, as
depicted in Figure 3(b). The node v42 is isolated after the interchange due to the fact that no edge
except e43 adjacent to v42 is equal to 12 in Figure 2(b).
It should be noted that the move of a boundary node vij E V7 into Py is rather simple than
the interchange because we need to check that only all nodes in P, adjacent to vij are connected
in their physical topology before the move.
3.2 Computing Gains
3.2.1 The Interchange
Assume that a pair of boundary nodes, vij E Vy and vkl E VV,, is feasible for the interchange.
Define the internal cost of node vij with respect to P, for the interchange to be
S
Ujjf EP1,uv, v
C ij;i'j + Cj'j;jj),
and the external cost of node vij with respect to Py for the interchange to be
Ee{(vj)
Vk'l' EPY ,vkl Vk'l
+ Ck'l';ij).
l6(vj)
Similarly, we define Ie(vkl) and Ee(vkl) for the boundary node vkl. If vij and vkl are interchanged,
then the gain g of the increase in cost is given by
g = (E(vij) + Ee(Vkl)) (Ie(vij) + I(vuk))
3.2.2 The Move
Assume that a node vij E Vy is feasible for the move. Define the internal cost of node vij with
respect to P, for the move to be
Im(vi)
v131 Cpx,v, v,3 7v
(Ci.:iY + Ci :i),
and the external cost of node vii with respect to Py for the move to be
E(v)= C (
vk'1'iEPY
+ Ck'l';ij).
If vij E P, is moved into Py, then the gain g of the increase in cost is given by
g = Em(Vkl) Im(vij).
3.3 Heuristic.1
Given a pair of adjacent clusters P, and Py, Heuristic.1 interchanges only one feasible pair of
boundary nodes with a positive maximum gain g between the two clusters. This process is repeated
with an updated boundary until no feasible pair produces a positive gain. Let k be the number of
feasible pairs interchanged. Then the the total gain with respect to the sum of the initial costs for
the two clusters is Gy, = k=1 gi. Note that a pair of nodes interchanged in the previous step is
not interchanged again at the next step because the positive maximum gain in the previous step
becomes the minimum gain with the same negative value in the next step.
Heuristic. 1(11)
1 for every pair of adjacent clusters P, and Py in II
2 begin
3 Determine the boundary node lists Vy and V,
4 forever
5 begin
6 Determine a set of feasible pairs
7 if there is no feasible pair
8 break
Compute gains for all feasible pair
if there is no feasible pair with a positive gain
break
Choose a feasible pair (vij, vkl) with the maximum gain
Interchange vij and vkl
Update the boundary node lists Vy and VVy
end
end
3.4 Heuristic.2
Once a set of feasible pairs of boundary nodes is determined from the boundary node lists Vy and
V,1, Heuristic.2 interchanges all feasible pairs with positive gains before updating the boundary
node lists Vy and VV,. Initially, all feasible pairs are unmarked. A feasible pair with the maximum
positive gain is first interchanged and then all remaining unmarked feasible pairs which involve
the boundary nodes of the interchanged pair are marked so that in the next step they cannot be
considered again. After marking, the gains for all unmarked feasible pairs are computed again and
then an unmarked feasible pair with the largest positive gain is next interchanged.
Heuristic.2(11)
1 for every pair of adjacent clusters P, and Py in 11
2 begin
3 Determine the boundary node lists V, and VV
4 forever
5 begin
6 Determine and unmark a set of feasible pairs
7 if there is no feasible pair
8 break
9 Compute gains for all feasible pairs
10 while there are unmarked feasible pairs with positive gains
11 begin
12 Choose an unmarked feasible pair (vij, vkl) with the largest gain
13 Interchange vij and vkl
14 Mark all unmarked feasible pairs involving vij or vkl
15 Compute gains for all unmarked feasible pairs
16 end
17 Update the boundary node lists Vy and VV/
18 end
19 end
3.5 Heuristic.3
Heuristic.3 is based on the observation that infeasible pairs of boundary nodes might become feasible
pairs after interchanging feasible pairs. For example, consider a pair of adjacent clusters P1 and P2
in the topology matrix of H3, where P1 = {voo, v02, v04} and P2 = {v20, v22, v24, v26}. Then V2 = P1
and V1, = P2. A pair of boundary nodes (v02, v22) is infeasible for the interchange because v02 and
v22 violate Condition 1. However, if a feasible pair (v04, v20) is interchanged, the infeasible pair
(v02, v22) becomes a feasible pair for the next interchange.
The algorithm of Heuristic.3 is obtained by slightly modifying that of Heuristic.2. Before
performing line 15 in Heuristic.2, infeasible pairs of boundary nodes which become feasible pairs
due to the interchange of nodes vij and vkl in line 13 are added to the set of unmarked feasible
pairs.
3.6 Heuristic.4
While the previous three heuristics interchange boundary nodes between a pair of adjacent clusters,
Heuristic.4 moves a boundary node in a cluster into the other cluster between a pair of adjacent
clusters. Since the move of a boundary node between a pair of adjacent clusters changes their
cluster sizes, the constraint on the cluster size is given by the minimum and maximum cluster sizes.
Given a pair of adjacent clusters, Heuristic.4 determines a feasible boundary node with a maxi
mum positive gain for each cluster and compares them to determine a boundary node to be moved.
Once the boundary node to be moved is determined, it is removed from its cluster and added to the
other cluster as long as its move does not violate the constraint on the cluster size. This process
is repeated until there is no feasible boundary node for the move or one of the clusters has the
minimum or maximum cluster size.
Heuristic.4(H)
1 for every pair of adjacent clusters P, and Py in H
2 begin
3 Determine the boundary node lists Vy and VV,
4 while the constraint on the cluster size is satisfied
5 begin
6 Determine a set of feasible nodes
7 if there is no feasible node
8 break
9 Compute gains for all feasible nodes
10 if there is no feasible node with a positive gain
11 break
12 Choose a feasible node vij with the maximum gain
13 Move vij into the other cluster
14 Update the boundary node lists V7y and VVy
15 end
16 end
3.7 Heuristic.5, Heuristic.6, and Heuristic.7
Heuristic.5, Heuristic.6, and Heuristic.7 are based on a repeated application of twophase optimiza
tion. In the first phase, the heuristics use Heuristic.1, Heuristic.2, and Heuristic.3 for interchanging
the boundary nodes between adjacent clusters, respectively. Their second phase uses Heuristic.4
for moving the boundary nodes between adjacent clusters to improve the gains obtained by their
first phase if possible. The improvement by their second phase implies the change of the boundary
nodes and the possibility that their first phase further improves their gains. Thus, the twophase
optimization is repeatedly applied until no improvement is possible by their second phase. The
basic idea of the heuristics is that changing the cluster size by their second phase might overcome
the limited improvement due to preserving the cluster sizes of the initial partition by their first
phase.
3.8 Heuristic.8, Heuristic.9, and Heuristic.10
Heuristic.8, Heuristic.9, and Heuristic.10 are also based on a repeated application of twophase
optimization. However, in the first phase, the heuristics use Heuristic.4 for moving the boundary
nodes between adjacent clusters. Their second phase uses respectively Heuristic.1, Heuristic.2, and
Heuristic.3 for interchanging the boundary nodes between adjacent clusters to improve the gains
obtained by their first phase if possible. The heuristics are derived from the fact that the limited
gain of increase in cost due to the constraint on the cluster size in their first phase might be further
improved by interchanging the boundary nodes without violating the constraint on the cluster size.
4 Experimental Testing and Analysis
We obtain the initial partition for experimental testing of the heuristics by two different methods:
random and , ,, "',oi The random partition is based on the topology matrix because it is only
concerned with the geographical arrangement of base stations and the cluster size constraint. On
the other hand, the centering partition is based on both the topology and relative cost matrices
because it further considers the traffic pattern among base stations.
The centering partition of m clusters is achieved by a twophase algorithm. In the first phase,
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91 Total
1 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0
4 17 6 2 1 17 8 1 2 54
5 84 48 47 37 76 56 50 486
6 84 51 43 37 86 80 70 45 496
7 85 51 47 33 ,. 71 70 45 487
8 70 59 39 41 76 62 57 49 453
9 70 62 41 40 76 63 60 52 464
10 70 61 43 38 75 62 61 52 462
Table 1: The Number of Times in 100 Trials a Heuristic Produced a Solution with Maximal Total
Cost with Respect to Other Heuristics When m = 3 and 1/2 x [n/rn] < Pil < 2 x [n/rn]
the m center nodes on which traffics are concentrated are selected by using the relative cost matrix.
Each center node forms an initial intermediate cluster. For every intermediate cluster, the second
phase identifies the adjacent nodes of the cluster which are not involved in other clusters by using
the topology matrix and chooses one of them by using the relative cost matrix, which produces a
maximum increase in cost when it is involved in the cluster. Then the maximum node is added
to the intermediate cluster. This process is repeated until all nodes are contained one of the m
clusters.
In experimental testing, the 100 instances of the relative cost matrix were randomly generated
for each of several values of n, where n = 19, 37, 61, and 91 for H3, H4, H5, and H6, respectively.
For each value of n, Heuristic.1 through Heuristic.10 were extensively tested on the 100 cost matrix
instances for each of the four cases which are the combinations of the two methods for obtaining
the initial partition, random or centering, and the number of clusters m = 3 or 4. The constraint
on the cluster size is given by 1/2 x [n/mr] < Pil < 2 x [n/mr].
Table 1 and Table 2 present the number of times a heuristic produces a solution that is maximal
with respect to the solutions produced by the other heuristics when m = 3 and 4, respectively. The
tables show that Heuristic.5 through Heuristic.10 which are the combinations of the techniques of
interchanging or moving boundary nodes greatly outperform the other heuristics using only one of
the techniques. This is due to the fact that although the cost between a pair of adjacent clusters
cannot be improved by interchanging their boundary nodes, moving their boundary nodes might
further improve the cost, which in turn changes the boundary between the adjacent clusters for
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91 Total
1 0 0 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0
4 8 2 2 2 7 2 2 1 26
5 43 46 41 33 73 71 58 52 417
6 43 45 43 42 73 72 61 52 431
7 64 53 56 45 84 71 63 60 496
8 68 59 34 27 80 74 66 41 449
9 68 59 36 27 80 75 71 45 461
10 68 61 37 29 80 76 66 43 460
Table 2: The Number of Times in 100 Trials a Heuristic Produced a Solution with Maximal Total
Cost with Respect to Other Heuristics When n = 4 and 1/2 x [n/rn] < Pil < 2 x [n/rn]
possible interchanging at the next step, and vice versa. The tables also show that as the number
of nodes n increases, the probability that a heuristic produces a maximal solution becomes lower.
It is interesting that Heuristic.1 through Heuristic.3 do not produce maximal solutions at all
even though Heuristic.4 produces a small number of solutions being maximal. This is because
interchanging boundary nodes does not change the number of nodes for each cluster in the initial
partition and so less flexible than moving boundary nodes. Thus, unlike general graph partitioning
problems, the algorithm which is only based on one of the techniques of interchanging or moving
boundary nodes is no longer useful for our problem which additionally considers the underlying
topology.
Table 3 and Table 4 present the percent of maximum difference in cost from the solution with
the maximal cost, while Table 5 and Table 6 present the percent of average difference in cost from
the solution with the maximal cost. The tables confirm the superiority of Heuristic.5 through
Heuristic.10 with respect to the other hueristics. In general, the maximum differences for the
centering partition of Heuristic.8 through Heuristic.10 are minimal with respect to the random
partition of Heuristic.8 through Heuristic.10 and both the random and centering partitions of
Heuristic.5 through Heuristic.7 with some exceptional test cases. The average differences for the
centering partition of Heuristic.8 through Heuristic.10 are also minimal in general.
Table 3: The Maximum Difference in Total Cost From Maximal Solution When m = 3 and 1/2 x
[n/mn] < Pi < 2 X [n/lm]
Table 4: The Maximum Difference in Total Cost From Maximal Solution When m = 4 and 1/2 x
[n/mn] < Pi < 2 X [n/lm]
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91
1 44.2 36.4 34.6 31.4 42.8 37.1 35.0 34.7
2 44.2 36.4 34.6 31.2 42.8 37.1 35.0 34.7
3 44.2 36.4 33.1 31.0 42.8 37.1 35.0 34.7
4 9.3 6.3 5.6 4.3 11.2 17.7 5.7 23.4
5 8.7 28.1 25.9 13.5 27.3 3.2 23.5 23.1
6 8.7 28.1 25.9 18.9 27.3 3.2 23.5 23.1
7 8.7 26.3 25.9 18.9 27.3 4.5 2.4 1.6
8 7.2 23.3 25.6 23.9 3.9 3.7 2.7 2.0
9 7.2 23.3 25.6 23.9 3.9 3.7 2.7 2.0
10 7.2 23.3 3.1 23.9 3.9 3.7 2.7 2.0
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91
1 40.2 33.1 29.5 26.3 40.6 34.6 31.5 29.9
2 40.2 33.1 29.7 26.1 40.6 34.6 31.5 29.9
3 40.2 33.1 29.7 26.0 40.6 34.6 31.5 29.9
4 16.0 8.2 5.5 5.0 23.3 17.9 9.4 5.4
5 11.2 5.2 10.3 4.5 14.0 17.0 12.8 3.7
6 11.2 5.2 10.3 3.5 14.0 17.0 12.8 3.7
7 9.8 5.2 2.7 4.1 6.8 4.5 3.3 2.1
8 13.1 5.4 3.9 4.3 6.8 4.7 3.6 3.9
9 13.1 5.4 3.9 4.3 6.8 4.7 3.6 3.9
10 13.1 5.4 3.9 4.3 6.8 4.7 3.6 3.9
Table 5: The Average Difference in Total Cost From Maximal Solution When m = 3 and 1/2 x
[n/ln] < Pi < 2 X [n/n]
Table 6: The Average Difference in Total Cost From Maximal Solution When m = 4 and 1/2 x
[n/ln] < Pi < 2 X [n/n]
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91
1 38.2 33.6 31.0 29.2 35.1 32.3 31.7 31.9
2 38.2 33.6 31.0 29.2 35.1 32.2 31.7 31.9
3 38.2 33.6 31.0 29.1 35.1 32.2 31.7 31.9
4 3.0 2.3 2.3 1.7 3.1 2.4 1.6 1.5
5 0.4 1.6 1.0 0.8 0.7 0.2 0.7 0.5
6 0.4 1.3 1.4 0.8 0.7 0.2 0.6 0.5
7 0.4 1.0 1.3 1.1 0.6 0.4 0.2 0.3
8 0.9 0.7 0.9 0.9 0.4 0.4 0.3 0.3
9 0.9 0.6 0.9 0.9 0.4 0.4 0.3 0.3
10 0.9 0.6 0.6 0.7 0.4 0.4 0.3 0.3
Random Centering
Heuristics n=19 n=37 n=61 n=91 n=19 n=37 n=61 n=91
1 31.3 29.2 27.5 24.2 31.8 29.3 26.1 25.4
2 31.3 29.2 27.5 24.2 31.8 29.3 26.1 25.5
3 31.3 29.2 27.4 24.2 31.8 29.3 26.1 25.4
4 5.9 3.3 2.8 2.1 5.2 3.2 2.2 2.0
5 2.3 0.7 0.7 0.7 1.1 0.7 0.5 0.4
6 2.3 0.6 0.7 0.5 1.1 0.7 0.5 0.4
7 0.9 0.5 0.4 0.4 0.5 0.5 0.3 0.3
8 0.9 0.5 0.7 0.6 0.6 0.5 0.2 0.5
9 0.9 0.5 0.8 0.6 0.6 0.4 0.2 0.5
10 0.9 0.5 0.7 0.6 0.6 0.4 0.2 0.5
5 Conclusion
In cellular systems, a hexagonal mesh of n base stations may generate multiple types of traffic
among themselves. We have considered the problem of finding a cover of disjoint clusters of base
stations, so as to minimize the total communication cost for the entire system. The problem differs
from general graph partitioning problems in that it additionally considers the underlying physical
topology among base stations. We have developed several heuristics based on the combinations of
the techniques of moving or interchanging the boundary nodes between adjacent clusters. These
heuristics produce optimal partitions with respect to the initial partition obtained randomly or
by centering. The heuristics are compared and shown to behave quite well through experimental
testing and analysis.
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