Optimal Partitioning of Cellular Networks Ioannis G. Tollis Telecommunications Research Laboratory and Department of Computer Science P.O. Box 830688, EC 31 The University of Texas at Dallas Richardson, TX 75083-0688 email:
[email protected] Home page: http://wwwpub.utdallas.edu/~tollis/
Abstract The problem of partitioning a cellular network into several disjoint parts is central to the design of cellular systems. The objective is to partition the network of cells into two or more parts such that the amount of handover between cells that belong to di erent parts (or any other cost function) is minimized. All the cells in each part are assigned to the same MSC. We present a model and fast algorithms that obtain partitions that minimize the handover (or any other cost function) between cells that belong to di erent MSCs.
1 Introduction The problem of partitioning is central to the design of wireless communications (cellular) networks, and it arises in several stages of the design. The handover function is a most frequently requested function in a cellular network and it has a direct impact on the perceived quality of service [3, 7]. In this paper we concentrate on partitioning techniques that nd applications in assigning each Base Station (BS), or Cell, to a Mobile Switching Center (MSC), also known as Mobile Telephone Switching Oce (MTSO), such that the total handover requests between cells that belong to different MSCs is minimized. We call this problem the interswitch handover minimization problem. We present a model and algorithms for solving this problem very eciently. Notice that our model and partitioning algorithms work for
any cost function between cells, and hence can be used for solving other design problems as well. The demand for mobile communications has been increasing monotonically recently, especially in the metropolitan areas. In order to respond to the demand for new subscribers and because of physical limitations (such as prede ned geographical service area, limitation of available radio frequency spectrum, etc.), the solution is to reduce the size of the cells taking advantage of the most important characteristic of the cellular model: reuse of frequencies. This action however increases the handover between cells. The handover between cells is naturally categorized into two categories [3]: (a) the handover between cells belonging to the same MSC, and (b) the handover between cells belonging to di erent MSCs. The latter adds signi cant extra overhead, since interswitch communication is very expensive. The expansion of the coverage area coupled with the reduction in cell size result in a rapidly growing number of cells, which in turn creates the need for the installation of new MSCs. This typically creates a dramatic increase of trac between cells belonging to di erent MSCs [3]. The signi cance of this fact is that it does not only a ect the quality of service but, most importantly, it increases the overhead cost, since the handover between cells that belong to different MSCs is very expensive. Hence, in order to increase the use of the existing capacity for the subscribers (as opposed to, for the overhead communications) minimizing the interswitch trac is of paramount importance. This problem will become even more important in the future with the advent of Personal Communication Systems (PCS) [3]. In this paper we present techniques that assign cells to MSCs
such that the interswitch trac is minimized. Clearly the above minimization problem can be modeled by a graph whose nodes are the base stations (cells) and its edges connect base stations that have handover trac, i.e., their cells share a common boundary. The handover trac between two cells will be the cost of the corresponding edge. The object of the optimization is to partition the nodes of the graph into k (almost) equal parts such that the sum of the costs of all the edges that have one node in one part and the other in a di erent part is minimized. The graph partitioning problem has received signi cant attention, due to its important applications in crucial areas such as the layout of integrated circuits and printed circuit boards [6]. The graph partitioning problem is NP-complete, even when k = 2 and all the costs are equal to one [2, 6]. This constrained version of the problem is called graph bisection. The NP-completeness of graph bisection implies that it is highly unlikely that there exist ecient algorithms for solving the problem optimally. Due to this, several heuristics have been developed to solve the graph bisection problem [1, 4, 6]. An empirical study of several heuristics can be found in [5]. If the restriction that the two parts have prespeci ed sizes is removed, then the problem is solvable in polynomial time using the max- ow min-cut theorem [6]. However, the restriction on prespeci ed sizes is crucial in most applications, including the minimization of handover, and the layout problems. In this paper we present a model and algorithms for solving the problem optimally (or in some cases almost optimally) by exploiting the special conditions imposed by the interswitch handover minimization problem.
2 The Model
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Figure 1: A regular grid-like cellular network. In partitioning the nodes of the network, we observe the following:
each part should contain (approximately) the same
number of vertices (same weight); we say approximately because we have found out through experimentation that if we allow a small tolerance in the balance of the partitions, the total cost can be lowered even further.
each part should be connected; if a part consists of two
Formally, the problem can be modeled as follows: Each base station is a node in a graph (network) and if there are handover requests between two base stations, then the corresponding nodes are connected with an edge. Hence, we have a graph G = (V; E ) with a node set V and an edge set E . Each node of G is connected with an edge to each of the vertices which correspond to the base stations of all its neighboring cells. If this is a \regular" network, then the graph is the hexagonal grid, see Fig. 1. The degree of a node is the number of edges that are incident upon the node. Each edge (v ; v ) of G has a cost associated with it, which is de ned in the following way: if vertices v and v are adjacent then the cost t(v ; v ) is simply the handover trac i
between the cells corresponding to nodes v and v . (If these vertices are not adjacent, then the cost is set to zero.) Additionally, each node v may have a weight w(v ) associated with it. The weight may correspond to the amount of traf c handled by the node, the average number of subscribers using this node per unit of time, etc. It will become apparent from the description of our algorithms that they can easily handle nodes with various weights. For simplicity of our presentation, we assume that each node has weight one. Later, we will describe how to extend the ideas to arbitrary weights.
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or more disconnected pieces, then it is not geographically contiguous, and its boundary is typically longer.
Based on the above observations, our task is to partition the nodes of G into k approximately equal parts so that: 1. the sum of the costs of all the edges which connect vertices belonging to di erent parts is minimum: min
P
2E t(vi ; vj )xij
(vi ;vj )
where E is the set of edges of graph G and x is 1 if vertices v and v belong to di erent parts, and 0 otherwise. ij
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2. if we delete the edges that connect vertices of di erent parts, the result is k disjoint connected subgraphs (i.e., each part is geographically contiguous). As discussed before, the problem of two way partitions and multiway partitions has been studied extensively in VLSI layout[6]. One could use a min-cost graph bisection heuristic such as the ones described in [1, 4, 5, 6]. We believe that the Kerninghan-Lin approach [4] is not as well suited for this kind of problems as the Fiduccia-Mattheyses [1] approach for the following three reasons: the Kerninghan-Lin heuristic (a) handles only exact bisections, (b) exchanges pairs of vertices, and (c) has a rather high time complexity. Instead, the Fiduccia-Mattheyses heuristic avoids the above problems. These techniques work only for graph bisection (i.e., expanding from one to two switches). However, the quality of the solutions obtained by the above heuristics (or any heuristic that works in a similar way as these heuristics) cannot be analyzed. Also, it is possible to obtain bisections where each part contains several disconnected pieces (since these heuristics do not take into account the existing topological information). Because of the above problems we decided to investigate other techniques that exploit the geographic properties of cellular networks. Notice that the graph that models our handover minimization problem is planar, that is no edges cross except for their common nodes. Furthermore, because of the connectivity requirement, each part in a k-way partition induces a connected planar subgraph of the original graph. In other words, we need to nd a set of edges (of minimum total cost) whose removal disconnects the planar graph into exactly k planar subgraphs, such that each subgraph has approximately the same number of nodes (or total node weight). This problem is better solved on the dual graph of our original (primal) planar graph. The dual graph is obtained by placing a new node in the middle of each face of the primal graph (along with some appropriate dual nodes on the outer face, called boundary nodes) and connecting two dual nodes if the corresponding faces share an edge. The hexagons in Fig. 1 de ne the dual graph, and the grey nodes are the dual nodes. Each edge of the primal that has a cost is thus intersected by a dual edge and we assign the same cost to the dual as the one on the primal edge. Thus our partitioning problem has been transformed into a problem of nding a collection of \short paths" in the dual graph.
3 Bisections and k-way Partitions In this section we discuss rst how to nd bisections of the given network. Namely, we explain our technique in the simple case where k = 2. Next we extend our ideas to handle k-way partitions. We will show how to solve the bisection problem in the primal graph by nding \shortest paths" from a boundary node to another boundary node (of the dual). Such paths separate the primal graph into two almost equal parts. The simplest way to nd a shortest path in the dual graph that separates the primal as described above, is as follows: 1. From each boundary node nd shortest paths to all the other boundary nodes. 2. Traverse the shortest paths of each boundary node until we nd such a path that splits the nodes of the primal as speci ed. This is our rst candidate path. 3. Continue searching through the other paths in order to nd a shorter path than the currently best. If found, replace the old candidate path with the new one, and continue the search until all shortest paths of every boundary node have been examined. In order to understand why the above algorithm works we have to consider the following two questions: 1. What is an acceptable partition? 2. Is there always a shortest path that gives such a partition? To answer the rst question we have de ned a parameter called tolerance. The tolerance is a user de ned parameter such that if the size of each part of the partition is between n=2 ? and n=2 + , then the bisection is considered acceptable. There is an interesting connection between the degree of the nodes of the primal graph, the minimum and maximum trac costs, and the tolerance. The answer to the second question is more complicated: Generally speaking, the answer is negative. However, our study has revealed a strong connection between the minimum and maximum trac costs (over all edges), the degree of the nodes, and the choice of the tolerance. In fact, the \correct" choice of tolerance guarantees that there is always such a shortest path that provides an acceptable partition.
The above algorithm is rather fast. Theoretically, the algorithm can be implemented so that its time complexity is proportional to about n3 2 log n (depending on the shape), or O(n3 2 log n), where n is the number of base stations (or cells). In practice our program found a solution in less than two minutes for networks with 210 cells. However, the solutions obtained by this simple algorithm are not guaranteed to be optimal. Better bisections can be obtained by considering the following variation of the above algorithm: =
=
Instead of nding shortest paths from all boundary nodes to all other boundary nodes, nd shortest paths from all interior nodes to all boundary nodes. Next, for each interior node v nd two shortest paths to two boundary nodes u and w such that the combined path from u to v and from v to w partitions the graph into two parts that are within the acceptable tolerance. From all these possible paths choose the shortest. This algorithm nds better partitions than the ones obtained by the previous algorithm. However, it is signi cantly slower. The time complexity is O(n3 ). Our experimental results show that with respect to the previous algorithm, the improvement obtained is between 5 and 10% but it takes more time to nish (for our examples it took more than four times as much time). The last algorithm can be modi ed in order to make it faster. Observe that each path that splits the area into two almost equal parts has to pass through a speci c region of the interior. We de ned that region to be an \X" and found shortest paths from any node of the \X" to all boundary vertices. The resulting solutions are of quality similar to the slower algorithm. However, the new algorithm takes about the same time as the original algorithm. The above ideas can be extended in order to obtain algorithms for creating k-way partitions. For example, in order to obtain 3-way partitions, we can nd a 1=3n - 2=3n split rst, and then nd a bisection of the larger (2=3n) part. In general, the shortest path that obtains the 1=3n - 2=3n split does not lead to the best overall partition. Instead, as we observed experimentally, the best overall partition is obtained when the 1=3n - 2=3n partition is not always the minimum. This implies that we have to check each possible 1=3n - 2=3n partition and for each one to nd a further split of the larger region. This technique works very well, but it is rather time consuming. However, we can speed up the algorithm if we are willing to accept a solution that is not
optimal, by reducing the search space. In order to obtain 4-way partitions, we employ the bisection algorithm recursively. In general, k-way partitions are obtained by combining the above techniques.
4 Experimental Results and Conclusions We have implemented our simple algorithm (for k = 2) and ran it on 110 examples in order to gain some practical understanding about the quality of the solutions produced. Speci cally, we used the regular topology (shown in Fig. 2) which contains 210 cells. We ran the algorithm on it 110 times, each time choosing the costs on the edges randomly. The algorithm produced a partition for each example within two minutes. Furthermore, the best partition was found within the rst 25% shortest paths explored. This implies that if time is of essence, we can interrupt the computation after exploring about 25% of shortest paths and expect to be very close to the nal best solution (if that is not already the best solution). This might be important during the intermediate stages of the design, when the designer is interactively using the tool. We recently implemented and performed experiments on the same regular topology using the second and third algorithm for bisections described above. We observed that the second algorithm took about the same amount of time as the rst one but it produced better solutions by about 2-10% (similar to the solutions produced by the second algorithm). We also performed experiments on an irregular topology (shown in Fig. 3). The algorithms performed very well. In Fig. 3 we show ve di erent partitions of the topology obtained from random assignments of costs on edges and di erent choices of tolerance ranging from zero to eight. In conclusion, we presented a new model and fast algorithms for solving the interswitch handover minimization problem. We are currently working on re ning our data structures in order to reduce the running time of the program. We are also planning to include an interactive graphical interface in the future.
Acknowledgement The author would like to thank Brian Lindstrom for working on a prototype of the algorithms described in this paper,
Figure 2: An example of the regular-topology cellular network with tolerance four. and Clark Foundation that provided nancial support for his work.
References [1] C. M. Fiduccia and R. M. Mattheyses, A Linear Time Heuristic for Improving Network Partitions, Proc. of the 19th Design Automation Conf., pp. 175-181, 1982. [2] M. R. Garey and D. S. Johnson, Computers and Intractability - A Guide to The Theory of NPCompleteness, W.H. Freeman, 1979.
[3] B. Jabbari, G. Colombo, A. Nakajima, and J. Kulkarni, Network Issues for Wireless Communications, IEEE Communications magazine, pp. 88-98, Jan. 1995. [4] B. W. Kernighan and S. Lin, An Ecient Heuristic Procedure for Partitioning Graphs, Bell Systems Technical Journal, 49(2), pp. 291-307, 1970. [5] Kevin Lang and Satish Rao, Finding Near-Optimal Cuts: An Empirical Evaluation, Proc. Symp. on Discrete Algorithms, pp. 212-221, 1993. [6] Thomas Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, B. G. Teubner - John Wiley & Sons, 1990.
Figure 3: Five examples of the irregular topology network with various values for tolerance. [7] S. Tekinay and B. Jabbari, Handover and Channel Assignment in Mobile Cellular Networks, IEEE Communications magazine, pp. 42-46, Nov. 1991.
