Heuristic and simulated annealing algorithms for wireless ATM

25 Pages·2006·0.45 MB·English

Heuristic and simulated annealing algorithms for
wireless ATM backbone network design problem
Der-Rong Diny
Department of Computer Science and Information Engineering,
National Changhua University of Education,
Changhua City, Taiwan R.O.C.
Abstract
Personal Communication Network (PCN) is an emerging wireless network that promises
many new services for the telecommunication industry. The high speed backbone network
(ATM or WDM) is one possible approach to provide broadband wireless transmission with
PCN’susingtheATMswitchesforinterconnectionofPCNcells. Thewireless ATM backbone
network design (WABND) problem is to allocate backbone links among ATM switches such
thatthee(cid:11)ectsofterminalmobilityontheperformanceofATM-basedPCN’scanbereduced.
In this paper, the WABND problem is formulated and studied. The goal of the WABND
is to minimize the location update cost under constraints. Since WABND is an NP-hard
problem, two heuristic algorithms and a simulated annealing algorithm were proposed and
used to (cid:12)nd the close-to-optimal solutions. The simulated annealing algorithm was able to
achieve good performance as indicated from the simulated results.
Keywords: wireless ATM, heuristic algorithm, simulated annealing, NP-hard, backbone net-
work design.
yTo whom all correspondences should be addressed,
E MAIL: [email protected],
Address: No. 1, Jin De Road, Changhua 500, Taiwan R.O.C.
Tel: [phone]
FAX: 886{4{7211081
1
1 Introduction
Personal Communication Network (PCN)[1, 2, 3] is an emerging wireless network that promises
many new services. Users may move from one place to another and can maintain transparent net-
work access through wireless links. Information exchanging between users, may be bidirectional,
which includes voice, data, and image.
In a PCN, the covered geographical area is typically partitioned into a set of cells. Each cell
has a base station(BS) used for exchanging radio signals with mobile terminals. Due to the limited
power of wireless transceivers, mobile users can communicate only with base stations that reside
within the same cell. Moreover, several coverage areas of cells are grouped and formed a location
area(LA). That is, an LA consists of an aggregation of coverage areas of cells forming a contiguous
geographical region.
A typical PCN architecture based on ATM switches is illustrated in Fig. 1(a). The covered
geographical area is partitioned into a set P = fP , P , ..., P g of m disjoint clusters (or LAs).
1 2 m
An ATM switch is allocated within each cluster and each BS in this cluster is connected to the
ATM switch. The ATM switch o(cid:11)ers the services of establishing/releasing channels for the mobile
terminals in the cluster. Two neighboring clusters can be interconnected via the associated ATM
switches. The links between the ATM switches are called backbone links, and the links between
ATM switches and BSs are called local links [3].
Whenasubscriberentersacellthatbelongstoadi(cid:11)erentLA,alocation update (LU)orhando(cid:11)
procedure that informs the network about the subscriber’s new location is performed. This will
generate network tra(cid:14)c overhead in PCN and consume scarce radio resources. Moreover, LU
also increases the load on distributed location databases and, thus, increases the complexity of
implementing the databases [4][5].
During the wireless environment, two types of hando(cid:11)s should be considered in the designing
of the network, they are inter-switch hando(cid:11) and intra-switch hando(cid:11). The intra-switch hando(cid:11)
involves only one switch and the inter-switch hando(cid:11) involves two switches. The inter-switch
hando(cid:11)s that occur between two cells, which connected to di(cid:11)erent switches, consume much more
network resources (therefore, are much more costly) than the intra-switch hando(cid:11) that occur
2
Figure 1: (a) A typical PCN architecture based on ATM switches, (b) H(S;F), (c) G(S;E) with
p=10, deg=3.
3
between cells, which connected to the same switch[6, 7, 8, 9]. Thus, the cost of intra-switch
hando(cid:11)s involving only one switch is negligible in designing the two-level wireless ATM network.
Consider the example shown in Fig. 1, where cells A and B are connected to switch s , and
1
cells C and D are connected to switch s . If the subscriber moves from cell B to cell A, switch
2
s will perform a hando(cid:11) for this call. This intra-switch hando(cid:11) is relatively simple and does
1
not involve any location update in the databases that record the position of the subscriber. The
hando(cid:11) also does not involve any network entity other than switch s . Now imagine that the
1
subscriber moves from cell B to cell C. Then the inter-switch hando(cid:11) involves the execution of a
fairly complicated protocol between switches s and s [6, 7, 8, 9].
1 2
In this paper, the wireless ATM network design (WABND) problem is studied. Given the
PCN network, the hando(cid:11) frequencies between cells, the local links between cells and switches,
the degree constraint, and the number of backbone links, the WABND problem is to (cid:12)nd a set of
backbone links which forms a connected network such that the total cost is minimized under the
degree constraint. In [10], the WABND problem is studied and formulated, a heuristic algorithm
and a genetic algorithm have proposed to (cid:12)nd the sub-optimal solution. Due to the complexity of
theWABNDprobleminawirelessATMnetwork,theprovisionofanoptimalsolutioninreasonable
time is not guaranteed. In this respect, the usual step is to devise an approximate algorithm for
solving this problem. Simulated annealing (SA) is a stochastic computational technique derived
from statistical mechanics for (cid:12)nding near globally-minimum-cost solutions to large optimization
problems. Kirkpatrick et al [11] were the (cid:12)rst to propose and demonstrate the application of
simulation techniques from statistical physics to problem of combinatorial optimization. In this
paper, two heuristic algorithms and a simulated annealing algorithm are proposed to solve it.
Theorganizationofthispaperisasfollows. Section2givesaformaldescriptionoftheWABND
problem. Section 3 describes the proposed heuristic algorithms. In Section 4, the details of the
proposed simulated annealing algorithm are presented. Simulation results are presented in Section
5, and some concluding remarks are given in Section 6.
4
2 Notations, Problem Formulation and Related Works
This section (cid:12)rst provides an overview of various terms and notations used in explaining the
concepts outlined in subsequent sections. There after, the formulation of the problem and the
related works are presented.
2.1 Notations and Assumptions
In this paper, the location of cells and switches are (cid:12)xed and known. Let there be n cells in PCN
network CG(C;L;w) where CG=fc , c , ..., c g. De(cid:12)ne Cell Graph CG(C;L;w), where C is a
1 2 n
(cid:12)nite set of cells with jCj =n and L is the set of edges such that L (cid:26) C (cid:2) C, all the edges are
undirected and with weight function w . Let f be the cost per unit time of the hando(cid:11)s that
ij
occurs between cell c and c , (i, j=1, 2, ..., n). Thus, f is proportional to the frequency of
i j ij
hando(cid:11)s that occur between cell c and c . Assume cells c and c are connected by an edge (c ,
i j i j i
c ) in CG with weight w , where w =f +f , w = w , and w = 0.
j ij ij ij ji ij ji ii
An ATM-based PCN topology can be represented by an undirected graph H(S;F;z); where
S is a (cid:12)nite set of switches and jSj = m. Each node s in S stands for a cluster P (or an
k k
ATM switch) and an edge e is in F if clusters P and P are adjacent in the given network with
kl k l
communication cost z . For example, the corresponding graph of the ATM-based PCN topology
kl
depicted in Fig. 1(a) is shown in Fig. 1(b) and a possible backbone network with 10 backbone links
for the PCN topology is given in Fig. 1(c). De(cid:12)ne that a sequence of switches is a path. For each
pair of neighboring switches in the path, there is a backbone link between the two corresponding
clusters. Therefore, any established call among clusters could be represented by a path in the
corresponding graph[3].
Let switch s , k=1, 2, ..., m in S and (s , s ) in F. Let d be the minimal communication cost
k k l kl
on the path between the switch s , and s . If cells c and c are assigned to di(cid:11)erent switches, then
k l i j
a (intra-switch) hando(cid:11) cost is incurred. Assume for each cell c in C, cell c has been connected
i i
to a unique switch sid(c ), that is, sid(c )=k if c 2 C is assigned to the switch s , k is called the
i i i k
sid of cell c .
i
With the given graph H, a backbone network G(S;E;w) could then be built. The backbone
5
network is built by utilizing the available edges in graph H. Typically, some limitation is placed
on the number of links that could to be laid in the backbone network as the cost of the network
is proportional to the number of links that are to be set up in the network. The objective is to
determine the link between switches so as to minimize the hando(cid:11) costs per unit time under the
degree constraint. Let deg(s) denote the degree of switch s and deg(G) denote the degree of graph
G which is the maximum degree of the switches in G. Thus deg(G) = max fdeg(s)js 2 S g.
For example, given the graph shown in Fig. 1(b), the corresponding graph of a possible backbone
network with 10 links and deg(G) = 3 is shown in Fig. 1(c).
2.2 Problem Formulation
To formulate the location update (or hando(cid:11)) cost, variables h i, j =1, 2, ..., n takes a value of
ij
1, if both cells c and c are connected to a common switch; 0, otherwise. That is, h =1 if and
i j ij
only if sid(c ) = sid(c ). With this de(cid:12)nition, it is easy to see that the cost of hando(cid:11)s per unit
i j
time is given by
n n
(1(cid:0)h )w (cid:2)d : (1)
XX ij ij sid(ci);sid(cj)
i=1j=1
Given m nonempty sets of cells P=fP , P , ..., P g, P is called a m{way cell partition of CG,
1 2 m
if P [P [:::[P =CG and P P = ;, where k 6= l, k, l=1, 2, ..., m. Without loss of generality,
1 2 m k l
T
assume the cells in set P are assigned to switch s , j=1, 2, ..., m. Let LUCS(i;l) = w ,
k k Pcj2Pl ij
if c not in P ; LUCS(i;l)=0, otherwise. Then for a given m-way cell partition P, the location
i l
update cost of the partition can be represented as
LUCS(i;l)(cid:2)d : (2)
X X sid(ci);l
ci2CGsl2S
Because the assignments of cells to switches are (cid:12)xed and known, the location update cost
between switches is also (cid:12)xed. Let LUSS(k;l) = LUCS(i;l), if k 6= l; LUSS(k;l) =0,
Pci2Pk
otherwise. Then the location update cost can be represented as
m m
LUSS(k;l)(cid:2)d : (3)
XX kl
k=1l=1
In this paper, the designing problem of allocating the backbone links among the ATM switches
withtheobjectivethatminimizesthecostoflocationupdateunderthedegreeconstraintisstudied.
6
With the above de(cid:12)nition, the wireless ATM backbone network design(WABND) problem can be
formally de(cid:12)ned as follows:
Wireless ATM backbone network design (WABND) problem[10]: Given a graph H =
(S;F;z) with jSj = m, the m(cid:2)m matrix LUSS, and positive integers deg and p, deg (cid:20) jSj and
jSj (cid:0) 1 (cid:20) p (cid:20) jFj, the WABND problem is to (cid:12)nd a connected subgraph G(S;E;z) of H with
jEj = p, such that the location update cost m m LUSS(k;l)(cid:2)d is minimized and satis(cid:12)es
k=1 l=1 kl
P P
the constraint deg(G) (cid:20) deg, where d is the minimal communication cost between s and s on
kl k l
G(S;E;z).
2.3 ILP formulation
The WABND problem can be formulated as integer program. De(cid:12)ne F =1, if the link (i, j)
ij
between switches s and s is in graph H; F =0, otherwise. Note that, m m F = jFj, and
i j ij i=1 j=1 ij
P P
the value F is known and (cid:12)xed. Let x =1, if the link (i, j) between switches s and s is selected
ij ij i j
as a backbone link; x =0, otherwise. Let c be the cost of the link (i, j) in F. Let ykl = 1, if the
ij ij ij
shortest path between switches s and s pass the link (i, j); ykl = 0, otherwise.
k l ij
The following is an integer programming formulation of the problem:
m m
Minimize : LUSS(k;l)(cid:2)d : (4)
XX kl
k=1l=1
Thus the minimal communication cost d between switches s and s is de(cid:12)ned as
kl k l
m m
d = min c ykl;8k;l = 1;2;:::;m: (5)
kl XX ij ij
i=1j=1
(cid:0)1; if j = k
8
Xyikjl (cid:0)Xyjkql = >< 1; if j = l (6)
8i 8q 0; otherwise:
>:
ykl (cid:20) x ;8k;l = 1;2;:::;m: (7)
ij ij
ykl 2 f0;1g;8k;l;i;j = 1;2;:::;m: (8)
ij
7
m m
x = p; (9)
XX ij
i=1j=1
m
x (cid:20) deg;8j = 1;2;:::;m; (10)
X ij
i=1
x (cid:20) F ;8i;j = 1;2;:::;m; (11)
ij ij
x 2 f0;1g;8i;j = 1;2;:::;m; (12)
ij
The backbone network must be a connected graph, the reachability from one switch to any
switch is ensured this way. The graph G is partitioned into two connected parts, X and Y.
Obviously there must be at least an edge going from X to Y for each feasible solution. Otherwise,
the switches in Y cannot be reached. The reachability of graph G is stated as
x (cid:21) 1;8 partition X and Y of G: (13)
X ij
i2X;j2Y
There are 2m ways to partition a graph, so the number of constraints is in the order of 2m.
Constraint (13) is satis(cid:12)ed by all feasible solutions, and vice versa, any solution in which there
exist unreachable switch must violate at least one of them.
2.4 Related Works
Given the PCN, hando(cid:11) frequencies of cells in PCN, the connected ATM backbone network, and
the capacities of switches, Merchant and Sengupta[6] formulated the cell assignment problem and
considered the problem of assigning cells to switches (determined local links) in wireless ATM
network and several extended studies has been explored[7, 8, 9].
HuangandWang[3]investigatedthedesignproblemofallocatingthebackbonelinksamongthe
ATM switches with the objective of reducing the e(cid:11)ect of terminal mobility on the performance of
wireless PCN’s. The problem, which is called the Link Allocation Problem (LAP) on ATM-based
PCN, is known NP-complete [3]. Formally, given a graph H(S;F;z = 1) and positive integers deg
and p, deg (cid:20) jSj and jSj(cid:0)1 (cid:20) p (cid:20) jFj, the LAP is to (cid:12)nd a connected subgraph G = (S;E) of
8
H with jEj = p, such that the diameter(= max fd g) of the graph G is minimized and
8sk;sl2S kl
deg(G) (cid:20) deg. Two e(cid:14)cient heuristic algorithms were proposed in [3][12] to solve the LAP.
The Optimal Communication spanning Tree (OCT) problem [13] is similar to LAP problem
and de(cid:12)ned as follows. Let H(S;F;z) be an undirected graph with nonnegative edge length
function z. Given the requirements (cid:21)(s ;s ) for each pair of nodes s and s . For any spanning
k l k l
tree T of H, the communication cost of T is de(cid:12)ned as (cid:21)(s ;s ) (cid:2) d . The goal of OCT
P8sk;sl k l kl
is to construct a spanning tree T with the minimal communication cost. Like other constrained
spanning tree problem, OCT problem is NP-hard[14]. The current best approximation ratio for
the OCT problem is due to Yair Bartal’s algorithms which approximate arbitrary metrics by tree
metrics[15]. Recently, in [16], genetic algorithm with tree chromosome was developed to (cid:12)nd the
near-optimal solution.
The Degree Constrained Minimum Spanning Tree (DCMST) on a graph is the problem of
generating a minimum cost spanning tree with degree constraints. By reducing it to an equivalent
symmetric TSP, Garey and Johnson[14] showed the DCMST problem is NP-hard. Several studies
about DCMST on complexity, performance of approximation algorithms and worst case perfor-
mance guarantees were described in [17, 18, 19]. In [18], three heuristics for the DCMST, including
simulated annealing, a genetic algorithm and a method based on problem space search were pro-
posed. Branch and bound algorithms have been developed by Gavish[20] and by Narula and
Ho[21]. A Lagrangian-based algorithm has been developed by Volgenant[22], while Yamamoto
[23] developed an algorithm based upon (cid:12)nding the minimum common basis of two matroids.
Savelsbergh and Volgenant [24] used a branch and bound method based on Lagrangian relaxation
to solve the problem. Zhou and Gen [25][26] present an approach to solve DCMSTs using a genetic
algorithm. Their method used the concept of Pru(cid:127)fer number, which is also extended and used in
this paper.
Since the OCT problem is a special case of the WABND problem, therefore the WABND is
NP-hard[10]. That is, (cid:12)nding an optimal solution for it is impractical due to exponential growth
in execution time. In this paper, the WABND problem is studied and two heuristic algorithms
and a simulated annealing algorithm are proposed to solve it.
9
3 Heuristic Algorithm for WABND
In this section, two heuristic algorithms are proposed to solve the WABND problem, they are
remove-based heuristic (RBH) and weight-median-based heuristic (WMBH) algorithms.
3.1 Remove-based Heuristic(RBH)
Given network H(S;F), since p (cid:20) jFj, jFj(cid:0)p edges should be removed from H(S;F) to construct
thebackbonenetworkG(S;E). De(cid:12)nebridge edge beanedgewhoseremovedisconnectedH(S;F).
After removing edges from H(S;F), the graph G(S;E) should be a connected and constraint-
satis(cid:12)ed (degree-constraint and edge-constraint) graph. Obviously, bridge edges in H(S;F) can
not be removed. Moreover, if the connected components of graph H(S;F) are given, bridge edge
be the edge which connects di(cid:11)erent components. Obviously, the connected-component (cid:12)nding
algorithm (in O(jFj+jVj) time) and a sequential testing procedure (in O(jFj) time) can be used
to (cid:12)nd the set BE of bridge edges in H(S;F). Let CC = fCC ;CC2;:::;CC g be the set of
H H 1 q
the connected components of the H(S;F).
Letdegree (s )bethedegreeofswitchs inH(S;F). Foreachswitchs ,de(cid:12)neCON (s )=1,
H k k k H k
if degree (s ) > deg; CON (s )=0, otherwise. For each edge e = (s ;s ) in CC , de(cid:12)ne
H k H k k l H
CON (e) = CON (s ) + CON (s ). Moreover, edges in F (cid:0) BE are partitioned into three
H H k H l H
groups FREE , CON1 , CON2 according to the CON value of edges. That is, FREE ,
H H H H H
CON1 , and CON2 be the set of edges in CC , whose CON value is equal to 0, 1, and 2,
H H H H
respectively.
0 0
Initially, let H (S;F ) be the same as H(S;F). A non-bridge edge e = (s ;s ) is found in
k l
0 0
(CON2 0, CON1 0 or FREE 0) which minimize (cid:1) = opt(H (cid:0)e)(cid:0)opt(H (cid:0));8e 2 CON2 0.
H H H H
That is, if CON2H0 is nonempty, (cid:12)nd an edge e = (sk;sl) in CON2H0; otherwise if CON1H0 is
nonempty (cid:12)nd an edge e = (sk;sl) in CON1H0; otherwise (cid:12)nd an edge e = (sk;sl) in FREEH0.
0
Then, remove e from H . Since edge e should be in CCi (cid:18) CCH0 for some i, remove e from
0
H may make the CC change. The component CC may be divided into serval smaller connected
i i
components and some new bridge edges may be generated. These may change BE 0 and CC 0;
H H
moreover, the sets FREE 0, CON1 0 and CON2 0 should be updated. The edge-removing
H H H
10