Computational analysis of an optimal routing strategy in a hierarchical computer communication network Debashis Saha* and Amitava Mukherjee+

The computational analysis of an optimal hierarchical routing to choose the best communication path in a large network is demonstrated in this paper. A scheme is also provided for selecting the optimal route in a hierarchical computer commu- nication network. The problem of optimal routing in a hierarchical network is formulated as a non-linear combina- torial problem solved using the Lagrangean relaxation and subgradient optimization techniques. A simulation study on application of the technique to a number of networks is presented. The results provide validation of the technique as well as its usefulness in network design.

Keywords: computer communication networks, hierarchical design, routing, subgradient optimization, clustering

A computer communication network (CCN) intercon-

nects a number of geographically distributed computers by means of communication links. The design of a

backbone network’ ~3 . IS a major consideration in the topological design of a CCN. Given a number of host

sites, traffic between every pair of sites, and the required node connectivity, the design problem is to find a

suitable network topology at a minimum cost. The complete design consists of two logically related subpro-

blems, viz. topological design’-3 and optimal routing4-s.

This paper addresses the second subproblem, optimal routing, for a hierarchical computer communication network’-‘*. It also provides a solution strategy with

the help of the Lagrangean relaxation and subgradient optimization technique2* 3*9.

Routing in CCN involves a collection of algo-

rithms”--‘6. First, routing requires coordination among

all the nodes in the network; second, the routing system must contend with the link and node failures, necessi-

*Department of Computer Science & Engineering, Jadavpur Univer- sity, Calcutta 700 032, India +Department of Electronics & Telecommunications Engineering, Jadavpur University, Calcutta 700 032, India Paper received: 3 Novcmher 1993; revised paper received: 7 March 1994

tating redirection of traffic and a subsequent update in

topology; third, to achieve high performance, the

routing algorithm may need to modify its routes when

some areas within the network become congested. All

three requirements are dependent on the reliabilities of the nodes, since a failure in the system can lead to a

disaster. The existing algorithms4 ‘, which address the

above-mentioned three requirements of the routing problem, are polynomial in nature. Moreover, if the backbone network interconnects a large number of

nodes, the size of the routing tables and the time taken to compute them by different routing algorithms

increases considerably. To tackle the situation, hierarch-

ical design 9 lo is used with a view to reducing the

complexity of the problem.

The nodes are grouped and interconnected to form clusters or subnetworks. At least one node is every cluster is identified as a gateway’ for that subnetwork.

If a message is destined to go out of a cluster, it must

be routed through one of the gateways. The gateways are also called intecface nodes or simply gate nodes. Each subnetwork at level H, having one or more

interface or gate nodes, is considered to be a single node of its immediate upper level (H + 1). The nodes at

level (H + 1) are again grouped into clusters, which act as nodes of level (H + 2), and so on. This procedure

stops at the level where all nodes come under a single cluster. Routing, based on this model, is called

hierarchical routing43 5. lo* ’ ‘, and usually involves multiple levels.

At each level of the design, there may be separate

routing algorithms for individual clusters that handle local intracluster traffic. On the other hand, remote intercluster traffic from every cluster is routed through its interface node(s). So, there will be another routing algorithm that determines the sequence of clusters and interface nodes that each intercluster traffic must follow. Basically, the low-level algorithms handle local (or intracluster) traffic, and the high-level algorithms

0140-3664/95/$09.50 0 1995-Elsevier Science B.V. All rights reserved

computer communications volume 18 number 7 july 1995 507

Optimal routing strategy: D Saha and A Mukherjee

control long-distance (or intercluster) traffic. Hierarch- ical routing is generally used for long-distance traffic in large network8 ‘, “* ’ ‘.

The complexities of routing algorithms are reduced significantly when the hierarchical scheme is used in a multilevel network design. In the case of a non- hierarchical approach, the size of routing tables and the computations required to update them become enormous, even for a moderate sized network, when the shortest path calculation is performed. Not only are these calculations minimized significantly in hierarchical routing, but also the complexities of other design algorithms in routing problems are reduced appreci- ably. It has been shown that the optimal number of levels for an N node network is In(w8. It has also been proved that the increase in effective mean path length caused by hierarchical routing is not objectionably large.

Once the design of the hierarchical backbone network’ has been accomplished, the routing of the messages between nodes will be as follows. The outgoing message from a node is first forwarded to the gate node of the cluster to which the node belongs. After identifying the intercluster nature of the message, the gate node routes them to the gate node of the supernode to which it belongs, and this process continues until the message reaches its proper ancestor gate node. The proper ancestor is determined by the source and destination addresses of the message. From this ancestor, the message starts its downward journey until it reaches the level-l gate node. From this last gate node, the message is distributed to the destination node.

This paper is organized as follows. The next section proposes a hierarchical scheme for a computer commu- nication network, followed by a section that defines the problem and discusses its solution. The subsequent section presents the computational result and summarizes the observations; the final section concludes the paper.

HIERARCHICAL SCHEME FOR CCN

We define a topology that possesses a hierarchical clustering structure. The hierarchical structure is obtained by partitioning the design problem into a set of smaller subproblems, with the aim of introdu- cing independence into the set of design variables of the subproblems. This independence will reduce considerably the computational complexity of the complete design problem’. The partitions of the problem are obtained through clustering of the set of nodes.

The hierarchical topology characterizes a network structure realized through a recursive interconnection of smaller networks. In a general H-level topology, the subnetworks that have (j- I)-level clusters as their members are referred to as j-level clusters. The actual network nodes are O-level clusters. Each j-level

cluster is itself a j-level topology, for all jE {1,2,... , H}. The j-level topology is created by interconnecting a collection of distinct (j - I)-level topologies. Applying this rule recursively, any H-level hierarchical topology can be constructed for any integral value of H 2 1.

The proposed hierarchical scheme is based on the fact that all intercluster traffic must pass through the gate nodes. As discussed earlier, for each node set (or cluster) at any level, there exists a set of gate nodes, called a gate set for that cluster at that level. The ith gate set in the jth level is defined as the subset of nodes (or set of gate nodes) taken from the ith node set (or cluster) of the jth level. They are directly connected to the gate nodes of some other pth node set at the same jth level, if both ith and pth node sets belong to a common (j + 1) level node set and p # i. The elements of all the gate sets of the jth level are called the jth level gate nodes.

The traffic between nodes in the ith node set at the jth level takes paths which are internal to that node set, i.e. paths that are contained in the corresponding ith node set. For the traffic between two different node sets, routing is done through gate nodes. For example, the traffic transmitted from an originating node in the ith node set at the jth level first reaches the gate node of that node set. It then takes the path connecting the gate node of the ith node set to the gate node of the pth node set at the jth level. It is then transmitted from the gate node of the pth node set to the destination.

PROBLEM FORMULATION AND SOLUTION

The formulation of the problem is given as follows:

Given. l a set of nodes, l interconnection among nodes, l traffic matrix, i.e. the trafftc flows from each

node to every other node, 0 the number of gate nodes per cluster, l the computational cost parameter that charac-

terizes the cost of evaluating the routes used by the communicating nodes, and

l the computational cost is defined in terms of computing resources expended to calculate the route’*.

Design objective: l minimize the route computation cost of a

hierarchical computer communication network which has clusters of about equal size at the same level.

We formulate the routing problem for a hierarchical computer communication network as a non-linear combinatorial optimization problem that can be solved using Lagrangean relaxation and subgradient optimiza- tion techniques.

508 computer communications volume 18 number 7 july 1995

Optimat routing strategy: D Saha and A Mukherjee

Notation Lagrangean relaxation

N: total number of nodes H: number of levels in the hierarchy nk: average size of cluster (i.e. number of nodes in the

cluster) at kth level, k E [l, Hj

bk: number of gate nodes per cluster at kth level c(k: computational cost parameter for kth level ;1: Lagrangean multipliers G: Total design cost of route computation.

The design discussed in this paper is called a non- uniform design. A non-uniform design strategy signifies that the number of gate nodes in each gate set at all levels of the hierarchy is different. Correspondingly, the value of cri is also different for all j, {j E 1, . . . , H}.

We remember that N number of O-level nodes are first grouped into 1st level clusters, then 1st level clusters are again grouped into 2nd level clusters, and so on. The repetition of this process results in a hierarchical structure which includes all lower level clusters.

The sizes of clusters in the hierarchy are represented by a size vector n = (n’, n2,. . . , nH). The product of all elements in the size vector gives the size of the network, i.e. (n’ x n2 x . . . x nH) = N. In the zeroth level all nodes are gate nodes. We also consider a gate vector /I = (/?, , p2,. _ , pH). For the 1st level of the hierarchy, 8, number of gate nodes are selected and, since all nodes of zero level are gate nodes, then p, = 1. In the 2nd level, the selection of gate nodes satisfies the conditions: b2 < B’n’. In this way, we follow the same rule of selection of gate nodes as for other higher levels of the hierarchy. The gate nodes are used for interconnection of the clusters at every level of the hierarchy.

The computational cost of the 1st level in the hierarchy is ny’n2.. . nH and that of the kth level in the

hierarchy is (nkfik)‘k fi nj, j= I

The above discussion is now expressed in terms of mathematical formulation. This formulation resembles a non-linear optimization problem given as:

Problem (P):

Minimize G = ny’ fi n, + 5 (nib;)” fi ni i=2 i=2 j=i+ I

+ hYBHYH

Subject to:

ni = N i= I

&,I dn,Pi (i E I,H- 1) (2)

Constraint (1) is known as the size constraint, and constraint (2) is called the gate constraint.

The solution of P is obtained by relaxation and subgradient optimization, as briefly discussed below.

The Lagrangean relaxation L(1) of the problem (P) subject to constraint (2) is as follows:

It can be solved using the subgradient optimization technique’. We simply present the solution here, and for detailed derivation the reader is referred to Appendix 1.

Solution of the problem

n, = (-1, /~,)‘/(~~ - ‘)

i-l i- I

ni = -1; n nk - ny’ n nk

k=l k=2

_ g (nkflk)“k E nk}/@;pT] “‘‘-’ where i > 2

To help the reader understand the implication of n’ and ni, let us take recourse to a simple situation. This will provide an insight into the usefulness of the results obtained in the subsequent section. We consider a small 2-level network of 10 nodes (i.e. N = 10 and H = 2) with one gate node per cluster at all levels (i.e. /3, = /I, = I). Assuming GIN = IX~ = 3, the Lagrangean multipliers Al and J2 are adjusted during subgradient optimization to generate an optimal solution:

n’ = (-A’/a’)“‘I-’ = 5.05 and

n2 = K-A2nl)la2P2 1 a2 I/(‘% - ‘1 = 2.06

The above values of nl and nz indicate that, in the first level, there will be (N/n,) = (10/5.05) N 2 clusters, each having five nodes, one of which is a gate node. In the second level, there is a single cluster of two nodes, one of which will be a gate node. Since this is a 2-level design, the routing algorithm stops here. So, strictly speaking, the value of n2 is redundant for the routing purpose in this case.

Again, if we take N = 22 and H = 3, keeping a; and p, unchanged, the optimal solution is obtained as:

n’ = (-l,/a’)“‘I-’ = 3.78

n2 = [(-i2n’)/cr2~2a’]“(a’ ‘) = 2.91 and

n3 = [(( -Agz’nz - (n2P2)“‘)/a~P3”‘]““’ _ ‘) = 2.06

So the design will be as follows. At the lowest level, 22 nodes will be divided into (22/n,) or six clusters, two of which will contain four nodes apiece while the rest will contain three nodes per cluster. Each cluster will have

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Optimal routing strategy: D Saha and A Mukherjee

one gate node. In the second level, those six gate nodes will be divided into (6/nz) or two clusters, each having three nodes and one gate node. Since it is a 3-level design, the routing algorithm can be stopped at this point as the value of n3 is not essential to the design. Rather, it verifies the optimality of the solution, i.e. (n’ x n2 x n3) 21 N. More results on larger networks can be found elsewhere.

The best lower bound is provided by the vector J.* that corresponds to L(n*) = max L(1). The condition

160

L(A*) = G,‘, states that the relationship that exists between L(,J*) and the continuous relaxation of problem P, where Gm’n is the objective (cost) function value obtained for problem P when the variable n; is allowed to take a fractional value. The details are given elsewhere”.

COMPUTATIONAL RESULTS AND OBSERVATIONS

Extensive numerical experiments have been performed with the problem posed in this paper. Several different topologies are used in computational experiments with this optimization problem”. In all cases, the number of nodes, the levels of hierarchy, the gate vector for the hierarchy and the computational parameters are provided. Some of the results are presented in Table 1.

The experiments are conducted with two purposes in mind: first, to test the performance of the cost optimization problem; and second, to examine the impact of various parameters, especially the computa- tional parameter, on the solutions generated.

Table 1 shows the results obtained in a network of 93 nodes for different values of c1 (all ui)s are assumed identical and equal to a). The plot of G,‘” versus u is shown in Figure

1. Since the computational parameter CI dominates the other parameters, an increase in CI always results in a higher computational cost of the network (Figure I).

CONCLUSION

A large network will certainly have a number of possible paths between any two nodes. Therefore, there is a need to choose the best route by which to transmit messages. This necessitates a suitable routing algorithm, of which there are many alternatives”-‘5. In selecting a routing algorithm, there are many factors, such as ease of use and performance, that must be considered. A hierarch- ical routing strategy for a large CCN has been proposed

Table I Computational results

DL N P Gmin

3 93 {I, I,11 6411.13 4 93 {I, I,11 33179.17 5 93 {l, 1.1) 184991.58 6 93 {l,l, 1) 962857.20

962857.1250

771567.9375

._ E’ 580278.7500

“’ 388989.5625

“7700~-~ a

Figure 1 G,,, versus OL

in this paper, and its computational efficiency has been thoroughly examined to verify its optimality.

It is important to note that, given the complexity of the original problem, the subgradient procedure is very effective in narrowing the gap between the lower and upper bounds. This justifies the use of the Lagrangean technique in this case. Moreover, for the cases when certain conditions on the values of input parameters hold true (for example, no significant variance among the elements of the gate vector), it is possible to further improve upon the quality of the solution because the Lagrangean bound incorporates this restriction as part of the solution procedure. The subgradient optimization procedure, used to estimate A’, the vector multipliers corresponding to the Lagrangean, provides the tightest lower bound always.

APPENDIX

Main steps of derivation

If we take H = 3, the simplified form of Lagrangean becomes:

L(ni,Jbi)= $’ n2n3 +(n2fi2)" n3 +(n3/33)"'

(Al)

+ A;(nl n2n3 -N) where i = I,3

Differentiating (Al) w.r.t. n’ and making the 1st order partial derivative equal to 0, we get:

aL aI-’ +I’ =() an, = a’n’ Solving this equation, we get:

n’ = (-+‘)‘/@ - ‘)

Again, differentiating (Al) w.r.t. n2 and taking the 1st order partial derivative equal to 0, we get:

aL -=na’n3+~~-’ an2 I ni'B2n3+A2nln3 =0

Solving this equation, we get:

n2 = (-AIn’ /(i~@))“(~~ _ ‘)

In this way differentiating the general expression of L(n;, A;) partially w.r.t. nj (i E I, H), we arrive at the

510 computer communications volume 18 number 7 july 1995

general form of ni that is already given in the main paper.

REFERENCES

Chattopadhyay, N G, Morgan, T W and Raghuram, A ‘An innovative technique for backbone network design’, IEEE Trans. SMAC, Vol 19 No 5 (September/October 1989) pp 1122-1132 Saha, D, Mukherjee, A and Dutta, S K ‘Design of computer communication networks under link reliability constraints’, Proc.

IEEE Tencon ‘93, Beijing, China (October 1993) Saha, D, Mukherjee, A and Dutta, S K ‘Capacity assignment in computer communication networks with unreliable links’, Comput. Commun., Vol 17 No I2 (December 1994) pp 871-875 Gavish, B and Hantler, S L ‘An algorithm for optimal route selection in SNA networks’, IEEE Trans. Commun., Vol 31 No lO(l983)pp 1151-1160 Klienrock, L and Kamoun, F ‘Hierarchical routing for large networks performance evaluation and optimization’. Comput.

Networks, Vol I (1977) pp 155-174 Bertesekas, D P ‘A class of optimal routing algorithms for communications networks’, Proc. Int Cotzf. Circuits Computer,

Atlanta, GA (November 1980) Gallager, R G ‘A minimum delay routing algorithm using

8

9

IO

II

I2

I3

I4

I5

16

I7

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distributed computation’, IEEE Trans. Commun., Vol 25 (1977) pp 73-85 Kamoun, F and Klienrock, L ‘Stochastic performance evaluation of hierarchical routing for large networks’, Comput. Networks, Vol3 (1979) pp 337-353 Saha, D, Mukherjee, A and Dutta, S K ‘Design of hierarchical computer communication network under node/link failure constraint’, Comput. Commun. (to appear) Garcia Lunq, J J ‘Routing management in very large scale networks’. Furure Generation Comput. Sj>st.. Vol 4 (1988) Tsuchiya, P ‘The landmark hierarchy: a new hierarchy for routing in very large networks’, Proc. ACM SIGCOMM (1988) Dandamudi. S P and Eager, D L ‘Hierarchical interconnection networks for multicomputer systems’. IEEE Trans. Cornput.. Vol 39 No 6 (June 1990) pp 786797 Krishnan, K R ‘Markov decision algorithms for dynamic routing’, IEEE Commun. Mug., Vol 28 (I 990) pp 6669 Watanabe. Y and Oda. T ‘Dynamic routing schemes for international networks’, /EEE Commun. Mug.. Vol 28 (1990) pp 70-75 Key, P B and Cope, G A ‘Distributed dynamic routing schemes’, IEEE Commun. Mug., Vol 28 (1990) pp 5664 Bertseks, D and Gallagar, R Datu Networks. Prentice Hall. Englewood Cliffs, NJ (1987) Saha, D, Mukherjee, A and Dutta. S K An optimal routing in u

hierarchical computer communication network. Technical report, Department of Computer Science, Jadavpur University, India (August 1993)

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