a hybrid genetic algorithm approach for ospf weight setting problem

2nd POLISH-GERMAN TELETRAFFIC SYMPOSIUM PGTS 2002

9th Polish Teletraffic Symposium

A HYBRID GENETIC ALGORITHM APPROACH FOR OSPF WEIGHT SETTING PROBLEM

Eueung Mulyana, Ulrich Killat

Department of Communication Networks, Technical University Hamburg-Harburg Denickestrasse 17, D-21073 Hamburg, Germany

phone: (+49) 40-42878-2925, fax: (+49) 40-42878-2941 E-mail: [email protected]

Abstract In this paper, a method based on hybrid genetic algorithm (GA), is presented to optimize administrative weights for OSPF routing. This method can be seen as an alternative to the local-search method in [1] or another GA-based method in [8,10]. However, the GA as well as the objective function we use are different. Instead of minimizing a convex cost function we prefer to minimize the maximum and average utilization directly. This objective function is similar as one proposed in [6,7]. In addition we combine the population-based search feature of the GA with a search heuristic representing an individual-based search method and propose a new objective function to minimize weight changes for an existing operational network. We will demonstrate our method in a case of a small network and compare the results with MIP-based (Mixed Integer Programming) method from [6,7]. Afterwards we will show the results for a bigger network with increasing traffic demands, compared with the results of some conventional weight settings as well as with the lower bound of general optimal routing (linear program / LP solution).

Keywords OSPF, genetic algorithm, traffic engineering, routing optimization

1. INTRODUCTION

OSPF Routing. OSPF (Open Shortest Path First) is the most popular interior gateway protocol (IGP) used in today's IP networks. OSPF calculates routes as follows. Each link is assigned a dimensionless metric, called cost or weight. The cost of a path is the sum of link costs. Paths are selected using Dijkstra's shortest path algorithm. Given a network topology and predicted traffic demands, the OSPF weight setting problem is to find a set of OSPF weights that optimizes network performance. The chosen arc weights determine the shortest paths, which in turn completely determine the routing of traffic flows, the loads on the arcs and the


value of the cost function. In the case of multiple shortest paths, some vendors have implemented OSPF with ECMP rule (Equal Cost MultiPath) so that it will use load balancing and split the traffic flow over several shortest paths roughly evenly.

Problem Formulation. A directed network G(V,E) is given, where V is the set of vertices (nodes) representing the network's routers and E is the set of edges (arcs) representing the network's links. Each link (i,j)∈E has a capacity cij. Furthermore, we have a demand fuv for each pair (u,v)∈V×V, giving the demand to be carried from source u to destination v. A real variable lij

uv is associated with the load on link (i,j) resulting from flow demand fuv. Thus for each source destination pair (u,v) and link (i,j), the variable lij

uv=0 if link (i,j) is not on a shortest path from u to v and because of even-splitting lim

uv=linuv if both (i,m) and (i,n) are on shortest

paths from u to v. As mentioned earlier, in OSPF routing we choose a weight wk for each arc, where k is a vectorized version of matrix index ij where i≠j and cij≠0. For a given demand fuv, ∀(u,v)∈V×V and a given set of weights we can compute the load distribution in the network. Note that this “computed” load distribution is not dependent on link capacities i.e. some weight configurations may cause a congestion as the total traffic to be routed on a certain link may exceed its capacity. In this case for the link utilization ( )∑ >

uv ijuvij cl 1/ holds. The optimization is

performed with no constraints to force a solution to have a utilization below 1, but we simply minimize a certain cost function. The desired result then is a set of weights which corresponds to the minimized cost function.

General Routing Problem. We will compare our results with a solution from so-called General Routing Problem [1] as lower bound. In this general routing problem, there are no limitations on how flows can be distributed along the paths from source to destination, and the problem can be formulated and solved in polynomial time as a multi commodity flow problem. Thus, the general routing problem optimization can be formulated as follows:

VVvuEji

cl

Eta

ij uv ij

uvij

t

×∈∀∈∀

+⋅ ∑∑

),(,),(

1)(min

(1)

VmnVVvu

lflfVm Vm

uvnmuvnv

uvmnuvun

∈∀×∈∀

+=+ ∑ ∑∈ ∈

,,),(

δδ (2)

Ejitcl

uv ij

uvij ∈∀≤∑ ),(, (3)

A Hybrid Genetic Algorithm Approach for OSPF Weight Setting Problem

VVvuEjil uvij ×∈∀∈∀≥ ),(,),(,0 (4)

Equation 1 is the objective function to minimize utilization t on the most utilized link (Eq. 3) and average utilization. A constant at is used to trade between these two components. Eq. 2 describes flow conservation constraints that ensure the desired traffic flow to be routed from source to destination.

2. MINIMIZING WEIGHT CHANGES

Weight changes have to be flooded in the nework, thus it should be avoided as much as possible for an operational network [2]. As the routers learn about the changes, they recompute their shortest paths to update their routing tables. The more weight changes we try to flood simultaneously, the more chaos we introduce in the network with packets being sent back and forth between routers. Thus a modification of administrative metric values used by OSPF or generally IGP is not desirable too often and should be confined to a medium or long term basis [11]. Also, if we want to modify the metric values to optimize network performance, it is worth to change only an amount as small as possible. In the following we introduce a different version of the objective function in order to minimize the changes to be performed.

+⋅ ∑

∈Ekkt y

Eta

1)( (5)

rkk

rkk

k wwww

y=≠

=,,

01

(6)

The last term in (5) measures similarities between current weights' configuration as reference ⟨w1

r, w2r, … , wk

r, … , w|E|r⟩ and a new configuration to

be evaluated ⟨w1, w2, … , wk, … , w|E|⟩.

3. A HYBRID HEURISTIC/GA (HGA) APPROACH

In order to apply a genetic algorithm, generally a suitable encoding of possible solutions in a vector (i.e. chromosome) representation is needed. In our case a chromosome is represented by a set of link weights ⟨w1, w2, … , wk, … , w|E|⟩ where wk∈[1, MAX] for each edge k= 1, ... ,|E| and the maximum value for MAX is 65535. Each chromosome has a fitness value according to Eq. 1 or Eq. 5 and corresponds to a certain link utilization resulting from a load distribution computed by Dijkstra's shortest path algorithm.


Figure 1 : The hybrid genetic algorithm (HGA) proposed

Figure 1 shows a block diagram from our hybrid Heuristic/GA implementation. The loop differs from the ordinary chain of selection, reproduction and mutation by featuring an additional block called "heuristic search". We begin with a randomly generated initial population of 50 chromosomes. At the beginning of each iteration some vectors of high quality are selected to produce new, hopefully better solutions. After this selection process there follow two genetic mechanisms "reproduction" and "mutation" to form some new chromosomes. Afterwards we construct the next population by substituting the least successful chromosomes of the previous iteration by the new ones. In parallel a heuristic search will be performed by inspecting always the best chromosome of a population provided that the fitness has been improved in the last iteration. We simply add (substract) a random number to the weight wk if the link utilization from an arc k is bigger (lower) than a particular treshold because we know that the bigger the weight, the lower the chance that traffic will get routed on that link and vice versa. The search process will be terminated if there are no more improvements for a given number of iterations. This heuristic is also performed to mutate the new chromosomes resulting from the reproduction process (depicted in Figure 2).

4. RESULTS

Minimizing changes. Figure 3 shows the results with the modified cost function (5). The figures at the top show the "current" network, in terms of the weight configurations (left) based on inverse capacities and the resulting link utilization (right). With this configuration we have an average utilization of 22.4% and a maximum utilization of 42.9%. The figures at the bottom show the optimized

yes

no Start

Population Exit Condition

Heuristic Search

Selection

Reproduction Mutation Add new

Population Selection


network configuration, with an average utilization of 22.7% and a maximum utilization of 35.7%; this is identical to the results from [6,7]. And this achieved by only 4 link costs’ changes namely (2,1), (3,4), (4,5) and (5,6).

Figure 2 : Forming a new generation (a) and population dynamics (b)

Convergence. In the following figures we will demonstrate the convergence characteristic of the proposed HGA. Figure 4 left shows the convergence of average fitness in the population. Figure 4 right shows the fitness convergence of best chromosome found in the population. The network used for this simulation is the AT&T network adopted from [12] and consisting of 29 nodes and 100 directed arcs with randomly generated traffic demand. Curves (1) result from 10 independent program runs, using only the reproduction strategy from [8] representing a standard GA and curves (2) are from our HGA by combining reproduction, additional mutation and a search heuristic. The average fitness from the population in our HGA-scenario converges faster than the other one. And of course if the average fitness in a population is better, then the chance that we have a better solution is also bigger.

Reproduction

„Targeted“Mutation

(a)

Best chromosome

Population 50 chromosomes

Selection (parents)8 chromosomes

Selection (remove 10%)

Population 45 chromosomes

Offsprings8 chromosomes

Search result (1 or 0 chromosome)

Population 53 or 54 chromosomes

Selection (best 50 chromosomes)

(b)

Heuristic Search

ReproductionMutation

P1 P2

O1 O2


Figure 3 : Result of minimizing changes

Figure 4 : Average population fitness and best chromosome fitness

Increasing traffic. Finally we present the results of our method for the case of increasing traffic demands. In Figure 5 we compare maximum utilization based on some common metrics to our results and to a lower bound resulting from a linear programming result for the general routing problem. In Figure 5 left we used Eq. 1 as objective function. In Figure 5 right we used the second objective function (Eq. 5). For comparison we considered hop count metric (denoted by UnitOSPF) and


inverse capacities metric (denoted by InvCapOSPF). HGAOSPF-X or HGAOSPF-X(Y%) denotes the result for X iterations and Y% relative changes to be performed from reference weight settings i.e. the last term of Eq. 5. For both types of calculations, the value of MAX was set to 99. The figures show that in comparison with InvCapOSPF and UnitOSPF, we can increase the network capacity by factors of about 35% and 500% respectively.

Figure 5 : Result of increasing traffic (AT&T model network consisting of 29 nodes and 100 arcs)

5. CONCLUSION

We have considered the problem of OSPF or generally IGP link costs setting with a hybrid genetic algorithm, using the objective function from [6,7] and proposing a new one in order to minimize changes to be made for an operational network. We proposed in addition a mutation and search heuristic to improve solution quality and to speed up convergence. The results show that our algorithm can find the same solution as the MIP method does [6,7] and by applying the second objective function we can minimize the changes and get almost the same result as from the first objective function. The application of our method to bigger networks as reported on in [1,8] is subject of our future research.

6. BIBLIOGRAPHY

[1] B. Fortz, M. Thorup: Internet Traffic Engineering by Optimizing OSPF Weights, Proc. IEEE Infocom, March 2000


[2] B. Fortz, M. Thorup: Optimizing OSPF/IS-IS Weights in a Changing World, to appear in IEEE JSAC Special Issue on Advances in Fundamentals of Network Management, Spring 2002

[3] D. Beckmann: Algorithmen zur Planung und Optimierung moderner Kommunikationsnetze, Dissertation, Technical University Hamburg-Harburg, 2001

[4] D. Beckmann, U. Killat: Routing and Wavelength Assignment in Optical Networks Using Genetic Algorithms, European Trans. Telecommunications, Vol. 10, Nr. 5, S. 537-544, September 1999

[5] D. Beckmann, U. Killat: Planning of Survivable ATM Networks Based on the Virtual Path Concept, In 2nd Int. Workshop on the Design of Reliable Communication Networks, S. 166-173, April 2000

[6] D. Staehle, S. Koehler, U. Kohlhaas: Optimization of IP Routing by Link Cost Specification, University of Wuerzburg, 2000

[7] D. Staehle, S. Koehler, U. Kohlhaas: Towards an optimization of the routing parameters for IP networks, Technical Report 258, University of Wuerzburg, May 2000

[8] M. Ericsson, M.G.C. Resende, P.M. Pardalos: A genetic algorithm for the weight setting problem in OSPF routing, to appear in J. of Combinatorial Optimization, preprint October 2001, http://www.research.att.com/~mgcr/papers.html

[9] P. Gajowniczek, M. Pioro, A. Szentesi, J. Harmatos, A. Juttner: Solving an OSPF Routing Problem with Simulated Allocation, First Polish-German Teletraffic Symposium, PGTS 2000

[10] P. Karas, M. Pioro: Optimisation Problems Related to the Assignment of Administrative Weights in the IP Networks' Routing Protocols, First Polish-German Teletraffic Symposium, PGTS 2000

[11] W. Ben-Ameur, N. Michel, B. Liau: Routing Strategies for IP-Networks, available at http://puck.nether.net/lists/irtf-rr/att-0029/.

[12] AT&T Backbone Network, available at http://www.ipservices.att.com/backbone/

a hybrid genetic algorithm approach for ospf weight setting problem

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