Logical Embeddings for Minimum Congestion Routing in Lightwave Networks

The problem considered in this paper is motivated by the independence between log ical and physical topology in Wavelength Division Multiplexing WDM based local and metropolitan lightwave networks This paper suggests logical embeddings of digraphs into multihop lightwave networks to maximize the throughput under nonuniform tra c conditions De ning congestion as the maximum ow carried on any link two perturbation heuristics are presented to nd a good logical embedding on which the routing problem is solved with minimum congestion A constructive proof for a lower bound of the problem is given and obtaining an optimal solution for integral routing is shown to be NP Complete The performance of the heuristics is empirically analyzed on various tra c models Sim ulation results show that our heuristics perform on the average from a computed lower bound Since this lower bound is not quite tight we suspect that the actual performance is better In addition we show that performance improvements can be obtained over the previous work Introduction The vast optical bandwidth of a ber confronted with the electro optic bottleneck moti vates the e ort on employing concurrency and parallelism in optical networks as well as new architectures The Wavelength Division Multiplexing technique realized by tunable opti cal transceivers has been proposed as a way of constructing multichannel multihop lightwave networks A station in such a network has electronic and optical components The optical com ponents are transmitters and receivers used to tap into the optical medium by tuning to a special wavelength whereas the electronic components constitute the rest to control the operation of the switch We consider networks whose underlying physical topology is based on sharing a single ber e g a linear bus a tree or a star topology In this case potentially all the stations can reach the medium by tapping into it Pairing a transmitter on one node with a receiver on another one by tuning them to the same wavelength establishes a logical connection between these two stations Therefore allocation of the wavelengths to the stations constructs a logical embedding into the network independent from the physical topology Although sharing is possible we consider unique end to end assign ments of the wavelengths no channel sharing The logical embeddings are constrained by some design parameters such as the number of available transceivers or wavelengths We refer to for a recent survey on WDM based architectures In this paper we propose logical embeddings of directed graphs called con gurations for achieving high throughput under non uniform tra c conditions Candidate con gurations are generated by perturbation heuristics and the routing problem is solved on these con gurations by Linear Programming The routing problem is formulated as a Multi Commodity Flow MCF problem with the objective function to minimize the maximum ow congestion on an edge This paper organized as follows In the following section we de ne the problem and state our approach more formally In section the algorithmic issues involved with the perturbation process are addressed In section and we present our algorithms and their computational performance analysis respectively A constructive proof for a lower bound of the problem by using special spanning trees and some complexity results on the problem are presented in sections respectively Problem De nition and Approach Given a tra c matrix T ts t for N stations such that an entry ts t is the amount of average tra c to be sent from a source station s to a destination t let integer i denote the number of incoming and outgoing links for each station i De ne a con guration CONF as a strongly connected digraph without self loops such that each node has in out degree exactly d A con guration is a logical representation of a net work topology in which each node corresponds to a station and each edge to a logical physical connection Given a tra c matrix T and a con guration CONF de ne routing R as an assignment of the tra c ts t to directed path s on CONF from source s to sink t for all s t pair De ne congestion zi j on a directed edge i j CONF as the total amount of tra c carried on this edge for routing the tra c in T Let Z max i j fzi jg For a given T and d we are interested in nding a con guration CONF and a routing R to minimize Z while satisfying the tra c T Note that this problem has two parts how to nd a good embedding how to nd a good routing algorithm A uni ed solution can be obtained by solving the following mixed integer programming prob lem by de ning a unique commodity for a station k such that P j tk j Denote by f k u v the ow on edge u v of commodity k Then Min Z such that Z P k f k u v k pair u v P i f k i j P i f k j i tk j k j where k j f u v P j tk j xu v P j xi j P j xj i i i j xi j f i u v There are two variables of interest the ow variable f i j the integer variable xi j which is if there is an edge from i to j otherwise We show in section that for an integral tra c matrix T nding a con guration CONF and an integral routing of the tra c with minimum congestion on this CONF is NP complete The conjecture is that the problem is still NP hard with the integrality relaxation on the routing problem due to remaining integrality constraint on the edge variables Therefore we introduce two perturbation heuristics for approximate solutions based on Variable Depth Local Search and Simulated Annealing techniques respectively Each heuristic performs a greedy search in an exponentially large con guration space the set of all CONFs The direction of the search is guided by the value of a cost function which is the amount of maximum congestion Value of congestion is computed by solving the routing problem formulated as a multicommodity ow problem with Linear Programming LP on the current con guration The Subroutine LP Given a con guration CONF the tra c matrix T and the degree bound d we formu late the problem of nding minimum congestion routing as an instance of Multicommodity ow problem MCF In this formulation each commodity corresponds to a source node leading to polynominal number of rows and columns in the Simplex Tableau Precisely de ne as a commodity a station k such that P j tk j Denote by f u v the ow on edge u v of commodity k Then Min Z such that Z P k f k u v commodity k and edge u v P u f k u v P u f k v u tk v k v where k v f u v The output of the LP solution is the minimized maximum congestion Z and the ow assignment f u v for all the edges u v and commodities k Although we relaxed the integrality constraint for the routing problem note here that if the entries of the tra c matrix are large real numbers a good rounding of the ow to obtain an integer solution to routing problem does not substantially change the value of the congestion Therefore LP solution approximates to the optimal integral routing as the tra c matrix consists of larger numbers Starting with an initial con guration the heuristics perform the following three basic steps Ask Subroutine LP for the value of maximum congestion for the current con guration Determine the direction of the search Modify the current con guration to obtain a new one Termination of the heuristics can be realized either externally by an input parameter or internally when no more decrease is achieved on the value of the cost function Choice of a starting con guration is important for nding a local optimum solution Starting points can be generated randomly as well as with greedy algorithms We give an empirical comparison of various starting points in section A candidate con guration for which the routing problem can be solved with less congestion is accepted and the search of the con guration space continues from that point An accepted CONF is modi ed to obtain a new candidate which is called valid if it maintains the CONF property Maintenance of CONF property is realized with a validity test prior to the perturbation operation which determines whether or not the CONF property would be destroyed after the operation The modi cation of the current con guration is achieved by perturbation operation which is based on a n change operation A set of edges with cardinality n is randomly chosen to be replaced by a new set of edges with the same cardinality to obtain a new con guration In other words one of the con gurations that can be obtained from the current one by n changes is chosen randomly Choice of perturbation determines the direction and the step size of the search For instance big changes cause big jumps in the search space therefore if the space has narrow valleys a nearby local optimum could be missed On the other hand small step size may require longer search process to nd a local optimum In the following section we introduce two di erent perturbation operations and then compare their performance in section Perturbation and Maintenance of CONF property In this section we present two perturbation operations node and edge perturbation and show how to test the validity in advance Node perturbation case change Two distinct nodes are randomly chosen among N nodes and their out going edges are exchanged as follows Choose randomly with probability N N a distinct pair of nodes u v from the con gu ration see gure