Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

On the Interaction of Bandwidth Constraints and Energy Efficiency 213source to destination (if we care only about particular source-destination pairsand not about the connectivity of nodes that have no traffic to send or receive).Hence, too low a τ i can result in a disconnected topology.For the sake of exposition, we will consider the problem of finding pathsbetween source-destination pairs in static wireless networks. We will be given atraffic matrix which indicates the traffic load between each source-destinationpair. The load is asymmetric (that is, the load from A to B is not necessarily equalto the load from B to A) and is non-zero for all source-destination pairs (butcan be arbitrarily small). Each source-destination pair will subsequently haveto be routed using multiple intermediate hops. If no capacity constraints arepresent, a source-destination pair can pick the lowest energy cost from source todestination using a conventional shortest path algorithm. That is, the shortestpath algorithm can be applied on a graph in which the costs stand for thedistance between the nodes raised to the loss exponent. For a single sourcedestinationpair, and if no other traffic or bandwidth constraints existed, thatwould be the optimum solution. We note that the unicast energy minimizationproblem is in fact a shortest path problem, compared to the multicast/broadcastenergy minimization problem, which is known to be NP-hard [6].Unfortunately, the general case of the problem (multiple source-destinationdemands, capacity constraints and wireless broadcast nature) is sufficiently complexto defy currently a simple answer. We note that the first two features (multiplesource-destination demands, capacity constraints) render the problem a caseof the node-capacitated multi-commodity flow class of problems. That is, eachnode has a capacity as a constraint while each node pair need to established aflow and deliver amount of traffic concurrently in a network. This problem is alsoshown to be NP hard [7], and many approximation algorithms have been proposed.Unfortunately, the literature on the topic does not consider the broadcastnature of the wireless medium, and thus the fact that τ i is not just a functionof the traffic intended to be received by i but also of the traffic of other nodesthat are “near” i - i.e., it depends on the geometric features of the topology. It isthus not surprising that results on node-capacitated multi-commodity flow arespecific to particular topologies, e.g., rings [8]. The general case appears to bean extremely complex case, even without the presence of mobility.We point out that several versions of the basic problem exist. For example,transmission and reception by a node are lumped into one capacity constraintonly. Clearly, separate channels can be used for transmitting and receiving, withseparate fixed capacities. Secondly, we assume knowledge of the traffic loadsbetween all source-destination. If the traffic load matrices present only knowledgeof the average load.2 AlgorithmsLet us assume that we have been given the costs D[u][v] between any two nodesu and v. The costs are determined as the Euclidian distance between the points,raised to the power of the loss exponent. The following two simple heuristics