Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...
Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...
Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...
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On the Interaction of Bandwidth Constra<strong>in</strong>ts and Energy Efficiency 213source to dest<strong>in</strong>ation (if we care only about particular source-dest<strong>in</strong>ation pairsand not about the connectivity of nodes that have no traffic to send or receive).Hence, too low a τ i can result <strong>in</strong> a disconnected topology.For the sake of exposition, we will consider the problem of f<strong>in</strong>d<strong>in</strong>g pathsbetween source-dest<strong>in</strong>ation pairs <strong>in</strong> static wireless networks. We will be given atraffic matrix which <strong>in</strong>dicates the traffic load between each source-dest<strong>in</strong>ationpair. 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-dest<strong>in</strong>ation pairs (butcan be arbitrarily small). Each source-dest<strong>in</strong>ation pair will subsequently haveto be routed us<strong>in</strong>g multiple <strong>in</strong>termediate hops. If no capacity constra<strong>in</strong>ts arepresent, a source-dest<strong>in</strong>ation pair can pick the lowest energy cost from source todest<strong>in</strong>ation us<strong>in</strong>g a conventional shortest path algorithm. That is, the shortestpath algorithm can be applied on a graph <strong>in</strong> which the costs stand for thedistance between the nodes raised to the loss exponent. For a s<strong>in</strong>gle sourcedest<strong>in</strong>ationpair, and if no other traffic or bandwidth constra<strong>in</strong>ts existed, thatwould be the optimum solution. We note that the unicast energy m<strong>in</strong>imizationproblem is <strong>in</strong> fact a shortest path problem, compared to the multicast/broadcastenergy m<strong>in</strong>imization problem, which is known to be NP-hard [6].Unfortunately, the general case of the problem (multiple source-dest<strong>in</strong>ationdemands, capacity constra<strong>in</strong>ts and wireless broadcast nature) is sufficiently complexto defy currently a simple answer. We note that the first two features (multiplesource-dest<strong>in</strong>ation demands, capacity constra<strong>in</strong>ts) render the problem a caseof the node-capacitated multi-commodity flow class of problems. That is, eachnode has a capacity as a constra<strong>in</strong>t while each node pair need to established aflow and deliver amount of traffic concurrently <strong>in</strong> 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 <strong>in</strong>tended to be received <strong>by</strong> 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 surpris<strong>in</strong>g that results on node-capacitated multi-commodity flow arespecific to particular topologies, e.g., r<strong>in</strong>gs [8]. The general case appears to bean extremely complex case, even without the presence of mobility.We po<strong>in</strong>t out that several versions of the basic problem exist. For example,transmission and reception <strong>by</strong> a node are lumped <strong>in</strong>to one capacity constra<strong>in</strong>tonly. Clearly, separate channels can be used for transmitt<strong>in</strong>g and receiv<strong>in</strong>g, withseparate fixed capacities. Secondly, we assume knowledge of the traffic loadsbetween all source-dest<strong>in</strong>ation. 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 determ<strong>in</strong>ed as the Euclidian distance between the po<strong>in</strong>ts,raised to the power of the loss exponent. The follow<strong>in</strong>g two simple heuristics