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

On the Interaction of Bandwidth Constraints and Energy Efficiency 21529: if (R[x] > capacity and D[x][y] >D[x][u] − D[u][v]) then30: E = E\(x, y)31: end if32: end for33: end for34: end if35: end if36: until (!reconstruct)37: T ← T \T i,j38: end while39: return (F EASIBLE)Psuedocode 1: Successive Minimum Energy PathsIn the first algorithm we create one (for each source-destination pair) minimumenergy path at a time. Such path construction can employ the shortestpath algorithm, e.g. the Bellman Ford algorithm, with the transmitting energyas the cost. We continue adding paths, until the capacity constraint of one nodeis exceeded. We will reduce the maximum transmission power level of each nodewhose transmission violated the capacity of the node in question. The reductionof the transmission power is captured by removing the corresponding edges and,subsequently, rerunning the shortest path. Hypothetically, it is still possible thatthe new path will influence the load of the removed edges, due to the nearbynodes relaying its traffic. Subsequently, we remove nodes as relays from considerationand re-run the shortest-path algorithm. The process continues until a pathcan be found that does not exceed the capacity constraints of any of the nodes ittraverses through and of any of the nodes that overhear its transmission. Therefore,the order of path construction is critical to the outcome. We also note thatcertain traffic load matrices are simply infeasible they cannot be accommodated,because one or more source-destination paths cannot be established.2.2 All Pairs Minimum Energy PathsAll_Pairs_Minimum_Energy_Paths(Input: G(V,E),D,P,T; Output Ps)1: All P airs Shortest P aths(G, P, P s)2: for all u ∈ V do3: C[u] ← capacity4: end for5: for all T i,j ∈ T do6: for all (u, v) ∈ Ps i,j do7: for all x ∈ V do8: if (P [u][v] ≤ P [u][x]) then9: C[x] ← C[x] − T i,j10: end if11: end for