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

214 T. Chu and I. Nikolaidisaim at producing paths among all source-destination pairs subject to the pernodebandwidth constraints and with the objective of minimizing the powerconsumption.The initial topology graph considered, G(V,E), is completely connected. Subsequentlyedges are removed to ensure that the capacity constraints are not violated.T is the set of source-destination demands, whereby T i,j is the demand(bits per second) from node i to node j. Ps is the set of all paths establishedwith the intention of minimizing energy cost. Shortest Path(G(V,E), D, source,destination, Path) is any straightforward implementation of the shortest pathalgorithm from source node to destination node on a graph whose connectivity iscaptured by G(V,E) and the edge costs are captured by D. The Shortest Path()returns true if a path is found and false if it could not, because the source anddestination are in two different connected components.2.1 Successive Minimum Energy PathsSuccessive_Minimum_Energy_Paths (Input: V, D, P, T; Output Ps)1: for all u ∈ V do2: C[u] ←03: end for4: G(V,E) ← completely connected graph of V nodes5: while T ≠ ∅ do6: select T i,j from T7: R ← C8: repeat9: reconstruct ← false10: if (!Shortest P ath(G, P, i, j, P ath)) then11: return (INFEASIBLE)12: else13: for all (u, v) ∈ P ath do14: for all x ∈ V do15: if D[u][v] ≤ D[u][x] then16: R[x] ← R[x]+T i,j17: end if18: if R[x] > capacity then19: reconstruct ← true20: end if21: end for22: end for23: if (!reconstruct) then24: C ← R25: Ps ← Ps∪ P ath26: else27: for all (u, v) ∈ P ath do28: for all (x, y) ∈ E do