Given a combination of unprotected and dedicated edge-disjoint path (1+1) protected connection requests and a finite set of fiber types, we consider the problem of allocating fibers on the links of a WDM network at minimum cost, such that all connection requests can be simultaneously realized. Each fiber type is characterized by its capacity and its cost per unit length, where costs reflect an economy of scale. It is known that a solution induced by "simply" routing each unprotected (respectively 1+1 protected) connection along the shortest path (respectively shortest pair of edge-disjoint paths) minimizes the total wavelength mileage, but may not minimize the total fiber cost. In this paper, we quantify the increase in fiber cost due to shortest path routing. In particular, we prove that the total cost of a shortest path based solution is guaranteed to lie within a certain factor of the minimum possible cost. This leads also to the fact that shortest path routing is asymptotically cost-optimal for a large total number of connection requests. Furthermore, for sparse topologies, e.g., the ring, the ShuffleNet and the mesh(-torus), we show that shortest path routing is asymptotically cost-optimal in large-scale networks supporting all-to-all communication. En route, we prove that by shortest path routing we obtain a provably optimal solution to the linear programming (LP-) relaxation of the problem. We have thus presented a provably good upper bound and a lower bound on the total fiber cost, that can be computed in polynomial-time. These bounds can be used as benchmarks against which heuristic approaches are compared.
Bibliographical noteFunding Information:
Manuscript received April 8, 2004; revised January 13, 2005. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) under a post doctoral fellowship, in part by the Canada Research Chair Program, and in part by the National science Foundation (NSF) under grant DMS-0312416.
- Integer linear programming
- Performance guarantee
- Survivable network design