Weighted Additive Spanners

Multi-level Weighted Additive Spanners

“…In this section, we provide experimental results involving the rounding-up framework described in Section 3. This framework needs a single level subroutine; we use the +2W subsetwise construction in Section 2 and the three pairwise +2W (•, •), +4W (•, •), +6W constructions provided in [3] 1 (see Appendix A). We generate multi-level instances and solve the instances using our exact algorithm and the four approximation algorithms.…”

“…All unweighted graphs have polynomially constructible +2 subsetwise spanners on O(n |S|) edges [30,20]. For weighted graphs, Ahmed et al [3] recently give a +4W subsetwise spanner construction, also using O(n |S|) edges. In this section we show how to generalize the +2 subsetwise construction [30,20] to the weighted setting by giving a construction which produces a +2W spanner of a weighted graph (with integer edge weights in [1, W ]) on O(nW |S|) edges.…”

“…This follows from applying the +8W construction of Ahmed et al [3] (Appendix A, Algorithm 3), except we use the above +2W subsetwise spanner instead of the +4W subsetwise spanner construction given in [3] as a subroutine.…”

“…We now consider the +4W (•, •) pairwise construction [3] (Algorithm 2) as a single level subroutine. We first describe the experimental results on Erdős-Rényi graphs w.r.t.…”

“…First, while the multiplicative spanner of Althöfer et al [7] works without issue for weighted graphs, the previous additive spanner constructions hold only for unweighted graphs, whereas the metrics that arise in applications often require edge weights. Addressing this, recent work of the authors [3] and in two papers of Elkin, Gitlitz, and Neiman [21,22] gave natural extensions of the classic additive spanner constructions to weighted graphs. For example, the +2 spanner bound becomes the following statement: for all n-vertex weighted graphs G, there is a subgraph H satisfying dist H (s, t) ≤ dist G (s, t) + 2W (s, t), where W (s, t) denotes the maximum edge weight along an arbitrary s t shortest path in G. The +4 spanner generalizes similarly, and the +6 spanner does as well with the small exception that the error increases to +(6 + ε)W (s, t), for ε > 0 arbitrarily small which trades off with the implicit constant in the spanner size.…”