New Greedy Spanners and Applications

Abstract

We present a simple greedy procedure to compute an $(α,β)$-spanner for a graph $G$. We then show that this procedure is useful for building fault-tolerant spanners, as well as spanners for weighted graphs. Our first main result is an algorithm that, given a multigraph $G$, outputs an $f$ edge fault-tolerant $(k,k-1)$-spanner $H$ of size $O(fn^{1+\frac1k})$ which is tight. To our knowledge, this is the first tight result concerning the price of fault tolerance in spanners which are not multiplicative, in any model of faults. Our second main result is a new construction of a spanner for weighted graphs. We show that any weighted graph $G$ has a subgraph $H$ with $O(n^{1+\frac{1}{k}})$ edges such that any path $P$ of hop-length $\ell$ in $G$ has a replacement path $P'$ in $H$ of weighted length $\leq w(P)+(2k-2)w^{(1/2)}(P)$ where $w(P)$ is the total edge weight of $P$, and $w^{(1/2)}$ denotes the sum of the largest $\lceil \frac{\ell}{2} \rceil$ edge weights along $P$. Moreover, we show such approximation is optimal for shortest paths of hop-length $2$. To our knowledge, this is the first construction of a spanner for weighted graphs that strictly improves upon the stretch of multiplicative $(2k-1)$-spanners for all non-adjacent vertex pairs, while maintaining the same size bound. Our technique is based on using clustering and ball-growing, which are methods commonly used in designing spanner algorithms, to analyze simple greedy algorithms. This allows us to combine the flexibility of clustering approaches with the unique properties of the greedy algorithm to get improved bounds. In particular, our methods give a very short proof that the parallel greedy spanner adds $O(kn^{1+\frac{1}{k}})$ edges, improving upon known bounds.

Figures (8)

• Figure 1: A big clique in the output of the $2\to 2k$ greedy spanner
• Figure 2: A path with an $R$-lateral cluster.
• Figure 3: A path with a cluster of $m_i$ almost contained in the lateral cluster of $y_i$. Observe that for any added path, two orange edges as depicted in the figure cannot share a vertex.
• Figure 4: Lateral clustering for weighted graphs. The edge $\{u,v\}$ was added in the "$T$-test" .
• Figure 5: Global path reduction by edges.

Theorems & Definitions (85)

• Definition 1.1: ($\alpha,\beta$)-spanner
• Definition 1.2: E/VFT Spanner
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Definition 1.6: $d \to r$ spanner
• Claim 1.7
• proof
• Lemma 1.8: Neighborhood exchange lemma
• Definition 1.9: EFT $(d\to r)$ spanners