The Gap Between Greedy Algorithm and Minimum Multiplicative Spanner

Abstract

The greedy algorithm adapted from Kruskal's algorithm is an efficient and folklore way to produce a $k$-spanner with girth at least $k+2$. The greedy algorithm has shown to be `existentially optimal', while it's not `universally optimal' for any constant $k$. Here, `universal optimality' means an algorithm can produce the smallest $k$-spanner $H$ given any $n$-vertex input graph $G$. However, how well the greedy algorithm works compared to `universal optimality' is still unclear for superconstant $k:=k(n)$. In this paper, we aim to give a new and fine-grained analysis of this problem in undirected unweighted graph setting. Specifically, we show some bounds on this problem including the following two (1) On the negative side, when $k<\frac{1}{3}n-O(1)$, the greedy algorithm is not `universally optimal'. (2) On the positive side, when $k>\frac{2}{3}n+O(1)$, the greedy algorithm is `universally optimal'. We also introduce an appropriate notion for `approximately universal optimality'. An algorithm is $(α,β)$-universally optimal iff given any $n$-vertex input graph $G$, it can produce a $k$-spanner $H$ of $G$ with size $|H|\leq n+α(|H^*|-n)+β$, where $H^*$ is the smallest $k$-spanner of $G$. We show the following positive bounds. (1) When $k>\frac{4}{7}n+O(1)$, the greedy algorithm is $(2,O(1))$-universally optimal. (2) When $k>\frac{12}{23}n+O(1)$, the greedy algorithm is $(18,O(1))$-universally optimal. (3) When $k>\frac{1}{2}n+O(1)$, the greedy algorithm is $(32,O(1))$-universally optimal. All our proofs are constructive building on new structural analysis on spanners. We give some ideas about how to break small cycles in a spanner to increase the girth. These ideas may help us to understand the relation between girth and spanners.

Figures (12)

• Figure 1: Different Bounds
• Figure 2: Construction of $G_k$. The dotted blue hexagon represents the $k$-cycle. The dotted orange curves are paths, and the green solid lines are edges. Two orange paths connected by a green edge form an arc. The specific positions of green edges on the arcs don't matter.
• Figure 3: The whole graph denotes $K$ and the middle circle denotes $SC_H$.
• Figure 4: Definitions of $a_i,b_i$
• Figure 5: An illustration for our analysis. On this subgraph of $H$, the path between $s$ and $t$ using $SC_H[a,b]$ is $p$, and the path between $s'$ and $t'$ using $SC_H[d,c]$ is $p'$. The orientation of the cycle $SC_H$ is clockwise.

Theorems & Definitions (75)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• proof
• Definition 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• proof