[CS.DS] Greedy Completion for Weighted $(\eta,\eta)$-Spanners

Abstract

We study $(\eta,\eta)$-spanners for weighted graphs. We propose a simple greedy completion procedure that starts from a sparse initial graph and repeatedly fixes pairs of vertices with a bad stretch, generalizing Knudsen's additive completion. As an application, we construct $(k,k-1)$-spanners for weighted graphs of size $\tilde{O}(n^{1+1/k})$, which were previously unknown.

Key Technical Details

The core of the greedy completion algorithm includes:

1. Sparse Initial Graph: Starting from a sparse graph ensures efficiency in the initial steps.
2. Vertex Pair Fixing: Continuously enhancing the graph's quality by correcting pairs of vertices with poor stretch.
3. Stretch Concept: In weighted graphs, stretch is defined as the ratio of the shortest path length between a pair of vertices to their original distance in the graph.

Algorithm Implementation

Here’s a pseudocode example demonstrating the basic greedy completion process:

function greedyCompletion(graph):
    while exists bad stretch pair (u, v) in graph:
        fixPair(u, v)
    return graph

This algorithm gradually achieves a better $(\eta,\eta)$-spanner by continuously correcting the stretch of vertex pairs.

Blogger's Review: This paper showcases the effective application of greedy algorithms in graph theory, particularly in the completion problem of weighted graphs. Through a straightforward greedy strategy, the author successfully constructs new $(k,k-1)$-spanners, providing an important foundation for future research.