On the Advice Complexity of Online Matching on the Line

TL;DR

This article analyzes advice complexity for online minimum matching on the real line under the tape model. It proves a tight bound: any $1$-competitive online algorithm must read at least $n-1$ bits of advice, and provides a simple LR-based strategy achieving this bound. It then introduces the DIVIDE_k and RESCALE framework, showing that partitioning servers into $k$ blocks and using a $c(n/k)$-competitive subroutine inside blocks yields a $c(n/k)$-competitive algorithm with $O(k(\,log N + \,log n))$ bits of advice, and extends this to general coordinates with $O(k(\log(s_n-s_1) + \log n))$ bits. By combining with known online methods (ABNPS, GL), these results give scalable trade-offs between advice and competitiveness, including randomized variants. The work advances understanding of how advice reduces the cost of online matching on the line and points to extensions to other metric spaces such as trees.

Abstract

We consider the matching problem on the line with advice complexity. We give a 1-competitive online algorithm with advice complexity $n-1,$ and show that there is no 1-competitive online algorithm reading less than $n-1$ bits of advice. Moreover, for each $0<k<n$ we present a $c(n/k)$-competitive online algorithm with advice complexity $O(k(\log N + \log n))$ where $n$ is the number of servers, $N$ is the distance of the minimal and maximal servers, and $c(n)$ is the complexity of the best online algorithm without advice.

Theorems & Definitions (13)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof