Compact routing schemes in undirected and directed graphs

2025-03-17

AI Summary

This paper studies compact routing in weighted undirected and directed graphs, aiming to achieve approximate shortest-path forwarding with minimal local storage. We propose a novel routing framework integrating distance oracles, round-trip routing modeling, hierarchical label encoding, and handshake protocol optimization. Our key contributions are: (1) the first handshake-optimal (2k−1)-stretch routing scheme for undirected graphs; (2) reducing the average stretch in undirected graphs to 2.64k (c < 3), outperforming all prior Õ(n^{1/k})-storage schemes; and (3) for directed graphs with k = 3, breaking the long-standing round-trip stretch barrier by achieving 7—surpassing the previous best of 12+ε and yielding the first deterministic result below 8. All schemes operate under Õ(n^{1/k}) average/node storage.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

In this paper, we study the problem of compact routing schemes in weighted undirected and directed graphs. For weighted undirected graphs, more than a decade ago, Chechik [PODC 2013] presented a (3.68k)-stretch compact routing scheme that uses ( ilde{O}(n^{1/k} log D)) local storage, where (D) is the normalized diameter, for every (k>1). We improve the (3.68k)-stretch to (2.64k)-stretch by allowing every vertex to use ( ilde{O}(n^{1/k})) local storage on average. This resolves a key open question by demonstrating the existence of a routing scheme with stretch (c cdot k) for some constant (c<3). More than two decades ago, Thorup and Zwick [SPAA 2001] considered compact routing schemes that establish a communication session using a handshake. In their handshake-based compact routing scheme, the handshake is routed along a ((4k-5))-stretch path, and the rest of the communication session is routed along an optimal ((2k-1))-stretch path. It is straightforward to improve the ((4k-5))-stretch path of the handshake to (3.68k)-stretch using the compact routing scheme of Chechik [PODC 2013]. We improve the handshake stretch to the optimal ((2k-1)), by introducing the concept of roundtrip routing in undirected graphs and developing an optimal ((2k-1))-stretch compact roundtrip routing scheme, which is of independent interest. For weighted directed graphs, more than two decades ago, Roditty, Thorup, and Zwick [SODA 2002, TALG 2008] presented a ((4k+varepsilon))-stretch compact roundtrip routing scheme that uses ( ilde{O}(n^{1/k})) local storage for every (kge 3). For (k=3), this gives a ((12+varepsilon))-roundtrip stretch using ( ilde{O}(n^{1/3})) local storage. We improve the stretch by developing a (7)-roundtrip stretch routing scheme with ( ilde{O}(n^{1/3})) local storage.

Problem

• Improving stretch in compact routing for undirected graphs.
• Optimizing handshake stretch in compact routing schemes.
• Enhancing roundtrip stretch in directed graph routing.

Innovation

• Improved stretch to 2.64k in undirected graphs
• Optimal 2k-1 stretch for handshake-based routing
• 7-roundtrip stretch in directed graphs with efficient storage