Distributed Approximation Algorithms for Weighted Shortest Paths (1403.5171v2)

Abstract: A distributed network is modeled by a graph having $n$ nodes (processors) and diameter $D$. We study the time complexity of approximating {\em weighted} (undirected) shortest paths on distributed networks with a $O(\log n)$ {\em bandwidth restriction} on edges (the standard synchronous \congest model). The question whether approximation algorithms help speed up the shortest paths (more precisely distance computation) was raised since at least 2004 by Elkin (SIGACT News 2004). The unweighted case of this problem is well-understood while its weighted counterpart is fundamental problem in the area of distributed approximation algorithms and remains widely open. We present new algorithms for computing both single-source shortest paths (\sssp) and all-pairs shortest paths (\apsp) in the weighted case. Our main result is an algorithm for \sssp. Previous results are the classic $O(n)$-time BeLLMan-Ford algorithm and an $\tilde O(n^{1/2+1/2k}+D)$-time $(8k\lceil \log (k+1) \rceil -1)$-approximation algorithm, for any integer $k\geq 1$, which follows from the result of Lenzen and Patt-Shamir (STOC 2013). (Note that Lenzen and Patt-Shamir in fact solve a harder problem, and we use $\tilde O(\cdot)$ to hide the $O(\poly\log n)$ term.) We present an $\tilde O(n^{1/2}D^{1/4}+D)$-time $(1+o(1))$-approximation algorithm for \sssp. This algorithm is {\em sublinear-time} as long as $D$ is sublinear, thus yielding a sublinear-time algorithm with almost optimal solution. When $D$ is small, our running time matches the lower bound of $\tilde \Omega(n^{1/2}+D)$ by Das Sarma et al. (SICOMP 2012), which holds even when $D=\Theta(\log n)$, up to a $\poly\log n$ factor.

• The paper presents a near-optimal (1 + o(1))-approximation SSSP algorithm that runs in ~O(n^(1/2)D^(1/4) + D) time, improving upon previous methods.
• The work introduces an ~O(n)-time APSP algorithm that meets theoretical lower bounds, enhancing efficiency in distributed network computations.
• It leverages lightweight bounded-hop procedures and shortcut techniques to minimize congestion and reduce network diameter in distributed settings.

Distributed Approximation Algorithms for Weighted Shortest Paths: An Analytical Overview

The paper "Distributed Approximation Algorithms for Weighted Shortest Paths" by Danupon Nanongkai addresses the inherently complex problem of computing shortest paths in distributed networks, focusing specifically on weighted undirected graphs. This paper explores the challenges associated with the bandwidth constraint imposed by the CONGEST model, which allows only $O(\log n)$ bits to be sent through each edge per communication round. Although the unweighted variant of this problem is well-documented, the weighted version presents additional complexities that this paper seeks to untangle.

Algorithmic Contributions

The paper primarily introduces novel algorithms for efficiently computing single-source shortest paths (SSSP) and all-pairs shortest paths (APSP) in a distributed environment where nodes have limited local knowledge. A notable contribution is the introduction of an algorithm that approximates the SSSP in $\tilde{O}(n^{1/2}D^{1/4} + D)$ time with a near-optimal approximation factor of $(1 + o(1))$.

1. Single-Source Shortest Path (SSSP) Algorithm: Nanongkai builds upon existing work, improving the approximation factor while achieving sublinear time complexity. The approximation guarantees a $(1 + o(1))$-factor while previous best approaches had large approximation numbers like $8k \lceil \log (k+1) \rceil - 1$ for a runtime of $\tilde{O}(n^{1/2+1/2k} + D)$.
2. All-Pairs Shortest Paths (APSP) Algorithm: The $\tilde{O}(n)$-time $(1 + o(1))$-approximate APSP algorithm is a significant step towards efficiently solving APSP on large networks. This algorithm aligns with the theoretical lower bound, making it both compelling and practically applicable.

Technical Innovations

The paper introduces two essential innovations:

• Light-Weight Bounded-Hop Algorithm: This algorithm not only speeds up the computation of $h$-hop SSSP from multiple sources but also does so efficiently, without causing congestion—a critical aspect in consistent distributed networking environments.
• Shortcut Procedures for Diameter Reduction: By strategically adding shortcut edges between nodes and their nearest neighbors, the paper effectively reduces the weighted shortest-path diameter, enabling faster calculations for SSSP and APSP without compromising path accuracy.

Implications

The implications of this research extend to both theoretical and practical computational arenas. Most notably, the near-optimal approximation factors and sublinear time complexity for SSSP reflect a substantial leap forward for distributed approximation algorithms within networks characterized by small diameters. Furthermore, the paper sets a precedent for pursuing lower time bounds on other complex network-based problems under bandwidth constraints, indicating potential pathways to achieving deterministic algorithms with similar efficiency benchmarks.

Future Directions

While this paper addresses immediate concerns in distributed shortest path computations, several areas warrant further exploration. One open question remains whether it is feasible to not only approximate but also compute shortest paths exactly in sublinear time, which could have immense ramifications for distributed computing paradigms. Additionally, extending these algorithms to scenarios involving asymmetric weight assignments offers fertile ground for future investigations—a pursuit that challenges current limitations and Asymmetric CONGEST model implications.

In summary, Danupon Nanongkai’s work provides profound insights into optimizing shortest paths within distributed networks, paving the way for more efficient algorithmic structures and potential theoretical breakthroughs in distributed computing.