[Paper Review] New and Simplified Distributed Algorithms for Weighted All Pairs Shortest Paths

Udit Agarwal, Vijaya Ramachandran|arXiv (Cornell University)|Oct 18, 2018

Complexity and Algorithms in Graphs14 references4 citations

TL;DR

This paper presents new deterministic and randomized distributed algorithms for computing all-pairs shortest paths (APSP) and k-source shortest paths in weighted graphs under the Congest model. By enhancing a pipelined algorithm with improved blocker set computation and leveraging weight transformations, it achieves faster sub-$$n^{3/2}$$-round bounds: $$\tilde{O}(\lambda^{1/4}n^{5/4})$$ for APSP with edge weights ≤ λ, and $$\tilde{O}(n(\Delta \log^2 n)^{1/3})$$ for bounded shortest path distances Δ, while also enabling zero-weight edge handling and $$(1+\epsilon)$$-approximation with $$\tilde{O}(n/\epsilon^2)$$ rounds.

ABSTRACT

We consider the problem of computing all pairs shortest paths (APSP) and shortest paths for k sources in a weighted graph in the distributed CONGEST model. For graphs with non-negative integer edge weights (including zero weights) we build on a recent pipelined algorithm to obtain $ ilde{O}(λ^{1/4}\cdot n^{5/4})$ in graphs with non-negative integer edge-weight at most $λ$, and $ ilde{O}(n \cdot \bigtriangleup^{1/3})$ rounds for shortest path distances at most $\bigtriangleup$. Additionally, we simplify some of the procedures in the earlier APSP algorithms for non-negative edge weights in [HNS17,ARKP18]. We also present results for computing h-hop shortest paths and shortest paths from $k$ given sources. In other results, we present a randomized exact APSP algorithm for graphs with arbitrary edge weights that runs in $ ilde{O}(n^{4/3})$ rounds w.h.p. in n, which improves the previous best $ ilde{O}(n^{3/2})$ bound, which is deterministic. We also present an $ ilde{O}(n/ε^2)$-round deterministic $(1+ε)$ approximation algorithm for graphs with non-negative $poly(n)$ integer weights (including zero edge-weights), improving results in [Nanongkai14,LP15] that hold only for positive integer weights.

Motivation & Objective

• To design faster deterministic distributed algorithms for APSP and k-source shortest paths in graphs with non-negative integer edge weights, including zero-weight edges.
• To simplify and improve upon prior randomized and deterministic APSP algorithms in the Congest model, particularly those handling zero-weight edges.
• To achieve improved round complexity bounds for APSP under moderate edge-weight and shortest-path distance constraints.
• To present a deterministic $$(1+\epsilon)$$-approximation algorithm for APSP that supports zero-weight edges, matching the best known round complexity for positive weights.
• To streamline key components of existing randomized APSP algorithms, such as the short-range and short-range-extension procedures, for better efficiency and clarity.

Proposed method

• Leverages a recent pipelined algorithm for $$(h,k)$$-SSP with bounded shortest path distances to derive improved APSP and k-SSP bounds.
• Introduces a refined deterministic blocker set computation algorithm to enhance the efficiency of the APSP pipeline.
• Applies a weight transformation strategy: setting zero-weight edges to 1 and scaling non-zero weights by $$n^2$$, enabling use of positive-weight approximation algorithms.
• Uses $$h$$ rounds of Bellman-Ford for both steps in the randomized algorithm, with a random set of centers of size $$\tilde{O}(\sqrt{n})$$ to reduce communication overhead.
• Combines pipelined shortest path propagation with efficient broadcasting of distance estimates from centers to sources to compute approximate distances.
• Employs a two-phase approach: first computing reachability via zero-weight paths in $$O(n)$$ rounds, then approximating non-zero-weight paths in the transformed graph.

Experimental results

Research questions

• RQ1 Can we achieve sub-$$n^{3/2}$$-round deterministic APSP for graphs with non-negative integer edge weights, including zero weights?
• RQ2 How can we simplify and optimize the randomized APSP algorithms from Huang et al. [8] for better round complexity and clarity?
• RQ3 Can we extend $$(1+\epsilon)$$-approximation algorithms for APSP to graphs with zero-weight edges while maintaining optimal round complexity?
• RQ4 What is the optimal trade-off between edge weight magnitude, shortest path distance, and round complexity in deterministic distributed APSP computation?
• RQ5 How can blocker set computation be optimized to improve the performance of pipelined shortest path algorithms?

Key findings

• The paper achieves a deterministic APSP bound of $$O(\lambda^{1/4} \cdot n^{5/4} \log^{1/2}n)$$ rounds for graphs with edge weights at most $$\lambda$$, improving upon the prior $$\tilde{O}(n^{3/2})$$ bound.
• For graphs with shortest path distances at most $$\Delta$$, the APSP algorithm runs in $$O(n(\Delta \log^2 n)^{1/3})$$ rounds, representing a significant improvement over previous deterministic bounds.
• A randomized exact APSP algorithm is presented with a round complexity of $$\tilde{O}(n^{4/3})$$ w.h.p., improving the previous best deterministic bound of $$\tilde{O}(n^{3/2})$$.
• A deterministic $$(1+\epsilon)$$-approximation algorithm for APSP with non-negative integer weights (including zero) runs in $$\tilde{O}(n/\epsilon^2)$$ rounds, matching the best-known bound for positive weights.
• The paper simplifies key procedures in Huang et al. [8]’s randomized APSP algorithm, particularly the short-range and short-range-extension components, through streamlined pipelined SSSP computation.
• An improved blocker set computation algorithm is developed, which contributes to the efficiency of the overall pipelined APSP framework and enables better round complexity bounds.