[Paper Review] Improved Distributed Steiner Forest Construction

Christoph Lenzen, Boaz Patt-Shamir|arXiv (Cornell University)|May 8, 2014

Complexity and Algorithms in Graphs11 references4 citations

TL;DR

This paper presents improved deterministic and randomized distributed algorithms for the Steiner Forest problem in the congest model. It achieves a (2+ε)-approximation in Õ(sk + √min{st,n}) rounds deterministically and an O(log n)-approximation in Õ(k + min{s,√n} + D) rounds with high probability, matching a tight lower bound of Õ(k + min{s,√n} + D).

ABSTRACT

We present new distributed algorithms for constructing a Steiner Forest in the CONGEST model. Our deterministic algorithm finds, for any given constant $ε>0$, a $(2+ε)$-approximation in $ ilde{O}(sk+\sqrt{\min(st,n)})$ rounds, where $s$ is the shortest path diameter, $t$ is the number of terminals, $k$ is the number of terminal components in the input, and $n$ is the number of nodes. Our randomized algorithm finds, with high probability, an $O(\log n)$- approximation in time $ ilde{O}(k+\min(s,\sqrt n)+D)$, where $D$ is the unweighted diameter of the network. We also prove a matching lower bound of $ ildeΩ(k+\min(s,\sqrt{n})+D)$ on the running time of any distributed approximation algorithm for the Steiner Forest problem. Previous algorithms were randomized, and obtained either an $O(\log n)$-approximation in $ ilde{O}(sk)$ time, or an $O(1/ε)$-approximation in $ ilde{O}((\sqrt{n}+t)^{1+ε}+D)$ time.

Motivation & Objective

• Address the gap in efficient distributed algorithms for the Steiner Forest problem in large-scale networks.
• Improve upon prior randomized algorithms that achieved O(log n)-approximation in Õ(sk) time or O(1/ε)-approximation in Õ((√n + t)^{1+ε} + D) time.
• Achieve a better trade-off between approximation ratio and round complexity, particularly for networks with small shortest path diameter or few terminal components.
• Establish a nearly matching lower bound to prove the optimality of the proposed algorithms in terms of round complexity.

Proposed method

• Use a deterministic moat-growing algorithm on a spanner of the terminal set to construct a (2+ε)-approximate solution.
• Construct a sparse spanner on the terminal set T ∪ S, where S is a random set of √n nodes, to reduce communication overhead.
• Apply a virtual tree embedding technique to simulate the solution on a smaller, structured graph while preserving approximation guarantees.
• Combine multiple runs of the randomized algorithm with a consensus step to select the best solution with high probability.
• Use the F-reduced instance technique to recursively solve the remaining connectivity requirements after an initial partial solution is found.
• Leverage the structure of the spanner and the shortest path diameter to bound the number of rounds required for computation and communication.

Experimental results

Research questions

• RQ1 Can a deterministic distributed algorithm achieve a (2+ε)-approximation for Steiner Forest with improved round complexity compared to prior randomized approaches?
• RQ2 Can the round complexity of randomized algorithms be reduced to Õ(k + min{s,√n} + D) while maintaining an O(log n)-approximation?
• RQ3 Is the proposed round complexity tight up to logarithmic factors, and can a matching lower bound be proven?
• RQ4 How does the performance of the algorithm scale with respect to the number of terminal components k, the shortest path diameter s, and the network diameter D?
• RQ5 Can the combination of spanner construction and virtual tree embedding be used to achieve both strong approximation ratios and low communication complexity in distributed settings?

Key findings

• The deterministic algorithm achieves a (2+ε)-approximation in Õ(sk + √min{st,n}) rounds, improving upon previous O(1/ε)-approximation algorithms that required Õ((√n + t)^{1+ε} + D) time.
• The randomized algorithm computes an O(log n)-approximate solution in Õ(k + min{s,√n} + D) rounds with high probability, which is optimal up to logarithmic factors.
• A matching lower bound of Õ(k + min{s,√n} + D) is proven for any distributed algorithm achieving a non-trivial approximation ratio, establishing the optimality of the randomized algorithm.
• For the case where input is given as connection requests between terminals, a lower bound of Õ(t + min{s,√n} + D) holds, which is tight up to logarithmic factors.
• The algorithm design leverages spanner construction on a random subset of √n nodes and a modified moat-growing algorithm to ensure low approximation ratio and communication efficiency.
• The combination of multiple randomized runs and a consensus step ensures that the best solution is selected with high probability, maintaining the O(log n)-approximation guarantee.