[Paper Review] Distributed Computation of Sparse Cuts

Atish Das Sarma, Anisur Rahaman Molla+1 more|arXiv (Cornell University)|Oct 21, 2013

Computational Geometry and Mesh Generation29 references3 citations

TL;DR

This paper presents two fast distributed algorithms for computing sparse cuts in undirected networks using random walks and PageRank, achieving an approximation ratio of Õ(√φ) for conductance in Õ(1/φ + n) rounds. The algorithms work without prior knowledge of the optimal conductance φ and can identify well-connected local clusters and critical bottleneck edges.

ABSTRACT

Finding sparse cuts is an important tool in analyzing large-scale distributed networks such as the Internet and Peer-to-Peer networks, as well as large-scale graphs such as the web graph, online social communities, and VLSI circuits. In distributed communication networks, they are useful for topology maintenance and for designing better search and routing algorithms. In this paper, we focus on developing fast distributed algorithms for computing sparse cuts in networks. Given an undirected $n$-node network $G$ with conductance $ϕ$, the goal is to find a cut set whose conductance is close to $ϕ$. We present two distributed algorithms that find a cut set with sparsity $ ilde O(\sqrtϕ)$ ($ ilde{O}$ hides $\polylog{n}$ factors). Both our algorithms work in the CONGEST distributed computing model and output a cut of conductance at most $ ilde O(\sqrtϕ)$ with high probability, in $ ilde O(\frac{1}{b}(\frac{1}ϕ + n))$ rounds, where $b$ is balance of the cut of given conductance. In particular, to find a sparse cut of constant balance, our algorithms take $ ilde O(\frac{1}ϕ + n)$ rounds. Our algorithms can also be used to output a {\em local} cluster, i.e., a subset of vertices near a given source node, and whose conductance is within a quadratic factor of the best possible cluster around the specified node. Both our distributed algorithm can work without knowledge of the optimal $ϕ$ value and hence can be used to find approximate conductance values both globally and with respect to a given source node. We also give a lower bound on the time needed for any distributed algorithm to compute any non-trivial sparse cut --- any distributed approximation algorithm (for any non-trivial approximation ratio) for computing sparsest cut will take $ ilde Ω(\sqrt{n} + D)$ rounds, where $D$ is the diameter of the graph.

Motivation & Objective

• Develop efficient distributed algorithms for computing sparse cuts in large-scale networks without prior knowledge of the optimal conductance.
• Enable identification of well-connected local clusters and critical bottleneck edges in distributed networks for improved routing and topology maintenance.
• Provide provable approximation guarantees for sparse cut computation under the CONGEST model with small message sizes.
• Establish a lower bound on the time complexity for any distributed sparse cut approximation algorithm.
• Support applications in network design, clustering, and congestion control through scalable and accurate sparse cut detection.

Proposed method

• Use random walks to estimate the probability distribution of node visitation, enabling approximation of conductance in a distributed manner.
• Apply PageRank-based techniques to compute local clusters around a given source node with low conductance, ensuring high intra-cluster connectivity.
• Design a two-phase algorithm: first, perform random walks from multiple source nodes to estimate cut sparsity; second, compute and evaluate partitions based on conductance.
• Leverage the CONGEST model, where each node can send O(1) messages per round, to ensure communication efficiency.
• Use a reduction from the spanning connected subgraph verification problem to derive a lower bound on the time complexity of sparse cut computation.
• Achieve high probability correctness by repeating random walk procedures O(log n) times to amplify accuracy.

Experimental results

Research questions

• RQ1 Can we design a distributed algorithm that computes a sparse cut with conductance within Õ(√φ) of the optimal in polylogarithmic time per node?
• RQ2 How can we compute a local cluster around a given source node with conductance close to the best possible cluster, without knowing the optimal φ?
• RQ3 What is the minimum time complexity required for any distributed algorithm to compute a non-trivial sparse cut in a network?
• RQ4 Can we avoid relying on prior knowledge of the optimal conductance φ while still achieving good approximation guarantees?
• RQ5 To what extent can random walks and PageRank be combined to improve the efficiency of distributed sparse cut computation?

Key findings

• The proposed algorithms compute a cut with conductance at most Õ(√φ) with high probability in Õ(1/φ + n) rounds, where φ is the conductance of the network.
• For cuts of constant balance, the running time is Õ(1/φ + n) rounds, which matches the lower bound up to polylogarithmic factors.
• The algorithms can compute local clusters around a source node with conductance within a quadratic factor of the optimal cluster, without prior knowledge of φ.
• A lower bound of Õ(√n + D) rounds is established for any distributed approximation algorithm for sparsest cut, where D is the graph diameter.
• The algorithms are communication-efficient, using only small messages per round, and are applicable in the standard CONGEST model.
• The results can be used to identify critical edges and well-connected components, supporting topology-aware network design and maintenance.