[Paper Review] A Local Algorithm for the Sparse Spanning Graph Problem

Christoph Lenzen, Reut Levi|arXiv (Cornell University)|Mar 15, 2017

Complexity and Algorithms in Graphs11 references3 citations

TL;DR

This paper presents the first centralized local algorithm for constructing sparse spanning subgraphs with stretch guarantees in arbitrary bounded-degree graphs. By combining Voronoi partitioning with random sampling and BFS-based cluster reconstruction, it achieves a query complexity of $\tilde{O}(\text{poly}(\Delta/\varepsilon)n^{2/3})$ and ensures that non-selected edges have paths of $O(\log n \cdot (\Delta + \log n)/\varepsilon)$ hops in the output, preserving approximate distances.

ABSTRACT

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most $(1+\varepsilon)n$ edges (where $n$ is the number of vertices and $\varepsilon$ is a given approximation/sparsity parameter). We achieve query complexity of $ ilde{O}(poly(Δ/\varepsilon)n^{2/3})$, ($ ilde{O}$-notation hides polylogarithmic factors in $n$). where $Δ$ is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanner, i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of $O(poly(Δ/\varepsilon)\log^2 n)$ hops in the output that connects its endpoints.

Motivation & Objective

• To design a centralized local algorithm that constructs a sparse spanning subgraph with at most $(1+\varepsilon)n$ edges, even when full graph access is infeasible.
• To ensure global consistency and independence from prior queries while minimizing local query complexity.
• To provide distance preservation guarantees by constructing a spanner, where non-selected edges have short replacement paths in the output subgraph.
• To achieve sublinear query complexity on arbitrary bounded-degree graphs, overcoming limitations of prior methods that failed on general graphs.

Proposed method

• Classify edges as expanding if their endpoints have $\Omega(n^{1/3})$ nodes within $O(\log n)$ hops, enabling efficient local processing.
• Sample $n^{2/3}$ random centers and partition nodes into Voronoi cells around them, ensuring each node is within $O\left(\log n\right)$ hops of its closest center with high probability.
• Construct spanning trees within each Voronoi cell using BFS rooted at the center, with query complexity $\tilde{O}(n^{1/3})$ for finding closest centers.
• Refine Voronoi cells into clusters of size $\tilde{O}(n^{1/3})$ by recursively identifying subtrees with more than $\tilde{O}(n^{1/3})$ nodes, using BFS and parent/child relationships.
• Use a randomized marking scheme on clusters to simulate a distributed spanner algorithm, with $\tilde{O}(n^{2/3})$ random bits sufficient due to $\tilde{O}(n^{1/3})$-wise independence.
• For any queried edge $\{u,v\}$, determine membership in the output subgraph by checking cluster membership, center proximity, and relative rank in adjacent edges, ensuring consistency and low query cost.

Experimental results

Research questions

• RQ1 Can a centralized local algorithm construct a sparse spanning subgraph with $O((1+\varepsilon)n)$ edges on arbitrary bounded-degree graphs with sublinear query complexity?
• RQ2 Can such an algorithm guarantee that non-selected edges have short replacement paths in the output, preserving approximate distances?
• RQ3 Is it possible to achieve global consistency in edge selection without shared memory or coordination, relying only on shared randomness and local queries?
• RQ4 Can the query complexity be bounded by $\tilde{O}(\text{poly}(\Delta/\varepsilon)n^{2/3})$ while maintaining stretch guarantees?

Key findings

• The algorithm achieves a query complexity of $\tilde{O}(\text{poly}(\Delta/\varepsilon)n^{2/3})$ per edge, which is the first such bound for arbitrary bounded-degree graphs in the centralized local model.
• The output subgraph is a spanner with stretch $O(\log n \cdot (\Delta + \log n)/\varepsilon)$, ensuring that for every non-selected edge, there exists a path of that length in the output.
• The algorithm ensures global consistency in edge selection: answers to queries are independent of prior queries and consistent with a single, fixed output subgraph.
• The construction relies on Voronoi partitioning with $n^{2/3}$ sampled centers and cluster refinement, enabling efficient local reconstruction of cluster and parent relationships.
• The use of $\tilde{O}(n^{2/3})$ random bits suffices due to $\tilde{O}(n^{1/3})$-wise independence, as each query depends only on a small local neighborhood.
• The algorithm successfully constructs a subgraph with at most $(1+\varepsilon)n$ edges with high probability, satisfying the relaxed spanning tree requirement while preserving distance approximations.