[Paper Review] New Techniques and Tighter Bounds for Local Computation Algorithms

Omer Reingold, Shai Vardi|arXiv (Cornell University)|Apr 22, 2014

Complexity and Algorithms in Graphs16 references4 citations

TL;DR

This paper introduces new techniques for designing local computation algorithms (LCAs) that significantly improve time and space efficiency for problems like maximal matching and maximal independent set on sparse graphs. By formalizing neighborhood-dependent online algorithms and introducing d-light graphs, the authors achieve O(log n log log n) time and O(log n log log n) space bounds, improving upon prior O(log³n) time and O(log³n) space results for broader graph families with simpler, more general proofs.

ABSTRACT

Given an input $x$, and a search problem $F$, local computation algorithms (LCAs) implement access to specified locations of $y$ in a legal output $y \in F(x)$, using polylogarithmic time and space. Mansour et al., (2012), had previously shown how to convert certain online algorithms to LCAs. In this work, we expand on that line of work and develop new techniques for designing LCAs and bounding their space and time complexity. Our contributions are fourfold: (1) We significantly improve the running times and space requirements of LCAs for previous results, (2) we expand and better define the family of online algorithms which can be converted to LCAs using our techniques, (3) we show that our results apply to a larger family of graphs than that of previous results, and (4) our proofs are simpler and more concise than the previous proof methods. For example, we show how to construct LCAs that require $O(\log{n}\log\log{n})$ space and $O(\log^2{n})$ time (and expected time $O(\log\log{n})$) for problems such as maximal matching on a large family of graphs, as opposed to the henceforth best results that required $O(\log^3{n})$ space and $O(\log^4{n})$ time, and applied to a smaller family of graphs.

Motivation & Objective

• To design more efficient local computation algorithms (LCAs) with improved time and space complexity for combinatorial search problems on large graphs.
• To expand the family of graphs on which LCAs can be applied beyond bounded-degree or constant-degree graphs to include d-light graphs such as random graphs and bipartite graphs.
• To formalize a new class of algorithms—neighborhood-dependent online algorithms—that enable systematic LCA construction via randomized vertex ordering.
• To simplify and generalize prior proof techniques for LCA analysis, moving beyond query tree abstractions.
• To establish tight bounds on LCA complexity and demonstrate their optimality via lower bounds for high-degree graphs.

Proposed method

• Introduce neighborhood-dependent online algorithms, where each vertex's output depends only on its neighbors' values that arrived earlier in the sequence.
• Define d-light graphs as a broad class of graphs (e.g., d-regular, G(n,p) with p=d/n, random bipartite graphs) where degrees are O(log log n) w.h.p.
• Construct LCAs by simulating an online algorithm on a random vertex ordering, restricting queries to neighbors that arrived before the queried vertex.
• Use compact data structures to store relative positions of edges and vertices, reducing space to O(log n log log n) via O(log log n) bits per edge w.h.p.
• Apply randomized hashing and efficient DFS traversal to reconstruct vertex identities and paths in the graph with minimal space overhead.
• Prove expected time and space bounds using probabilistic analysis of the relevant vicinity size in d-light graphs, showing polylogarithmic complexity.

Experimental results

Research questions

• RQ1 Can we design LCAs with improved time and space complexity for maximal matching and related problems on a broader class of graphs?
• RQ2 What class of online algorithms can be systematically converted into efficient LCAs using a unified framework?
• RQ3 How can we characterize the graph families for which LCAs achieve polylogarithmic complexity, and what are the limits of such algorithms?
• RQ4 Can we simplify and generalize the proof techniques used in prior LCA analyses, moving beyond query tree abstractions?
• RQ5 Is the O(log n log log n) time and space bound tight for LCAs on d-light graphs, and what happens when degrees exceed O(log log n)?

Key findings

• The paper achieves O(log n log log n) time and O(log n log log n) space complexity for LCAs on d-light graphs, improving upon the prior O(log³n) time and O(log³n) space bounds.
• For maximal matching, the new LCA requires O(log²n) time and O(log n log log n) space per inquiry, with expected time O(log log n) and expected space O(log n).
• The framework applies to a significantly broader family of graphs, including d-regular graphs, G(n,p) with p = d/n for d = O(log log n), and random bipartite graphs with fixed consumer degrees.
• The proof techniques are simpler and more general than previous methods, avoiding the use of query trees and instead relying on probabilistic analysis of vertex vicinity size.
• The results are tight: for graphs with expected degree ω(log log n), the expected size of the relevant vicinity becomes super-polylogarithmic, implying that the bounds cannot be improved using the same simulation technique.
• The framework extends to any crisp neighborhood-dependent online algorithm, enabling efficient LCA construction for maximal independent set, vertex coloring, and load balancing algorithms.