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Locally Computing Edge Orientations

Authors Slobodan Mitrović, Ronitt Rubinfeld, Mihir Singhal

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Slobodan Mitrović
  • University of California, Davis, CA, USA
Ronitt Rubinfeld
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Mihir Singhal
  • University of California, Berkeley, CA, USA

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Slobodan Mitrović, Ronitt Rubinfeld, and Mihir Singhal. Locally Computing Edge Orientations. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.89

Abstract

We consider the question of orienting the edges in a graph G such that every vertex has bounded out-degree. For graphs of arboricity α, there is an orientation in which every vertex has out-degree at most α and, moreover, the best possible maximum out-degree of an orientation is at least α - 1. We are thus interested in algorithms that can achieve a maximum out-degree of close to α. A widely studied approach for this problem in the distributed algorithms setting is a "peeling algorithm" that provides an orientation with maximum out-degree α(2+ε) in a logarithmic number of iterations.
We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form "What is the orientation of edge (u,v)?" by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires Ω(n) probes per query on an n-vertex graph. In the case where G has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree r must use Ω(√ n/r) probes to G per query in the worst case, even if G is known to be a forest (that is, α = 1). We also show several algorithms with sublinear probe complexity when G has unbounded degree. When G is a tree such that the maximum degree Δ of G is bounded, we demonstrate an algorithm that uses Δ n^{1-log_Δ r + o(1)} probes to G per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which 4-colors any tree using sublinear probes per query.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • local computation algorithms
  • edge orientation
  • tree coloring

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