Locally computing edge orientations
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singhal-mihirs-meng-eecs-2023-thesis.pdf
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Thesis PDF
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Author(s)
Singhal, Mihir
Advisor(s)
Rubinfeld, Ronitt
Date Issued
June 2023
Publisher
Massachusetts Institute of Technology
Abstract
We consider the question of orienting the edges in a graph ๐บ such that every vertex has bounded out-degree. For graphs of arboricity ๐ผ, there is an orientation in which every vertex has out-degree at most ๐ผ, and moreover, this is the best possible. We are thus interested in algorithms that can achieve a maximum out-degree of close to ๐ผ. A widely studied approach for this problem in the distributed algorithms setting is a โpeeling algorithmโ that provides an orientation with maximum out-degree ๐ผ(2 + ๐) in a logarithmic number of iterations.
We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form โWhat is the orientation of edge (๐ข, ๐ฃ)?โ 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 ฮฉ(๐) probes per query on an ๐-vertex graph. In the case where ๐บ has unbounded degree, we show that any LCA which orients its edges to yield maximum out-degree ๐ must use ฮฉ( โ ๐/๐) probes to ๐บ per query in the worst case, even if ๐บ is known to be a forest (that is, ๐ผ = 1). We also show several algorithms with sublinear probe complexity when ๐บ has unbounded degree. When the maximum degree ฮ of ๐บ is bounded, we demonstrate an algorithm that uses [formulation] probes to ๐บ 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 result to demonstrate an LCA which can 4-color any tree using sublinear probes per query.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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