# Constant-Time Local Computation Algorithms

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## Abstract

Local computation algorithms (LCAs) produce small parts of a single (possibly approximate) solution to a given search problem using time and space sublinear in the size of the input. In this work we present LCAs whose time complexity (and usually also space complexity) is independent of the input size. Specifically, we give (1) a (1 − 𝜖)-approximation LCA to the maximum weight acyclic edge set, (2) LCAs for approximating multicut and integer multicommodity flow on trees, and (3) a local reduction of weighted matching to any unweighted matching LCA, such that the running time of the weighted matching LCA is d times (where d is the maximal degree) the running time of the unweighted matching LCA, (and therefore independent of the edge weight function).

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## Notes

1. We assume the standard uniform-cost RAM model [1], in which the word size is O(log n) bits, where n is the input size.

2. We typically assume that vertex degrees are bounded by a constant.

3. In the LOCAL model [19], at the beginning of the algorithm’s execution, each vertex knows only its ID and the IDs of its neighbors. In each round, each vertex is allowed to send an unbounded message to all of its neighbors and perform an unbounded amount of computation. The goal is to minimize the maximal number of rounds a vertex requires to compute its own portion of the output.

4. We note that while our algorithm for MWM runs in constant time, independently of the size of the graph and of the edge weights, its approximation guarantee is much worse than that of [4], whose approximation factor is (1 − 𝜖).

5. In case of a randomized algorithm, expectation is over its random choices.

6. Note that the LCA is not allowed to deviate from the enduring memory bound.

7. Algorithm 6 inherits it running time, space complexity and failure probability from theLCA it uses as a subroutine.

8. In order to simulate Algorithm 1 on a vertex at distance i from v for j rounds, we need to discover vertices at distance i + j from v.

9. Technically, this is a global algorithm, which we later show how to implement as an LCA.

10. The ratio is $$\frac 14$$ when all capacities are even, and it tends to $$\frac 14$$ as c min. For c min = 1 the approximation ratio is 0.

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## Acknowledgements

The authors would like to thank the anonymous reviewers for their useful feedback.

Yishay Mansour is supported in part by a grant from the Israel Science Foundation, by a grant from United States-Israel Binational Science Foundation (BSF), by a grant from the Israeli Ministry of Science (MoS) and the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Boaz Patt-Shamir is supported in part by the Israel Science Foundation (grant No. 1444/14) and by the Israel Ministry of Science and Technology. Shai Vardi is supported in part by the Google Europe Fellowship in Game Theory.

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Correspondence to Shai Vardi.

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Mansour, Y., Patt-Shamir, B. & Vardi, S. Constant-Time Local Computation Algorithms. Theory Comput Syst 62, 249–267 (2018). https://doi.org/10.1007/s00224-017-9788-3