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Incremental Edge Orientation in Forests

Authors Michael A. Bender, Tsvi Kopelowitz, William Kuszmaul, Ely Porat, Clifford Stein

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Author Details

Michael A. Bender
  • Stony Brook University, NY, USA
Tsvi Kopelowitz
  • Bar-Ilan University, Ramat Gan, Israel
William Kuszmaul
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Ely Porat
  • Bar-Ilan University, Ramat Gan, Israel
Clifford Stein
  • Columbia University, New York, NY, USA

Funding

Supported in part by ISF grants no. 1278/16 and 1926/19, by a BSF grant no. 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064). This research was sponsored in part by National Science Foundation Grants XPS-1533644 and CCF-1930579, CCF-1714818, and CCF-1822809, CCF-2106827, CCF-1725543, CSR-1763680, CCF-1716252, and CNS-1938709. The research was also sponsored in part by the United States Air Force Research Laboratory and was accomplished under Cooperative Agreement Number FA8750-19-2-1000. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the United States Air Force or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

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Michael A. Bender, Tsvi Kopelowitz, William Kuszmaul, Ely Porat, and Clifford Stein. Incremental Edge Orientation in Forests. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.12

Abstract

For any forest G = (V, E) it is possible to orient the edges E so that no vertex in V has out-degree greater than 1. This paper considers the incremental edge-orientation problem, in which the edges E arrive over time and the algorithm must maintain a low-out-degree edge orientation at all times. We give an algorithm that maintains a maximum out-degree of 3 while flipping at most O(log log n) edge orientations per edge insertion, with high probability in n. The algorithm requires worst-case time O(log n log log n) per insertion, and takes amortized time O(1). The previous state of the art required up to O(log n / log log n) edge flips per insertion. We then apply our edge-orientation results to the problem of dynamic Cuckoo hashing. The problem of designing simple families ℋ of hash functions that are compatible with Cuckoo hashing has received extensive attention. These families ℋ are known to satisfy static guarantees, but do not come typically with dynamic guarantees for the running time of inserts and deletes. We show how to transform static guarantees (for 1-associativity) into near-state-of-the-art dynamic guarantees (for O(1)-associativity) in a black-box fashion. Rather than relying on the family ℋ to supply randomness, as in past work, we instead rely on randomness within our table-maintenance algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Dynamic graph algorithms
Keywords
  • edge orientation
  • graph algorithms
  • Cuckoo hashing
  • hash functions

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