Searching, Sorting, and Cake Cutting in Rounds
Simina Br\^anzei, Dimitris Paparas, Nicholas Recker

TL;DR
This paper explores the complexity of searching, sorting, and fair division problems in rounds, establishing new bounds and protocols that connect cake cutting with sorting and search problems under round-based interaction models.
Contribution
It introduces a novel connection between proportional cake cutting and sorting with rank queries, and provides new bounds for unordered and ordered search in round-based models.
Findings
Proportional cake cutting in rounds is equivalent to sorting with rank queries.
Expected query complexity for unordered search is approximately $np(\frac{k+1}{2k})$ for randomized algorithms.
Ordered search complexity is roughly $pk \cdot n^{1/k}$, similar for randomized and deterministic algorithms.
Abstract
We study searching and sorting in rounds motivated by a fair division question: given a cake cutting problem with players, compute a fair allocation in at most rounds of interaction with the players. Rounds interpolate between the simultaneous and the fully adaptive settings, also capturing parallel complexity. We find that proportional cake cutting in rounds is equivalent to sorting with rank queries in rounds. We design a protocol for proportional cake cutting in rounds, while lower bounds for sorting in rounds with rank queries were given by Alon and Azar. Inspired by the rank query model, we then consider two basic search problems: ordered and unordered search. In unordered search, we get an array and an element promised to be in . We have access to an oracle that receives queries of the form "Is at location ?" and answers…
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Taxonomy
TopicsOptimization and Search Problems · Machine Learning and Algorithms · Complexity and Algorithms in Graphs
