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Searching, Sorting, and Cake Cutting in Rounds

18 marzo 2021 ore 17:00 - 18:00

Aula Luiss Research Seminars, Sede di Luiss

Speaker: Simina Brânzei, Purdue University

Abstract

We study sorting and searching in rounds motivated by a cake cutting problem. The search problem we consider is: we are given an array x=(x1,…,xn) and an element z promised to be in the array. We have access to an oracle that answers comparison queries: "How is xi compared to xj?", where the answer can be "<", "=", or ">". The goal is to find the location of z with success probability at least p∈[0,1] in at most k rounds of interaction with the oracle. The problem is called ordered or unordered search, depending on whether the array x is sorted or unsorted, respectively.For ordered search, we show the expected query complexity of randomized algorithms is Θ(k⋅p⋅n1/k) in the worst case. In contrast, the expected query complexity of deterministic algorithms searching for a uniformly random element is Θ(k⋅p1/k⋅n1/k). The uniform distribution is the worst case for deterministic algorithms.For unordered search, the expected query complexity of randomized algorithms is np(k+12k)±1 in the worst case, while the expected query complexity of deterministic algorithms searching for a uniformly random element is np(1−k−12kp)±1.We also discuss the connections of these search problems to the rank query model, where the array x can be accessed via queries of the form "Is rank(xi)≤k?". Unordered search is equivalent to Select with rank queries (given q, find xi with rank q) and ordered search to Locate with rank queries (given xi, find its rank). We show an equivalence between sorting with rank queries and proportional cake cutting with contiguous pieces for any number of rounds, as well as an improved lower bound for deterministic sorting in rounds with rank queries.