Computer Science > Data Structures and Algorithms
[Submitted on 1 Dec 2020 (v1), last revised 19 Nov 2023 (this version, v5)]
Title:Searching, Sorting, and Cake Cutting in Rounds
View PDFAbstract:We study searching and sorting in rounds motivated by a fair division question: given a cake cutting problem with $n$ players, compute a fair allocation in at most $k$ 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 $\vec{x}=(x_1, \ldots, x_n)$ and an element $z$ promised to be in $\vec{x}$. We have access to an oracle that receives queries of the form "Is $z$ at location $i$?" and answers "Yes" or "No". The goal is to find the location of $z$ with success probability at least $p$ in at most $k$ rounds of interaction with the oracle.
We show the expected query complexity of randomized algorithms on a worst case input is $np\bigl(\frac{k+1}{2k}\bigr) \pm O(1)$, while that of deterministic algorithms on a worst case input distribution is $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm O(1)$. These bounds apply even to fully adaptive unordered search, where the ratio between the two complexities converges to $2-p$ as the size of the array grows.
In ordered search, we get sorted array $\vec{x}=(x_1, \ldots, x_n)$ and element $z$ promised to be in $\vec{x}$. We have access to an oracle that gets comparison queries. Here we find that the expected query complexity of randomized algorithms on a worst case input and deterministic algorithms on a worst case input distribution is essentially the same: $p k \cdot n^{\frac{1}{k}} \pm O(1+pk)$.
Submission history
From: Simina Brânzei [view email][v1] Tue, 1 Dec 2020 18:55:51 UTC (1,965 KB)
[v2] Wed, 9 Dec 2020 19:58:02 UTC (1,966 KB)
[v3] Wed, 27 Sep 2023 04:00:50 UTC (135 KB)
[v4] Tue, 14 Nov 2023 16:42:38 UTC (183 KB)
[v5] Sun, 19 Nov 2023 22:50:27 UTC (182 KB)
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