Searching, Sorting, and Cake Cutting in Rounds

We study sorting and searching in rounds motivated by a cake cutting problem. The search problem we consider is: we are given an array $x = (x_1, \ldots, x_n)$ and an element $z$ promised to be in the array. We have access to an oracle that answers comparison queries: "How is $x_i$ compared to $x_j$?", where the answer can be "$$". The goal is to find the location of $z$ with success probability at least $p \in [0,1]$ 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 $\Theta\bigl(k\cdot p \cdot n^{1/k}\bigr)$ in the worst case. In contrast, the expected query complexity of deterministic algorithms searching for a uniformly random element is $\Theta\bigl(k\cdot p^{1/k} \cdot n^{1/k}\bigr)$. The uniform distribution is the worst case for deterministic algorithms. For unordered search, the expected query complexity of randomized algorithms is $np\bigl(\frac{k+1}{2k}\bigr) \pm 1$ in the worst case, while the expected query complexity of deterministic algorithms searching for a uniformly random element is $np \bigl(1 - \frac{k-1}{2k}p \bigr) \pm 1$. 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$(x_i) \leq k$?". Unordered search is equivalent to Select with rank queries (given $q$, find $x_i$ with rank $q$) and ordered search to Locate with rank queries (given $x_i$, 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.Comment: 33 pages, 4 figure

Combinatorial search in two and more rounds

In a combinatorial search problem we wish to identify an unknown element by binary tests, where the edges of a hypergraph specify the available tests. We show that, for rather general cases of this problem, the worst-case minimum number of tests, even if adaptive testing is permitted, can already be achieved in a small number of rounds of parallel tests. In particular, the maximum number of necessary rounds grows only as the square root of the number of elements, and two rounds are enough if, e.g., the test number is close to the number of elements, or the hypergraph is a graph. We also provide polynomial-time, hardness, and parameterized results on the computational complexity of finding optimal strategies for some cases, including graphs and tree hypergraphs