On Multistage Learning a Hidden Hypergraph

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph $H_{un}=H(V,E)$ by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family $F(t,s,\ell)$ of localized hypergraphs for which the total number of vertices $|V| = t$, the number of edges $|E|\le s$, $s\ll t$, and the cardinality of any edge $|e|\le\ell$, $\ell\ll t$. Our goal is to identify all edges of $H_{un}\in F(t,s,\ell)$ by using the minimal number of tests. We develop an adaptive algorithm that matches the information theory bound, i.e., the total number of tests of the algorithm in the worst case is at most $s\ell\log_2 t(1+o(1))$. We also discuss a probabilistic generalization of the problem.Comment: 5 pages, IEEE conferenc

Finding Weighted Graphs by Combinatorial Search

We consider the problem of finding edges of a hidden weighted graph using a certain type of queries. Let $G$ be a weighted graph with $n$ vertices. In the most general setting, the $n$ vertices are known and no other information about $G$ is given. The problem is finding all edges of $G$ and their weights using additive queries, where, for an additive query, one chooses a set of vertices and asks the sum of the weights of edges with both ends in the set. This model has been extensively used in bioinformatics including genom sequencing. Extending recent results of Bshouty and Mazzawi, and Choi and Kim, we present a polynomial time randomized algorithm to find the hidden weighted graph $G$ when the number of edges in $G$ is known to be at most $m\geq 2$ and the weight $w(e)$ of each edge $e$ satisfies \ga \leq |w(e)|\leq \gb for fixed constants \ga, \gb>0. The query complexity of the algorithm is $O(\frac{m \log n}{\log m})$, which is optimal up to a constant factor

Error-Tolerant Exact Query Learning of Finite Set Partitions with Same-Cluster Oracle

This paper initiates the study of active learning for exact recovery of partitions exclusively through access to a same-cluster oracle in the presence of bounded adversarial error. We first highlight a novel connection between learning partitions and correlation clustering. Then we use this connection to build a R\'enyi-Ulam style analytical framework for this problem, and prove upper and lower bounds on its worst-case query complexity. Further, we bound the expected performance of a relevant randomized algorithm. Finally, we study the relationship between adaptivity and query complexity for this problem and related variants.Comment: 28 pages, 2 figure

Mapping Networks via Parallel kth-Hop Traceroute Queries

?(v,w), which return the name of the kth vertex on a shortest path from v to w, where ?(v,w) is the distance between v and w, that is, the number of edges in a shortest-path from v to w. The traceroute command is often used for network mapping applications, the study of the connectivity of networks, and it has been studied theoretically with respect to biases it introduces for network mapping when only a subset of nodes in the network can be the source of traceroute queries. In this paper, we provide efficient network mapping algorithms, that are based on kth-hop traceroute queries. Our results include an algorithm that runs in a constant number of parallel rounds with a subquadratic number of queries under reasonable assumptions about the sampling coverage of the nodes that may issue kth-hop traceroute queries. In addition, we introduce a number of new algorithmic techniques, including a high-probability parametric parallelization of a graph clustering technique of Thorup and Zwick, which may be of independent interest