We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node q, the algorithm learns either that q is the target, or is given an edge out of q that lies on a shortest path from q to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p > 1 2 (a known constant), and an (adversarial) incorrect one with probability 1 − p.Our main positive result is that when p = 1 (i.e., all answers are correct), log 2 n queries are always sufficient. For general p, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1−δ) log 2 n 1−H(p) +o(log n)+O(log 2 (1/δ)) queries, and identifies the target correctly with probability at least 1−δ. Here, H(p) = −(p log p+(1−p) log(1−p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm.Even for p = 1, we show several hardness results for the problem of determining whether a target can be found using K queries. Our upper bound of log 2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semiadaptive version, in which one may query r nodes each in k rounds, we show membership in Σ 2k−1 in the polynomial hierarchy, and hardness for Σ 2k−5 .

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function g from the n-dimensional hypercube to the bounded interval [−1, 1], and an n × m matrix A with bounded entries, maximize g(Ax) over x in the m-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling).We present an approximation algorithm for this problem when g simultaneously satisfies two "smoothness" properties: Lipschitz continuity with respect to the L ∞ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which lowprobability -and possibly correlated -errors in the inputs of g can impact its output. The approximation guarantee of our algorithm degrades gracefully as a function of the Lipschitz continuity and noise stability of g. In particular, when g is both O(1)-Lipschitz continuous and O(1)-stable, we obtain an (additive) polynomial-time approximation scheme (PTAS) for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive fully polynomial-time approximation scheme (FPTAS).We apply our algorithm for mixture selection to a number of different game-theoretic applications, focusing on problems from mechanism design and optimal signaling. In particular, we make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan [DHN14]; we resolve the problem of revenuemaximizing signaling in Bayesian second-price auctions posed by Emek et al. [EFG + 12] and Miltersen and Sheffet [BMS12]; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi [Dug14]; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and Câmara [AC14].

Mobile ad hoc networks (MANETs) have become very interesting during last years, but the security is the most important problem they suffer from. Asymmetric cryptography is a very useful solution to provide a secure environment in multihop networks where intermediate nodes are able to read, drop or change messages before resending them. However, storing all keys in every node by this approach is inefficient, if practically possible, in large-scale MANETs due to some limitations such as memory or process capability. In this paper, we propose a new probabilistic key management algorithm for large-scale MANETs. To the best of our knowledge, this is the first method which probabilistically uses asymmetric cryptography to manage the keys in MANETs. In this algorithm, we store only a few keys in each node instead of all. We analytically prove that the network will remain connected with a high probability more than 99.99%. Furthermore, we analytically calculate the average path length in the network and show that this parameter will not have a significant increment using our algorithm. All analytical results are also validated by simulation to make them dependable.

In many applications of clustering (for example, ontologies or clusterings of animal or plant species), hierarchical clusterings are more descriptive than a flat clustering. A hierarchical clustering over n elements is represented by a rooted binary tree with n leaves, each corresponding to one element. The subtrees rooted at interior nodes capture the clusters. In this paper, we study active learning of a hierarchical clustering using only ordinal queries. An ordinal query consists of a set of three elements, and the response to a query reveals the two elements (among the three elements in the query) which are "closer" to each other than to the third one. We say that elements x and x are closer to each other than x if there exists a cluster containing x and x , but not x .When all the query responses are correct, there is a deterministic algorithm that learns the underlying hierarchical clustering using at most n log 2 n adaptive ordinal queries. We generalize this algorithm to be robust in a model in which each query response is correct independently with probability p > 1 2 , and adversarially incorrect with probability 1 − p. We show that in the presence of noise, our algorithm outputs the correct hierarchical clustering with probability at least 1 − δ, using O(n log n + n log(1/δ)) adaptive ordinal queries. For our results, adaptivity is crucial: we prove that even in the absence of noise, every non-adaptive algorithm requires Ω(n 3 ) ordinal queries in the worst case.