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We define the following parameter of connected graphs. For a given graph G = (V, E) we place one agent in each vertex v ∈ V . Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of G (not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge become acquainted, and the process continues. We define the acquaintance time of a graph G, denoted by AC(G), to be the minimal number of rounds required until every two agents are acquainted. We first study the acquaintance time for some natural families of graphs including the path, expanders, the binary tree, and the complete bipartite graph. We also show that for all n ∈ N and for all positive integers k ≤ n 1.5 there exists an n-vertex graph G such that k/c ≤ AC(G) ≤ c · k for some universal constant c ≥ 1. We also prove that for all n-vertex connected graphs G we have AC(G) = O( n 2 log(n)/ log log(n) ), thus improving the trivial upper bound of O(n 2 ) achieved by sequentially letting each agent perform depth-first search along some spanning tree of G. Studying the computational complexity of this problem, we prove that for any constant t ≥ 1 the problem of deciding that a given graph G has AC(G) ≤ t or AC(G) ≥ 2t is N P-complete. That is, AC(G) is N P-hard to approximate within multiplicative factor of 2, as well as within any additive constant factor. On the algorithmic side, we give a deterministic algorithm that given an n-vertex graph G with AC(G) = 1 finds a strategy for acquaintance that consists of n/c matchings in time n c+O(1) . We also design a randomized polynomial time algorithm that given an n-vertex graph G with AC(G) = 1 finds with high probability an O(log(n))-rounds strategy for acquaintance.

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We initiate the study of testing properties of images that correspond to sparse 0/1-valued matrices of size n×n. Our study is related to but different from the study initiated by Raskhodnikova (Proceedings of RANDOM, 2003 ), where the images correspond to dense 0/1-valued matrices. Specifically, while distance between images in the model studied by Raskhodnikova is the fraction of entries on which the images differ taken with respect to all n 2 entries, the distance measure in our model is defined by the fraction of such entries taken with respect to the actual number of 1's in the matrix. We study several natural properties: connectivity, convexity, monotonicity, and being a line. In all cases we give testing algorithms with sublinear complexity, and in some of the cases we also provide corresponding lower bounds.

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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is "far" (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is known. Width-2 OBDDs generalize two classes of functions that have been studied in the context of property testing -linear functions (over GF (2)) and monomials. In both these cases membership can be tested in time that is linear in 1/ . Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number, n, of variables in the tested function, we show that (one-sided error) testing for computability by a width-2 OBDD requires Ω(log(n)) queries, and give an algorithm (with one-sided error) that tests for this property and performs O(log(n)/ ) queries.

We study a basic problem of approximating the size of an unknown set S in a known universe U . We consider two versions of the problem. In both versions, the algorithm can specify subsets T ⊆ U . In the first version, which we refer to as the group query or subset query version, the algorithm is told whether T ∩ S is nonempty. In the second version, which we refer to as the subset sampling version, if T ∩ S is nonempty, then the algorithm receives a uniformly selected element from T ∩ S . We study the difference between these two versions in both the case that the algorithm is adaptive and the case in which it is nonadaptive. Our main focus is on a natural family of allowed subsets, which correspond to intervals, as well as variants of this family.