A given Boolean function has its input distributed among many parties. The aim is to determine which parties to tMk to and what information to exchange with each of them in order to evaluate the function while minimizing the total communication. This paper shows that it is possible to obtain the Boolean answer deterministically with only a polynomial increase in communication with respect to the information lower bound given by the nondeterministic communication complexity of the function.

We study the question of determining whether an unknown function has a particular property or is ffl-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being k-colorable or having a ae-clique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edge-queries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...

We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NP-complete problems the current upper and lower bounds lie exponentially ...

this technical aspect (as in the bounded-degree model the closest graph having the property must have at most dN edges and degree bound d as well).

In this paper we give approximation algorithms for the following problems on metric spaces: Furthest Pair, k- median, Minimum Routing Cost Spanning Tree, Multiple Sequence Alignment, Maximum Traveling Salesman Problem, Maximum Spanning Tree and Average Distance. The key property of our algorithms is that their running time is linear in the number of metric space points. As the full specification o`f an n-point metric space is of size \Theta(n 2 ), the complexity of our algorithms is sublinear with respect to the input size. All previous algorithms (exact or approximate) for the problems we consider have running time\Omega\Gamma n 2 ). We believe that our techniques can be applied to get similar bounds for other problems. 1 Introduction In recent years there has been a dramatic growth of interest in algorithms operating on massive data sets. This poses new challenges for algorithm design, as algorithms quite efficient on small inputs (for example, having quadratic running time) ...

We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.

We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We consider (randomized) algorithms which may query the function at arguments of their choice, and seek algorithms which query the function at relatively few places. We focus on combinatorial properties, and specifically on graph properties. The two standard representations of graphs -- by adjacency matrices and by incidence lists -- yield two different models for testing graph properties. In the first model, most appropriate for dense graphs, distance between N-vertex graphs is measured as the fraction of edges on which the graphs disagree over N 2 . In the second model, most appropriate for bounded-degree graphs, distance between N-vertex d-degree graphs is measured as the fraction of edges on ...

this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex). There have been other adaptations of Morse Theory that can be applied to combinatorial spaces. For example, a Morse theory of piecewise linear functions appears in [26] and the very powerful \Stratied Morse Theory" was developed by Goresky and MacPherson [19],[20]. These theories, especially the latter, have each been successfully applied to prove some very striking results

Abstract. A quorum system is a collection of sets (quorums) every two of which intersect. Quorum systems have been used for many applications in the area of distributed systems, including mutual exclusion, data replication, and dissemination of information. When the elements may fail, a user of a distributed protocol needs to quickly find a quorum all of whose elements are alive or evidence that no such quorum exists. This is done by probing the system elements, one at a time, to determine if they are alive or dead. This paper studies the probe complexity PC(S) of a quorum system S, defined as the worst case number of probes required to find a live quorum or to show its nonexistence in S, using the best probing strategy. We show that for large classes of quorum systems, all n elements must be probed in the worst case. Suchsystems are called evasive. However, not all quorum systems are evasive; we demonstrate a system where O(log n) probes always suffice. Then we prove two lower bounds on the probe complexity in terms of the minimal quorum cardinality c(S) and the number of minimal quorums m(S). Finally, we show a universal probe strategy which never makes more than c(S) 2 − c(S) +1 probes; thus any system with c(S) ≤ √ n is nonevasive.

We study the problem of combinatorial search for graphs under the additive model. The main result concerns the reconstruction of bounded degree graphs, i.e. graphs with the degree of all vertices bounded by a constant d. We show that such graphs can be reconstructed in O(dn) non-adaptive queries, that matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. In particular, a new upper bound is obtained for d-separating matrices, that settles an open question stated by Lindstr#m in [16]. Finally, we consider several particular classes of graphs. We show how an optimal non-adaptive solution of O(n²/log n) queries for general graphs can be obtained.