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Given a weighted undirected graph, our basic goal is to represent all pair wise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse...
Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed base point. After O(|G|) time preprocessing independent of c and d, our algorithms answer the homotopy test in O(|c| + |d|) time, where |G|, |c| and |d| are the respective...
There has been considerable interest lately in the complexity of distributions. Recently, Lovett and Viola (CCC 2011) showed that the statistical distance between a uniform distribution over a good code, and any distribution which can be efficiently sampled by a small bounded-depth AC0 circuit, is inverse-polynomially close to one. That is, such distributions are very far from each other. We strengthen...
Spectral partitioning is a simple, nearly-linear time, algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee of the quality of the approximation found by the algorithm. Local graph partitioning algorithms [ST08, ACL06, AP09] run in time that is nearly linear in the size of the output set, and their approximation guarantee is worse than the guarantee provided by...
We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an $\varepsilon$-approximate solution in $O(m(m+\log n)\log(MUm/\varepsilon))$ arithmetic...
In this work, we study the problem of constructing interactive protocols that are robust to noise, a problem that was originally considered in the seminal works of Schulman (FOCS '92, STOC '93), and has recently regained popularity. Robust interactive communication is the interactive analogue of error correcting codes: Given an interactive protocol which is designed to run on an error-free channel,...
We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near-optimal seed-length even in the low-error regime: We get seed-length...
A strong direct product theorem for a problem in a given model of computation states that, in order to compute k instances of the problem, if we provide resource which is less than k times the resource required for computing one instance of the problem with constant success probability, then the probability of correctly computing all the k instances together, is exponentially small in k. In this paper,...
In this paper we show that for any mechanism design problem with the objective of maximizing social welfare, the exponential mechanism can be implemented as a truthful mechanism while still preserving differential privacy. Our instantiation of the exponential mechanism can be interpreted as a generalization of the VCG mechanism in the sense that the VCG mechanism is the extreme case when the privacy...
Asynchronous task allocation is a fundamental problem in distributed computing in which p asynchronous processes must execute a set of m tasks. Also known as write-all or do-all, this problem been studied extensively, both independently and as a key building block for various distributed algorithms. In this paper, we break new ground on this classic problem: we introduce the To-Do Tree concurrent...
We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is ``intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T, with each tile of T represented...
The problem of \textit{Text Indexing} is a fundamental algorithmic problem in which one wishes to preprocess a text in order to quickly locate pattern queries within the text. In the ever evolving world of dynamic and on-line data, there is also a need for developing solutions to index texts which arrive on-line, i.e.~a character at a time, and still be able to quickly locate said patterns. In this...
The introduction of a non-black-box simulation technique by Barak (FOCS 2001) has been a major landmark in cryptography, breaking the previous barriers of black-box impossibility. Barak's techniques were subsequently extended and have given rise to various powerful applications. We present the first non-black-box simulation technique that does not rely on Barak's technique (or on non-standard assumptions)...
Convex relaxations based on different hierarchies of linear/semi-definite programs have been used recently to devise approximation algorithms for various optimization problems. The approximation guarantee of these algorithms improves with the number of {\em rounds} in the hierarchy, though the complexity of solving (or even writing down the solution for) the $r$'th level program grows as $n^{\Omega(r)}$...
We consider the problem of finding a minimum edge cost sub graph of an undirected or a directed graph satisfying given connectivity requirements and degree bounds b(·) on nodes. We present an iterative rounding algorithm for this problem. When the graph is undirected and the connectivity requirements are on the element-connectivity with maximum value k, our algorithm computes a solution that is an...
In the Edge-Disjoint Paths with Congestion problem (\EDPwC), we are given an undirected $n$-vertex graph $G$, a collection $\mset=\set{(s_1, t_1), \ldots, (s_k, t_k)}$ of demand pairs and an integer $c$. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by $c$. When the maximum allowed...
We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V \subseteq \R^d$, we compute a $(1+\epsilon)$-factor approximation to deterministically in time $\poly(d)\cdot |V|^{O_\epsilon(1)}$. Previously, only a constant factor deterministic polynomial time approximation...
In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NP-hard, even when centers have no capacity restrictions...
We present a quantum algorithm solving the k-distinctness problem in a less number of queries than the previous algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler.
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