4,080 research outputs found

    Balanced Allocation on Graphs: A Random Walk Approach

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    In this paper we propose algorithms for allocating nn sequential balls into nn bins that are interconnected as a dd-regular nn-vertex graph GG, where d3d\ge3 can be any integer.Let ll be a given positive integer. In each round tt, 1tn1\le t\le n, ball tt picks a node of GG uniformly at random and performs a non-backtracking random walk of length ll from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that GG has a sufficiently large girth and d=ω(logn)d=\omega(\log n). Then we establish an upper bound for the maximum number of balls at any bin after allocating nn balls by the algorithm, called {\it maximum load}, in terms of ll with high probability. We also show that the upper bound is at most an O(loglogn)O(\log\log n) factor above the lower bound that is proved for the algorithm. In particular, we show that if we set l=(logn)1+ϵ2l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor, for every constant ϵ(0,1)\epsilon\in (0, 1), and GG has girth at least ω(l)\omega(l), then the maximum load attained by the algorithm is bounded by O(1/ϵ)O(1/\epsilon) with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on dd-regular graph with d[3,O(logn)]d\in[3, O(\log n)] and sufficiently large girth

    Clustering, Hamming Embedding, Generalized LSH and the Max Norm

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    We study the convex relaxation of clustering and hamming embedding, focusing on the asymmetric case (co-clustering and asymmetric hamming embedding), understanding their relationship to LSH as studied by (Charikar 2002) and to the max-norm ball, and the differences between their symmetric and asymmetric versions.Comment: 17 page

    Heavy Hitters and the Structure of Local Privacy

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    We present a new locally differentially private algorithm for the heavy hitters problem which achieves optimal worst-case error as a function of all standardly considered parameters. Prior work obtained error rates which depend optimally on the number of users, the size of the domain, and the privacy parameter, but depend sub-optimally on the failure probability. We strengthen existing lower bounds on the error to incorporate the failure probability, and show that our new upper bound is tight with respect to this parameter as well. Our lower bound is based on a new understanding of the structure of locally private protocols. We further develop these ideas to obtain the following general results beyond heavy hitters. \bullet Advanced Grouposition: In the local model, group privacy for kk users degrades proportionally to k\approx \sqrt{k}, instead of linearly in kk as in the central model. Stronger group privacy yields improved max-information guarantees, as well as stronger lower bounds (via "packing arguments"), over the central model. \bullet Building on a transformation of Bassily and Smith (STOC 2015), we give a generic transformation from any non-interactive approximate-private local protocol into a pure-private local protocol. Again in contrast with the central model, this shows that we cannot obtain more accurate algorithms by moving from pure to approximate local privacy

    The number of independent sets in a graph with small maximum degree

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    Let ind(G){\rm ind}(G) be the number of independent sets in a graph GG. We show that if GG has maximum degree at most 55 then ind(G)2iso(G)uvE(G)ind(Kd(u),d(v))1d(u)d(v) {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} (where d()d(\cdot) is vertex degree, iso(G){\rm iso}(G) is the number of isolated vertices in GG and Ka,bK_{a,b} is the complete bipartite graph with aa vertices in one partition class and bb in the other), with equality if and only if each connected component of GG is either a complete bipartite graph or a single vertex. This bound (for all GG) was conjectured by Kahn. A corollary of our result is that if GG is dd-regular with 1d51 \leq d \leq 5 then ind(G)(2d+11)V(G)2d, {\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d}, with equality if and only if GG is a disjoint union of V(G)/2dV(G)/2d copies of Kd,dK_{d,d}. This bound (for all dd) was conjectured by Alon and Kahn and recently proved for all dd by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most 33 the search could be done by hand, but for the case of maximum degree 44 or 55, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

    Compressibility and probabilistic proofs

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    We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lov\'asz local lemma.Comment: Invited talk for CiE 2017 (full version

    On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

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    We study classic streaming and sparse recovery problems using deterministic linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the latter also being known as l1-heavy hitters), norm estimation, and approximate inner product. We focus on devising a fixed matrix A in R^{m x n} and a deterministic recovery/estimation procedure which work for all possible input vectors simultaneously. Our results improve upon existing work, the following being our main contributions: * A proof that linf/l1 sparse recovery and inner product estimation are equivalent, and that incoherent matrices can be used to solve both problems. Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms by making use of the Fast Johnson-Lindenstrauss transform. Both our running times and number of measurements improve upon previous work. We can also obtain better error guarantees than previous work in terms of a smaller tail of the input vector. * A new lower bound for the number of linear measurements required to solve l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude. * A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of measurements required to solve deterministic norm estimation, i.e., to recover |x|_2 +/- eps|x|_1. For all the problems we study, tight bounds are already known for the randomized complexity from previous work, except in the case of l1/l1 sparse recovery, where a nearly tight bound is known. Our work thus aims to study the deterministic complexities of these problems