760 research outputs found

    Augmenting graphs to minimize the diameter

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    We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.Comment: 15 pages, 3 figure

    Optimal Color Range Reporting in One Dimension

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    Color (or categorical) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a \emph{color}. While the answer to an orthogonal point reporting query contains all points in the query range QQ, the answer to a color reporting query contains only distinct colors of points in QQ. In this paper we describe an O(N)-space data structure that answers one-dimensional color reporting queries in optimal O(k+1)O(k+1) time, where kk is the number of colors in the answer and NN is the number of points in the data structure. Our result can be also dynamized and extended to the external memory model

    Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

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    Let GG be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex vv and a label λ\lambda, returns a (1+ε)(1+\varepsilon)-approximation of the distance from vv to the closest vertex with label λ\lambda in GG. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements

    Dynamic Range Majority Data Structures

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    Given a set PP of coloured points on the real line, we study the problem of answering range α\alpha-majority (or "heavy hitter") queries on PP. More specifically, for a query range QQ, we want to return each colour that is assigned to more than an α\alpha-fraction of the points contained in QQ. We present a new data structure for answering range α\alpha-majority queries on a dynamic set of points, where α(0,1)\alpha \in (0,1). Our data structure uses O(n) space, supports queries in O((lgn)/α)O((\lg n) / \alpha) time, and updates in O((lgn)/α)O((\lg n) / \alpha) amortized time. If the coordinates of the points are integers, then the query time can be improved to O(lgn/(αlglgn)+(lg(1/α))/α))O(\lg n / (\alpha \lg \lg n) + (\lg(1/\alpha))/\alpha)). For constant values of α\alpha, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for d2d \ge 2, as well as dynamic arrays, in which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201

    Cross-Document Pattern Matching

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    We study a new variant of the string matching problem called cross-document string matching, which is the problem of indexing a collection of documents to support an efficient search for a pattern in a selected document, where the pattern itself is a substring of another document. Several variants of this problem are considered, and efficient linear-space solutions are proposed with query time bounds that either do not depend at all on the pattern size or depend on it in a very limited way (doubly logarithmic). As a side result, we propose an improved solution to the weighted level ancestor problem

    Lower bounds in the quantum cell probe model

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    We introduce a new model for studying quantum data structure problems --- the "quantum cell probe model". We prove a lower bound for the static predecessor problem in the 'address-only' version of this model where, essentially, we allow quantum parallelism only over the 'address lines' of the queries. This model subsumes the classical cell probe model, and many quantum query algorithms like Grover's algorithm fall into this framework. We prove our lower bound by obtaining a round elimination lemma for quantum communication complexity. A similar lemma was proved by Miltersen, Nisan, Safra and Wigderson for classical communication complexity, but their proof does not generalise to the quantum setting. We also study the static membership problem in the quantum cell probe model. Generalising a result of Yao, we show that if the storage scheme is 'implicit', that is it can only store members of the subset and 'pointers', then any quantum query scheme must make \Omega(\log n) probes. We also consider the one-round quantum communication complexity of set membership and show tight bounds

    Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search

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    The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distribution H\mathcal{H} over locality-sensitive hash functions that partition space. For a collection of nn points, after preprocessing, the query time is dominated by O(nρlogn)O(n^{\rho} \log n) evaluations of hash functions from H\mathcal{H} and O(nρ)O(n^{\rho}) hash table lookups and distance computations where ρ(0,1)\rho \in (0,1) is determined by the locality-sensitivity properties of H\mathcal{H}. It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to O(log2n)O(\log^2 n), leaving the query time to be dominated by O(nρ)O(n^{\rho}) distance computations and O(nρlogn)O(n^{\rho} \log n) additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework, making it a viable replacement in practice. Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to O(nρ)O(n^\rho).Comment: 15 pages, 3 figure

    String Indexing for Patterns with Wildcards

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    We consider the problem of indexing a string tt of length nn to report the occurrences of a query pattern pp containing mm characters and jj wildcards. Let occocc be the number of occurrences of pp in tt, and σ\sigma the size of the alphabet. We obtain the following results. - A linear space index with query time O(m+σjloglogn+occ)O(m+\sigma^j \log \log n + occ). This significantly improves the previously best known linear space index by Lam et al. [ISAAC 2007], which requires query time Θ(jn)\Theta(jn) in the worst case. - An index with query time O(m+j+occ)O(m+j+occ) using space O(σk2nlogklogn)O(\sigma^{k^2} n \log^k \log n), where kk is the maximum number of wildcards allowed in the pattern. This is the first non-trivial bound with this query time. - A time-space trade-off, generalizing the index by Cole et al. [STOC 2004]. We also show that these indexes can be generalized to allow variable length gaps in the pattern. Our results are obtained using a novel combination of well-known and new techniques, which could be of independent interest

    Dynamic Set Intersection

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    Consider the problem of maintaining a family FF of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given S,SFS,S'\in F, report every member of SSS\cap S' in any order. We show that in the word RAM model, where ww is the word size, given a cap dd on the maximum size of any set, we can support set intersection queries in O(dw/log2w)O(\frac{d}{w/\log^2 w}) expected time, and updates in O(logw)O(\log w) expected time. Using this algorithm we can list all tt triangles of a graph G=(V,E)G=(V,E) in O(m+mαw/log2w+t)O(m+\frac{m\alpha}{w/\log^2 w} +t) expected time, where m=Em=|E| and α\alpha is the arboricity of GG. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in O(mα)O(m \alpha) time. We provide an incremental data structure on FF that supports intersection {\em witness} queries, where we only need to find {\em one} eSSe\in S\cap S'. Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected time, where N=SFSN=\sum_{S\in F} |S|. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using MM words of space, each update costs O(MlogN)O(\sqrt {M \log N}) expected time, each reporting query costs O(NlogNMop+1)O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1}) expected time where opop is the size of the output, and each witness query costs O(NlogNM+logN)O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N) expected time.Comment: Accepted to WADS 201
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