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检索条件"丛书名=Proceedings in Applied Mathematics"

共 197 条记录，以下是41-50 订阅

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On the Optimality of Spiral Search

On the Optimality of Spiral Search

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Langetepe, Elmar Univ Bonn Inst Comp Sci 1 D-53117 Bonn Germany

ISBN: (纸本)9780898717013

Searching for a point in the plane is a well-known search game problem introduced in the early eighties. The best known search strategy is given by a spiral and achieves a competitive ratio of 17,289... It was shown by Gal [14] that this strategy is the best strategy among all monotone and periodic strategies. Since then it was unknown whether the given strategy is optimal in general. This paper settles this old open fundamental search problem and shows that spiral search is indeed optimal. The given problem can be considered as the continuous version of the well-known m-ray search problem and also appears in several non-geometric applications and modifications. Therefore the optimality of spiral search is an important question considered by many researchers in the last decades. We answer the logarithmic spiral conjecture for the given problem. The lower bound construction might be helpful for similar settings, it also simplifies existing proofs on classical m-ray search.

关键词： Search games computational geometry motion planning spiral search m-ray search competitive analysis lower bound

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A Space-Time Tradeoff for Permutation Problems

A Space-Time Tradeoff for Permutation Problems

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Koivisto, Mikko Parviainen, Pekka Univ Helsinki HIIT FIN-00014 Helsinki Finland

ISBN: (纸本)9780898717013

Many combinatorial problems such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem can be formulated as finding a feasible permutation of n elements Typically, such problems can be solved by dynamic programming in time and space O* (2(n)), by divide and conquer in tune O*(4(n)) and polynomial space, or by a. combination of the two in tune O* (4(n)2(-s)) and space O*(2(s)) [or s = n, n/2, n/4, Here, we show that, one can improve the tradeoff to time O (T-n) and space O*(S-n) with TS < 4 at any root 72 < S < 2 The idea is to find a small faintly of 'thin" partial orders on the n elements such that every linear order is an extension of one member of the family Our constriction is optimal within a natural class of partial order families

关键词： Dynamic programming

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Deletion Without Rebalancing in Balanced Binary Trees

Deletion Without Rebalancing in Balanced Binary Trees

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Sen, Siddhartha Tarjan, Robert E. Princeton Univ Princeton NJ 08544 USA Princeton Univ HP Lab Princeton NJ 08544 USA

ISBN: (纸本)9780898717013

We address the vexing issue of deletions in balanced trees Rebalancing after a deletion is generally more complicated than rebalancing after an insertion Textbooks neglect deletion rebalancing, and many database systems do not do it We describe a relaxation of AVL trees in which rebalancing is done after insertions but not after deletions, yet access time remains logarithmic in the number of insertions For many applications of balanced trees, our structure offers performance competitive with that of classical balanced trees With the addition of periodic rebuilding, the performance of our structure is theoretically superior to that of many if not all classic balanced tree structures Our structure needs O(log log m) bits of balance information pet node, where m is the number of inset Lions, or O(log log n) with periodic rebuilding, where it is the number of nodes. An insertion takes up to two rotations and O(1) amortized tune Using an analysis that relies on an exponential potential function, we show that. rebalancing steps occur with a frequency that is exponentially small in the height of the affected node

关键词： Periodic structures

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Towards a Calculus for Non-Linear Spectral Gaps

Towards a Calculus for Non-Linear Spectral Gaps

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Mendel, Manor Naor, Assaf Open Univ Israel Raanana Israel

ISBN: (纸本)9780898717013

Given a finite regular graph G = (V, E) and a inciti ic space (X, d(X)), let gamma+(G, X) denote the smallest. constant gamma+ > 0 such that for all f,g . V --> X we have. 1/|V|(2) Sigma(t,u is an element of V) d(X)((f(x)g(y))(2) <= gamma+/|E| Sigma(xy is an element of E) d(X) (f(x), g(y))(2) In the special case X ilk this quantity coincides with the reciprocal of the absolute spectral gap of C but hi the' geometries the paramete, gamma+(G, X), which we still think of as *** mg the non-linear spectral gap of C with respect to X (even though there is no actual specti urn present bele), can behave veiy diffeiently Non-linear spectral gaps arise often in the theory of metric ernbedclings, and in the present papei we systematically study the theory of non-linear specti al gaps, partially in older to obtain a combinatorial construction of super-expander - a family of bounded-degree giaphs G(i) = (V-i, E-i), with lim(i-->infinity) = |V-i| = infinity, which do not admit a coal se embedding into any uniformly convex lioi med space In addition, the hi-Lipschitz distortion of in any uniformly convex Banach space is Omega(log |V-i|), which is the woist possible behavior clue to Boulgam's embedding theolem [3] Such remarkable giaph families were previously known to exist due to a toui de force algebraic construction of Lafforgue [11] Our constiuction is different and combinatorial, relying on the zigzag product of Remgold-Vadhan-Wigderson [28] We show that. non-linear spectral gaps behave submultiplicatively under zigzag products a fact that amounts to a simple iteration of the inequality above This yields as a special ca-se a very simple (linear algebra free) proof of the Reingold-Vadhan-Wigclerson theorem winch states that zigzag products mese' ve the pi operty of having an absolute spectral gap (with quantitative control on the size of the gal)) The zigzag iteration of Reingold-Vadhan-Wigderson also involves taking graph poweis, which is trivial to analyze in the classic

关键词： Graph theory

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Counting Stars and Other Small Subgraphs in Sublinear Time

Counting Stars and Other Small Subgraphs in Sublinear Time

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Gonen, Mira Ron, Dana Shavitt, Yuval Tel Aviv Univ Sch Elect Engn IL-69978 Tel Aviv Israel

ISBN: (纸本)9780898717013

Detecting and counting the number of copies of certain subgraphs (also known as network motifs or gniphlets), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web Several polynomial-time algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design sub/mum-tune algorithms for approximating the number of copies of certain constant-size subgraphs in a graph C 'That is, our algorithms do not read the whole graph, but rather query parts of the graph Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length-2 paths and more generally on counting the number of stars of a certain size. Specifically, we design an algorithm that, given an approximation parameter 0 < epsilon < 1 and query access to a graph C, outputs an estimate (nu) over cap (s), such that with high constant probability, (1-epsilon)v(s)(G) <= nu(s) <= (1+ epsilon)vs(G) where nu(s)(G) denotes the number of stars of size 8+1 in the graph The expected query complexity and running time of the algorithm are O (n / (v(s)(G)_(1/ n+1) + min {n(1)s-(1/2)(v(s)(G))(1-1/s})) poly(logn, 1/epsilon). We also prove lower bounds showing that this algorithm is tight up to polylogarithmic factors in 71 and the dependence on E Our work extends the work of Feige (SIAM Journal on Computing, 2006) and Coldreich and Ron (Random Structures and Algorithms, 2008) on approximating the number of edges (or average degree) in a graph Combined with these results, our result can be used to obtain an estimate on the variance of the degrees in the graph and corresponding higher moments. In addi

关键词： Stars

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Cell-Probe Lower Bounds for Succinct Partial Sums

Cell-Probe Lower Bounds for Succinct Partial Sums

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21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Patrascu, Mihai Viola, Emanuele AT and T Labs. United States Northeastern University United States

ISBN: (纸本)9780898717013

The partial sums problem in succinct data structures asks to preprocess an array A[1..n] of bits into a data structure using as close to n bits as possible, and answer queries of the form RANK(k) = Sigma(k)(i=1) A[i]. The problem has been intensely studied, and features as a subroutine in a number of succinct data structures We show that, if we answer RANK(k) queries by probing t cells of omega bits, then the space of the data structure must be at least n+n/omega O(t) bits. This redundancy/probe trade-off is essentially optimal: Patrascu [FOCS'08] showed how to achieve n + n/(w/t)(Omega(t)) bits. We also extend our lower bound to the closely related SELECT queries, and to the case of sparse arrays.

关键词： Data structures

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Speeding up random walks with neighborhood exploration

Speeding up random walks with neighborhood exploration

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Berenbrink, Petra Cooper, Colin Elsaesser, Robert Radzik, Tomasz Sauerwald, Thomas Simon Fraser Univ Burnaby BC V5A 1S6 Canada Kings Coll London London WC2R 2LS England Univ Paderborn Paderborn Germany

ISBN: (纸本)9780898717013

We consider the following marking process (RW-RAND) made by a random walk on an undirected graph Upon arrival at a vertex v, it marks v if unmarked and otherwise it marks a randomly chosen unmarked neighbor of v. We also consider a variant of this process called RW-R-RANK. Here each vertex is assigned a, global random rank first and then in each step, the walk marks the lowest ranked unmarked neighbor of the currently visited vertex Depending on the degree and the expansion of the graph, we prove several upper bounds on the time required by these processes to mark all vertices. For instance, if G is a hypercube or random graph, our processes mark all vertices in time O(n), significantly speeding up the circle minus(n, log n)-cover time of standard random walks

关键词： Undirected graphs

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The scaling window for a random graph with a given degree sequence 10

The scaling window for a random graph with a given degree se...

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Hatami, Hamed Molloy, Michael Univ Toronto Dept Comp Sci Toronto ON M5S 1A1 Canada

ISBN: (纸本)9780898717013

We consider a random graph on a given degree sequence D, satisfying certain conditions. We focus on two parameters Q = Q(D), R = R(D). Molloy and Reed proved that Q = 0 is the threshold for the random graph to have a giant component. We prove that if vertical bar Q vertical bar = O(n(-1/3) R-2/3) then, with high probability, the size of the largest component of the random graph will be of order circle minus(n(2/3) R-1/3) If Q is asymptotically larger/smaller that n(-1/3) R-2/3 then the size of the largest component is asymptotically larger/smaller than n(2/3) R-1/3. In other words, we establish that vertical bar Q vertical bar = O(n(-1/3) R-2/3) is the scaling window.

关键词： Quality factor Line graph Random LARGE-COMPONENT Reed asymptotic Scale formation

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A Polynomial Time Approximation Scheme for k-Consensus Clustering

A Polynomial Time Approximation Scheme for <i>k</i>-Consensu...

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Coleman, Tom Wirth, Anthony Univ Melbourne Melbourne Vic 3010 Australia

ISBN: (纸本)9780898717013

This paper introduces a polynomial time approximation scheme for the metric CORRELATION CLUSTERING problem, when the number of clusters returned is bounded (by k). CONSENSUS CLUSTERING is a fundamental aggregation problem, with considerable application, and it is analysed here as a metric variant of the CORRELATION CLUSTERING problem. The PTAS exploits a connection between CORRELATION CLUSTERING and the k-CUT problems. This requires the introduction of a new rebalancing technique, based on minimum cost perfect matchings, to provide clusters of the required sizes. Both CONSENSUS CLUSTERING and CORRELATION CLUSTERING have been the focus of considerable recent study. There is an existing dichotomy between the k-restricted CORRELATION CLUSTERING problems and the unrestricted versions The former, in general, admit a PTAS, whereas the latter are, in general, APX-hard. This paper extends the dichotomy to the metric case, responding to the result that CONSENSUS CLUSTERING is APX-hard to approximate.

关键词： Polynomial approximation

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A locality-sensitive hash for real vectors

A locality-sensitive hash for real vectors

引用

21st Annual ACM/SIAM Symposium on Discrete Algorithms

作者： Neylon, Tyler

ISBN: (纸本)9780898717013

We present a simple and practical algorithm for the c approximate near neighbor problem (c-NN): given a points P subset of R-d and radius R. build a data structure which, given q is an element of R-d, can with probability 1 - delta return a point p is an element of P with dist(p,q) <= cR if there is any p* is an element of P with dist(p*,q) <= R. For c = d + 1, our algorithm deterministically (delta = 0) preprocesses in time O(nd log d), space O(dn), and answers queries in expected time O(d(2));this is the first known algorithm to deterministically guarantee an O(d)-NN solution in constant time with respect to n for all l(p) metrics. A probabilistic version empirically achieves useful c values (c < 2) where c appears to grow minimally as d -> infinity. A query time of O(d log d) is available, providing slightly less accuracy. These techniques can also be used to approximately find (pointers between) all pairs x, y is an element of P with dist(x,y) <= R in time O(nd log d). The key to the algorithm is a locality-sensitive hash a mapping h : R-d -> U with the property that h(x) = h(y) is much more likely for near by x, y We introduce a somewhat regular simplex which tessellates R-d, and efficiently hash each point in any simplex of this tessellation to all d corners;any points in neighboring cells will be hashed to a shared cornet and noticed as nearby points. This method is completely independent of dimension reduction, so that additional space and time savings are available by first reducing all input vectors.

关键词： Vector spaces

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