1,721,071 research outputs found
Minimum Dominating Sets of Intervals on Lines
No full textWe study the problem of computing minimum dominating sets of n intervals on lines in three cases: (1) the lines intersect at a single point, (2) all lines except one are parallel, and (3) one line with t weighted points on it and the minimum dominating set must maximize the sum of the weights of the points covered. We propose polynomial-time algorithms for the first two problems, which are special cases of the minimum dominating set problem for path graphs which is known to be NP-hard. The third problem requires identifying the structure of minimum dominating sets of intervals on a line so as to be able to select one that maximizes the weight sum of the weighted points covered. Assuming that presorting has been performed, the first problem has an O(n) time solution, while the second and the third problems are solved by dynamic programming algorithms, requiring O(n log n) and O(n + t) time, respectively. Contact author: Siu-Wing Cheng, email [email protected] y Research partially sup..
A Fast Algorithm for Computing Optimal Rectilinear Steiner Trees for Extremal Point Sets (Extended Abstract)
No full textSiu-Wing Cheng and Chi-Keung Tang Department of Computer Science The Hong Kong University of Science and Technology Clear Water Bay, Hong Kong Contact author e-mail: [email protected]. Abstract. We present a fast algorithm to compute an optimal rectilinear Steiner tree for extremal point sets. A point set is extremal if each point lies on the boundary of a rectilinear convex hull of the point set. Our algorithm can be used in homotopic routing in VLSI layout design and it runs in O(k 2 n) time, where n is the size of the point set and k is the size of its rectilinear convex hull. 1 Introduction Routing in VLSI layout design calls for connecting terminals in the same plane with horizontal and vertical wiring. The terminals are points that sit on the boundaries of isothetic modules which are impenetrable rectilinear polygons. The terminals are divided into groups known as nets [9] and the terminals in each net are to be wired together in the routing region outside the modules. The con..
Extensions of Self-Improving Sorters
Get PDFAilon et al. (SICOMP 2011) proposed a self-improving sorter that tunes its performance to the unknown input distribution in a training phase. The distribution of the input numbers x_1,x_2,...,x_n must be of the product type, that is, each x_i is drawn independently from an arbitrary distribution D_i, and the D_i's are independent of each other. We study two extensions that relax this requirement. The first extension models hidden classes in the input. We consider the case that numbers in the same class are governed by linear functions of the same hidden random parameter. The second extension considers a hidden mixture of product distributions
Adaptive Planar Point Location
Get PDFWe present a self-adjusting point location structure for convex subdivisions. Let n be the number of vertices in a convex subdivision S. Our structure for S uses O(n) space and processes any online query sequence sigma in O(n + OPT) time, where OPT is the minimum time required by any linear decision tree for answering point location queries in S to process sigma. The O(n + OPT) time bound includes the preprocessing time. Our result is a two-dimensional analog of the static optimality property of splay trees. For connected subdivisions, we achieve a processing time of O(|sigma| log log n + n + OPT)
Restricted Max-Min Allocation: Approximation and Integrality Gap
Get PDFAsadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most 4. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a (6 + delta)-approximation algorithm where delta can be any positive constant, and there is still a gap of roughly 2. In this paper, we narrow the gap significantly by proposing a (4+delta)-approximation algorithm where delta can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is poly(m,n)* n^{poly(1/(delta))} where n is the number of players and m is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to 3 + 21/26 =~ 3.808
Dynamic Distribution-Sensitive Point Location
Get PDFWe propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in ℝ², and we are given an oracle that can return in O(1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision S with n vertices such that each query is answered in O(OPT) expected time, where OPT is the minimum expected time of the best linear decision tree for point location in S. The space and construction time are O(n log² n). An update of S as a mixed sequence of k edge insertions and deletions takes O(k log⁵ n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log⁵ n) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time
Finding faithful boyce-codd normal form decompositions
No full textIt is well known that faithful (i.e. dependency preserving) decompositions of relational database schemas into Boyce-Codd Normal Form (BCNF) do not always exist, depending on the set of functional dependencies given, and that the corresponding decision problem is NP-hard. The only algorithm to guarantee both faithfulness and BCNF (if possible) proposed so far in [Os79] is a brute-force approach which always requires exponential time. To be useful in practice, e.g. in automated design tools, we require more efficient means. In this paper we present an algorithm which always finds a faithful BCNF decomposition if one exists, and which is usually efficient, and exponential only in notorious cases
Approximate Nearest Neighbor for Polygonal Curves Under Fréchet Distance
Get PDFWe propose κ-approximate nearest neighbor (ANN) data structures for n polygonal curves under the Fréchet distance in ℝ^d, where κ ∈ {1+ε,3+ε} and d ≥ 2. We assume that every input curve has at most m vertices, every query curve has at most k vertices, k ≪ m, and k is given for preprocessing. The query times are Õ(k(mn)^{0.5+ε}/ε^d+ k(d/ε)^O(dk)) for (1+ε)-ANN and Õ(k(mn)^{0.5+ε}/ε^d) for (3+ε)-ANN. The space and expected preprocessing time are Õ(k(mnd^d/ε^d)^O(k+1/ε²)) in both cases. In two and three dimensions, we improve the query times to O(1/ε)^O(k) ⋅ Õ(k) for (1+ε)-ANN and Õ(k) for (3+ε)-ANN. The space and expected preprocessing time improve to O(mn/ε)^O(k) ⋅ Õ(k) in both cases. For ease of presentation, we treat factors in our bounds that depend purely on d as O(1). The hidden polylog factors in the big-Õ notation have powers dependent on d
Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions
Get PDFWe propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The n input points are drawn from a hidden mixture of product distributions; we are only given an upper bound m = o(√n) on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of Ω(1/n). For any ε ∈ (0,1), after spending O(mn log^O(1)(mn) + m^ε n^(1+ε) log(mn)) time in a training phase, our algorithm achieves an O(1/ε n log m + 1/ε n 2^O(log^* n) + 1/ε H) expected running time with probability at least 1 - O(1/n), where H is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to O(1/ε n log m + 1/ε H)
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