Solving query-retrieval problems by compacting Voronoi diagrams

Aggarwal, Alok; Hansen, Mark D.; Leighton, Tom

doi:10.1145/100216.100260

Cited by 36 publications

(24 citation statements)

References 28 publications

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“…For d = 2, the problem has been solved optimally by Chazelle, Guibas and Lee . For d = 3, Chazelle and Preparata in [Chazelle and Preparata 1985] gave a solution with nearly linear space and O(log n + k) query time, while Aggarwal, Hansen and Leighton [Aggarwal et al 1990] gave a solution with a more expensive preprocessing but O(n log n) space. When the number of dimensions is greater than 4, i.e., d ≥ 4, Clarkson and Shor [Clarkson and Shor.…”

Section: Reporting Points In a Halfspacementioning

confidence: 99%

Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Tsourakakis

Peng

Tsiarli

et al. 2011

ACM J. Exp. Algorithmics

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The development of cancer is largely driven by the gain or loss of subsets of the genome, promoting uncontrolled growth or disabling defenses against it. Denoising array-based Comparative Genome Hybridization (aCGH) data is an important computational problem central to understanding cancer evolution. In this work we propose a new formulation of the denoising problem which we solve with a "vanilla" dynamic programming algorithm which runs in O(n 2 ) units of time. Then, we propose two approximation techniques. Our first algorithm reduces the problem into a well-studied geometric problem, namely halfspace emptiness queries, and provides an additive approximation to the optimal objective value inÕ(n 4 3 +δ log ( U )) time, where δ is an arbitrarily small positive constant and U = max{..,n } (P = (P 1 , P 2 , . . . , Pn), P i ∈ R, is the vector of the noisy aCGH measurements, C a normalization constant). The second algorithm provides a (1± ) approximation (multiplicative error) and runs in O(n log n/ ) time. The algorithm decomposes the initial problem into a small (logarithmic) number of Monge optimization subproblems which we can solve in linear time using existing techniques.Finally, we validate our model on synthetic and real cancer datasets. Our method consistently achieves superior precision and recall to leading competitors on the data with ground truth. In addition, it finds several novel markers not recorded in the benchmarks but supported in the oncology literature.Availability: Code, datasets and results available from the first author's web page

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Section: Reporting Points In a Halfspacementioning

confidence: 99%

Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Tsourakakis

Peng

Tsiarli

et al. 2011

ACM J. Exp. Algorithmics

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“…Aggarwal, Hansen, and Leighton [3] provide a data-structure that supports circular range reporting queries in O (log n +k) time where k is the number of items reported. The structure can be built in O (n log 2 n log log n) time and uses O (n log n) space.…”

Section: Large Fixed σ and D =mentioning

confidence: 99%

Sigma-local graphs

Bose

Collette

Langerman

et al. 2010

Journal of Discrete Algorithms

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We introduce and analyze σ -local graphs, based on a definition of locality by Erickson [J. Erickson, Local polyhedra and geometric graphs, Computational Geometry: Theory and Applications 31 (1-2) (2005) 101-125]. We present two algorithms to construct such graphs, for any real number σ > 1 and any set S of n points. These algorithms run in time O (σ d n + n log n) for sets in R d and O (n log 3 n log log n + k) for sets in the plane, where k is the size of the output.For sets in the plane, algorithms to find the minimum or maximum σ such that the corresponding graph has properties such as connectivity, planarity, and no-isolated vertex are presented, with complexities in O (n log O (1) n). These algorithms can also be used to efficiently construct the corresponding graphs.

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“…We will use a data structure, called partition tree, being proposed by Matoušek [16] that is based on simplicial partition for a set of points, can be build in O(n log n) time and O(n) space and answers queries in O(n 1−1/d+ε + k) time. We notice, that there are more efficient data structures with faster optimal O(log n + k) query time, see [3,10]; however the output produced by these data structures can not be represented as a collection of disjoint canonical subsets which is a main ingredient of our algorithm. In contrary, in our case, for a query halfspace h in R 3 , a partition tree allows to select a set W of O(n 2/3 ) nodes from the tree with the property that the subset of points in h is the disjoint union of the canonical subsets of the selected nodes.…”

Section: Linear Space Shortest Euclidean Pathmentioning

confidence: 99%

On Bounded Leg Shortest Paths Problems

Roditty

Segal

2009

Algorithmica

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Let V be a set of points in a d-dimensional l p -metric space. Let s, t ∈ V and let L be a real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set has length of at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s-t Bounded Leg Shortest Path (st-BLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and a real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper we obtain the following results:1. An algorithm for the apBLSP problem in any l p -metric which, for any fixed ε > 0, computes in O(n 3 (log 3 n +log 2 n ·ε −d )) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1 + ε). It requires O(n 2 log n) space and distance queries are answered in O(log log n) time. 2. An algorithm for the stBLSP problem that, given s, t ∈ V and a real number L, computes in O(n · polylog(n)) the exact L-bounded shortest path from s to t. This algorithm works in l 1 and l ∞ metrics. In the Euclidean metric we also obtain an exact algorithm but with a running time of O(n 4/3+ε ), for any ε > 0.M. Segal was partially supported by REMON (4G) consortium.Algorithmica ( 2011) 59: 583-600 3. For any weighted directed graph we give a data structure of size O(n 2.5 log n) which is capable of answering path queries with a multiplicative error of (1 + ε) in O(log log n + ) time, where is the length of the reported path.Our results improve upon the results given by Bose et al. (Comput. Geom. Theory Appl. 29:233-249, 2004). Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of independent interest.

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Solving query-retrieval problems by compacting Voronoi diagrams

Cited by 36 publications

References 28 publications

Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Sigma-local graphs

On Bounded Leg Shortest Paths Problems

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