Space-Efficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting

JáJá, Joseph F.; Mortensen, Christian Worm; Shi, Qingmin

doi:10.1007/978-3-540-30551-4_49

Cited by 58 publications

(36 citation statements)

References 13 publications

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“…In Figure 2, critical points dominated by point (4,11) are represented by curves whose both endpoints fit between the two vertical lines, corresponding to index values i = 4 and j = 11. Note that there are exactly two such curves, and that A(4, 11) = 11 − 4 − 2 = 5.…”

Section: Definitionmentioning

confidence: 99%

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Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs

Tiskin

2006

Combinatorial Pattern Matching

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Abstract. For two strings a, b, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS. In a previous paper, we defined a generalisation, called "the all semi-local LCS problem", for which we proposed an efficient output representation and an efficient algorithm. In this paper, we consider a restriction of this problem to strings that are permutations of a given set. The resulting problem is equivalent to the all local longest increasing subsequences (LIS) problem. We propose an algorithm for this problem, running in time O(n 1.5 ) on an input of size n. As an interesting application of our method, we propose a new algorithm for finding a maximum clique in a circle graph on n nodes, running in the same asymptotic time O(n 1.5 ). Compared to a number of previous algorithms for this problem, our approach presents a substantial improvement in worst-case running time.

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Section: Definitionmentioning

confidence: 99%

“…Time and memory asymptotics given in the theorem can be improved by using a more advanced data structure due to JaJa et al [11]; however, Theorem 2 is sufficient for our current purposes.…”

Section: Definitionmentioning

confidence: 99%

Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs

Tiskin

2006

Combinatorial Pattern Matching

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“…As above, but the range tree is replaced by the asymptotically more efficient data structure of [8].…”

Section: Theorem 2 For An Arbitrary Integer Pointmentioning

confidence: 99%

“…Individual output lengths can be obtained from this representation by dominance counting queries. This leads to a data structure of size O(m+n), that allows to query an individual output length in time O log(m+n) log log(m+n) , using a recent result by JaJa, Mortensen and Shi [8]. The described approach presents a substantial improvement in query efficiency over previous approaches.…”

Section: Introductionmentioning

confidence: 99%

All Semi-local Longest Common Subsequences in Subquadratic Time

Tiskin

2006

Computer Science – Theory and Applications

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Abstract. For two strings a, b of lengths m, n respectively, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS. In this paper, we define a generalisation, called "the all semi-local LCS problem", where each string is compared against all substrings of the other string, and all prefixes of each string are compared against all suffixes of the other string. An explicit representation of the output lengths is of size Θ (m+n) 2 . We show that the output can be represented implicitly by a geometric data structure of size O(m + n), allowing efficient queries of the individual output lengths. The currently best all string-substring LCS algorithm by Alves et al. can be adapted to produce the output in this form. We also develop the first all semi-local LCS algorithm, running in time o(mn) when m and n are reasonably close. Compared to a number of previous results, our approach presents an improvement in algorithm functionality, output representation efficiency, and/or running time.

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“…Adaptive Range Counting. In this paper, we investigate the possibility of obtaining better query times Reference Problem Space Query Time [10] Reporting O(n) O((1 + k) log n) [10] Reporting O(n log log n) O((1 + k) log log n) [10] Reporting O(n log n) O(log log n + k) [22] Counting O(n) O(log w n) for 2-D orthogonal range counting when the output count k is small. We give an adaptive data structure whose query time is sensitive to k: with O(n log log n) space, the query time is O(log log n + log w k).…”

Section: Introductionmentioning

confidence: 99%

Adaptive and Approximate Orthogonal Range Counting

Chan¹,

Wilkinson²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present three new results on one of the most basic problems in geometric data structures, 2-D orthogonal range counting. All the results are in the w-bit word RAM model.• It is well known that there are linear-space data structures for 2-D orthogonal range counting with worstcase optimal query time O(log w n).We give an O(n log log n)-space adaptive data structure that improves the query time to O(log log n + log w k), where k is the output count. When k = O(1), our bounds match the state of the art for the 2-D orthogonal range emptiness problem [Chan, Larsen, and Pȃtraşcu, SoCG 2011].• We give an O(n log log n)-space data structure for approximate 2-D orthogonal range counting that can compute a (1 + δ)-factor approximation to the count in O(log log n) time for any fixed constant δ > 0. Again, our bounds match the state of the art for the 2-D orthogonal range emptiness problem.• Lastly, we consider the 1-D range selection problem, where a query in an array involves finding the kth least element in a given subarray. This problem is closely related to 2-D 3-sided orthogonal range counting. Recently, Jørgensen and Larsen [SODA 2011] presented a linear-space adaptive data structure with query time O(log log n + log w k). We give a new linear-space structure that improves the query time to O(1 + log w k), exactly matching the lower bound proved by Jørgensen and Larsen.

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Space-Efficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting

Cited by 58 publications

References 13 publications

Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs

Longest Common Subsequences in Permutations and Maximum Cliques in Circle Graphs

All Semi-local Longest Common Subsequences in Subquadratic Time

Adaptive and Approximate Orthogonal Range Counting

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