Combinatorial solutions of multidimensional divide-and-conquer recurrences

Monier, Louis

doi:10.1016/0196-6774(80)90005-x

Cited by 24 publications

(9 citation statements)

References 2 publications

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“…Instead, our solution follows the ideas of Cabello and Knauer [6] for constant treewidth, much like in [1]. All that was needed was a better understanding of the asymptotics of bivariate functions, rediscovering a 40-year old analysis of spatial data structures [15] (see the discussion in Sec. 3.3), and using a recent algorithm for approximate tree decompositions [5].…”

Section: Discussionmentioning

confidence: 99%

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

2020

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We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(n · k+ log n k · 2 k k 2 log n), where k is the treewidth of the graph. For every > 0, this bound is n 1+ exp O(k), which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log d n to d+ log n d , as originally observed by Monier (J. Alg. 1980).We also investigate the parameterization by vertex cover number. ACM Subject ClassificationTheory of computation → Shortest paths, Parameterized complexity and exact algorithms, Computational geometry. Mathematics of computing → Paths and connectivity problems.In fact, this holds under the potentially weaker Orthogonal Vectors conjecture, see [18] for an introduction to these arguments. Thus, under this assumption, the dependency on tw(G) in Theorem 1 cannot be significantly improved, even if the dependency on n is relaxed from just above linear to just below quadratic. Our analysis encompasses the Wiener index, an important structural graph parameter left unexplored by [1].Perhaps surprisingly, the main insight needed to establish Theorem 1 has nothing to do with graph distances or treewidth. Instead, we make-or re-discover-the following observation about the running time of d-dimensional range trees: Lemma 2 ([15]). A d-dimensional range tree over n points supporting orthogonal range queries for the aggregate value over a commutative monoid has query time O(2 d · B(n, d)) and can be built in time O (nd · B(n, d)This is a more careful statement than the standard textbook analysis, which gives the query time as O(log d n) and the construction time as O(n log d n). For many values of d, the asymptotic complexities of these bounds agree-in particular, this is true for constant d and for very large d, which are the main regimes of interest to computational geometers. But crucially, B(n, d) is always n exp O(d) for any > 0, while log d n is not.After Lemma 2 is realised, Theorem 1 follows via divide-and-conquer in decomposable graphs, closely following the idea of Cabello and Knauer [6] and augmented with known arguments [1,5]. We choose to give a careful presentation of the entire construction, as some of the analysis is quite fragile.Using known reductions, this implies that the following multivariate lower bound on orthogonal range searching is tight: K. Bringmann and T. Husfeldt and M. Magnusson 23:3Theorem 3 (Implicit in [1]). A data structure for the orthogonal range query problem for the monoid (Z, max) with construction time n · q (n, d) and query time q (n, d), where q (n, d) = n 1− exp o (d) for some > 0, refutes the Strong Exponential Time hypothesis.We a...

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Section: Discussionmentioning

confidence: 99%

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

2020

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“…When the remaining dimensions become less than three, a specialcase algorithm is applied that has complexity O(N log N ). If the dimension is not fixed then its complexity is bounded by O(MN 2 ) [4,20]. These algorithms exhibit many un-necessary comparisons which increases with the number of objectives [12].…”

Section: Related Workmentioning

confidence: 99%

Best Order Sort

Roy

Islam

Deb

2016

Proceedings of the 2016 on Genetic and Evolutionary Computation Conference Companion

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Finding the non-dominated sorting of a given set vectors has applications in Pareto based evolutionary multi-objective optimization (EMO), finding convex hull, linear optimization, nearest neighbor, skyline queries in database and many others. Among these, EMOs use this method for survival selection. The worst case complexity of this problem is found to be O(N log M −1 N) when the number of objectives M is constant and the size of solutions N is varying. But this bound becomes too large when M depends on N. In this paper we are proposing a new algorithm with worst case complexity O(MN log N + MN 2), however, with reduced running time in many objective cases. This algorithm can make use of the faster implementation of sorting algorithms. It removes unnecessary comparisons among the solutions which improves the running time. The proposed algorithm is compared with four other competing algorithms on three different datasets. Experimental results show that our approach, namely, best order sort (BOS) is computationally more efficient than all other compared algorithms with respect to running time.

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“…However, there are two problems in applying existing work to the resulting recurrence: nonhomogeneity and dimensions. Techniques are scarce and hard to apply for k>2, and for k ≤ 2 complicated summations must still be evaluated to obtain solutions by existing methods (see [14,19]). This approach cannot be applied to recurrences (1) and (2).…”

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confidence: 99%

General Techniques for Analyzing Recursive Algorithms with Applications

Verma¹

1997

SIAM J. Comput.

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The complexity of divide-and-conquer algorithms is often described by recurrences of various forms. In this paper, we develop general techniques and master theorems for solving several kinds of recurrences, and we give several applications of our results. In particular, almost all of the earlier work on solving the recurrences considered here is subsumed by our work. In the process of solving such recurrences, we establish interesting connections between some elegant mathematics and analysis of recurrences. Using our results and improved bipartite matching algorithms, we also improve existing bounds in the literature for several problems, viz, associative-commutative (AC) matching of linear terms, associative matching of linear terms, rooted subtree isomorphism, and rooted subgraph homeomorphism for trees.

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Combinatorial solutions of multidimensional divide-and-conquer recurrences

Cited by 24 publications

References 2 publications

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Best Order Sort

General Techniques for Analyzing Recursive Algorithms with Applications

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