We propose a new computational model for the study of massive data processing. Our model measures the complexity of reading the input data in terms of their very large size N and analyzes the computational cost in ter...
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We propose a new computational model for the study of massive data processing. Our model measures the complexity of reading the input data in terms of their very large size N and analyzes the computational cost in terms of a parameter k that characterizes the computational power provided by limited local computing resources. We develop new algorithmic techniques for solving well-known computational problems on the model. In particular, randomized algorithms of running time O (N + g1(k)) and space O (k2), with very high probability, are developed for the famous graph matching problem on unweighted and weighted graphs. More specifically, our algorithm for unweighted graphs finds a k -matching (i.e., a matching of k edges) in a general unweighted graph in time O (N + k2.5), and our algorithm for weighted graphs finds a maximum weighted k-matching in a general weighted graph in time O (N + k3 log k). (c) 2022 Elsevier Inc. All rights reserved.
We show how to test in lineartime whether an outerplanar graph admits a planar rectilinear drawing, both if the graph has a prescribed plane embedding that the drawing has to respect and if it does not. Our algorithm...
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We show how to test in lineartime whether an outerplanar graph admits a planar rectilinear drawing, both if the graph has a prescribed plane embedding that the drawing has to respect and if it does not. Our algorithm returns a planar rectilinear drawing if the graph admits one. (C) 2021 Elsevier B.V. All rights reserved.
A ladder lottery, known as "Amidakuji" in Japan, is a common way to decide an assignment at random. A ladder lottery L of a given permutation is optimal if L has the minimum number of horizontal lines. In th...
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A ladder lottery, known as "Amidakuji" in Japan, is a common way to decide an assignment at random. A ladder lottery L of a given permutation is optimal if L has the minimum number of horizontal lines. In this paper, we investigate a reconfiguration problem of optimal ladder lotteries. The reconfiguration problem on a set of optimal ladder lotteries asks, given two optimal ladder lotteries L, L' of a permutation pi, to find a sequence of (L-1, L-2, ... , L-k) of optimal ladder lotteries of pi such that (1) L-1 = L and L-k = L' and (2) L-i for i = 2, 3, ... , k is obtained from Li-1 by moving a bar in Li-1 locally. An existing result implies that any two optimal ladder lotteries of a permutation pi have a reconfiguration sequence of length O(n3), where n is the number of elements in pi. In this paper, we characterize the minimum length of reconfiguration sequences between two optimal ladder lotteries. Moreover, we present a linear-time algorithm that computes the minimum length. (C) 2021 Elsevier B.V. All rights reserved.
A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-col...
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A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph G and an integer p, the COLOURFUL COMPONENTS problem asks whether there exist at most p edges whose removal makes G colourful and the COLOURFUL PARTITION problem asks whether there exists a partition of G into at most p colourful components. In order to refine our understanding of the complexity of the problems on trees, we study both problems on k-caterpillars, which are trees with a central path P such that every vertex not in P is within distance k from a vertex in P. We prove that COLOURFUL COMPONENTS and COLOURFUL PARTITION are NP-complete on 4-caterpillars with maximum degree 3, 3-caterpillars with maximum degree 4 and 2caterpillars with maximum degree 5. On the other hand, we show that the problems are linear-time solvable on 1-caterpillars. Hence, our results imply two complexity dichotomies on trees: COLOURFUL COMPONENTS and COLOURFUL PARTITION are linear-time solvable on trees with maximum degree d if d <= 2 (that is, on paths), and NP-complete otherwise;COLOURFUL COMPONENTS and COLOURFUL PARTITION are linear-time solvable on k-caterpillars if k <= 1, and NP-complete otherwise. We leave three open cases which, if solved, would provide a complexity dichotomy for both problems on k-caterpillars, for every non-negative integer k, with respect to the maximum degree. We also show that COLOURFUL COMPONENTS is NP-complete on 5-coloured planar graphs with maximum degree 4 and on 12-coloured planar graphs with maximum degree 3. Our results answer two open questions of Bulteau et al. mentioned in Bulteau et al. (2019) [6]. (C) 2021 Elsevier B.V. All rights reserved.
An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differs by at most one. In this work, we show that every interval graph ...
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An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differs by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k >= (Delta(G) + 1)/2, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree -k-coloring for a given integer k. For disjoint union of split graphs, or K-1,K-r-free interval graphs with r >= 4, we prove that it is W[1]-hard to decide whether there is an equitable tree -k-coloring when parameterized by number of colors, or by treewidth, number of colors and maximum degree, respectively. On the positive side, we propose a quadratic 2-approximation algorithm for the equitable tree-coloring problem in chordal graphs. Moreover, it is proved that there is no alpha-approximation algorithm for any alpha < 3/2 . (c) 2020 Elsevier B.V. All rights reserved.
A module of a fault tree is an independent subtree that has no input from the rest of the tree and no output to the rest, except the top events. Modularization is an important technique to reduce the computation cost ...
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A module of a fault tree is an independent subtree that has no input from the rest of the tree and no output to the rest, except the top events. Modularization is an important technique to reduce the computation cost for large, complex fault tree analysis. This article presents a new linear-time algorithm that is more efficient and easier to code for finding modules existing in fault trees. Two main stages are included in the proposed algorithm: branching and transforming. To demonstrate the efficiency and applicability of the proposed algorithm, comparisons are performed between the proposed algorithm and other linear-time algorithms for finding modules in fault trees. Results have shown the superiority and effectiveness of the proposed algorithm.
In this paper, the power domination problem on trees is studied and we give a linear-time algorithm to it by using the labeling method. Our algorithm is simpler and easier to understand than those in [3,7]. & COPY...
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In this paper, the power domination problem on trees is studied and we give a linear-time algorithm to it by using the labeling method. Our algorithm is simpler and easier to understand than those in [3,7]. & COPY;2023 Elsevier Inc. All rights reserved.
A graph G =(V, E) is distance hereditary if every induced path of Gis a shortest path. In this paper, we show that the eccentricity function e(v) = max{d(v, u) : u is an element of V} in any distance-hereditary graph ...
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A graph G =(V, E) is distance hereditary if every induced path of Gis a shortest path. In this paper, we show that the eccentricity function e(v) = max{d(v, u) : u is an element of V} in any distance-hereditary graph Gis almost unimodal, that is, every vertex vwith e(v) > rad(G) + 1 has a neighbor with smaller eccentricity. Here, rad(G) = min{e(v) : v is an element of V} is the radius of graph G. Moreover, we use this result to fully characterize the centers of distance-hereditary graphs. Several bounds on the eccentricity of a vertex with respect to its distance to the center of Gor to the ends of a diametral path are established. Finally, we propose a new lineartimealgorithm to compute all eccentricities in a distance-hereditary graph. (c) 2020 Elsevier B.V. All rights reserved.
A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the c...
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A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases F = 1 and F = 2. For F = 1, the max-min and min-max problems are solvable in lineartime. The case F = 2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in lineartime if m = n - 1, and a solution always exists in this case, in contrast to F = 1. Moreover, the problem is fixed-parameter tractable in the parameter 2m - n. (Note that this parameter measures the number of agents above the trivial threshold m = n/2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximationpreserving reduction from our original splitting problem with fragmentation F = 2 to this averaging problem, and some approximation results in cases when m is close to either n or n/2.
A previously unstudied optimization problem concerning the summation of elements of numerical sequences and of respective lengths N and q <= N is considered. The task is to minimize the sum of differences between w...
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A previously unstudied optimization problem concerning the summation of elements of numerical sequences and of respective lengths N and q <= N is considered. The task is to minimize the sum of differences between weighted convolutions of sequences of variable length (of at least). In each difference, the minuend is a nonweighted autoconvolution of the sequence extended to a variable length (by multiple repeats of its elements) and the subtrahend is a weighted convolution of this extended sequence and a subsequence of. The variant of the problem with an optimized number of summed differences is analyzed. It is shown that the problem is equivalent to a problem of approximating the sequence by an element of an exponential-size set of sequences. This set consists of all sequences of length that include, as subsequences, a variable number of admissible quasi-periodic (fluctuation) repeats of. Each quasi-periodic repeat is generated by admissible transformations of. These transformations are (i) a shift of by a variable quantity that does not exceed T-max <= N between neighboring repeats, and (ii) a variable extension mapping of into a sequence of variable length defined in the form of repeats of elements of with the multiplicity of these repeats being variable. The approximation criterion is the minimum of the sum of squared distances between the elements of the sequences. It is proved that the considered optimization problem, together with the approximation problem, is solvable in polynomial time. More specifically, it is shown that there exists an exact algorithm finding the solution of the problem in O(T-max(3) N) time. If T-max is a fixed parameter of the problem, then the running time of the algorithm is linear. Examples of numerical simulation are used to illustrate the applicability of the algorithm for solving model application problems of noiseproof processing of ECG-like and PPG-like quasi-periodic signals (electrocardiogram-like and photoplethysmogram-like sig
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