A time- and cost-optimal algorithm for interlocking sets-with applications

Olariu, Stephan; Zomaya, Albert Y.

doi:10.1109/71.539733

Cited by 9 publications

(6 citation statements)

References 23 publications

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“…We remark that this will not be necessary in the application of this lemma, as predecessors and successors in this order can be maintained during a traversal of C i . Note also that the work in [44] describes mainly a nonsequential variant of the algorithm; for a simple sequential variant, Lemmas 4.1 and 4.2 in [44] suffice.…”

Section: Reduction To Overlapping Intervalsmentioning

confidence: 97%

“…A simple sweep-line algorithm due to Olariu and Zomaya [44] computes in time O(t) a spanning subgraph G s of the overlap graph of Y (but not of the merged overlap graph) such that G s has at most 2t edges and exactly the same connected components as the overlap graph of Y . The algorithm assumes the end points of intervals to be sorted.…”

Section: Reduction To Overlapping Intervalsmentioning

confidence: 99%

“…A sequence of overlaps from I(C i ) to every other created interval exists if and only if the overlap graph (i. e., the graph with intervals as vertices and edges between overlapping intervals) is connected. Simple sweep-line algorithms for constructing the connected components of the overlap graph are known [19] (Lemmas 4.1 and 4.2 suffice), run in time O(t) for t intervals and, thus, ensures the efficient computation of the reduction.…”

Section: A Linear-time Algorithmmentioning

confidence: 99%

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Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Schmidt

2013

SIAM J. Comput.

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One of the most noted construction methods of 3-vertex-connected graphs is due to Tutte and is based on the following fact: Any 3-vertex-connected graph G = (V, E) on more than 4 vertices contains a contractible edge, i.e., an edge whose contraction generates a 3-connected graph. This implies the existence of a sequence of edge contractions from G to the complete graph K 4 , such that every intermediate graph is 3-vertex-connected. A theorem of Barnette and Grünbaum gives a similar sequence using removals on edges instead of contractions. We show how to compute both sequences in optimal time, improving the previously best known running times of O(|V | 2 ) to O(|E|). This result has a number of consequences; an important one is a new linear-time test of 3-connectivity that is certifying; finding such an algorithm has been a major open problem in the design of certifying algorithms in recent years. The test is conceptually different from well-known linear-time 3-connectivity tests and uses a certificate that is easy to verify in time O(|E|). We show how to extend the results to an optimal certifying test of 3-edge-connectivity.1. Introduction. The class of 3-connected (i.e., 3-vertex-connected) graphs has been studied intensively for many reasons in the past 50 years. Algorithmic applications include problems in graph drawing (see [41] for a survey), problems related to planarity [8,27], and online problems on planar graphs (see [7] for a survey). From a complexity point of view, 3-connectivity is in particular important for problems dealing with longest paths, because it lies, somewhat surprisingly, on the borderline of NP-hardness: Finding a Hamiltonian cycle is NP-hard for 3-connected planar graphs [24] but becomes solvable in linear running time [13] for higher connectivity, as 4-connected planar graphs are Hamiltonian [62].From a top-level view, we design an efficient algorithm from an inductively defined construction of a graph class. For a given graph class C, such constructions start with a set of base graphs and apply iteratively operations from a finite set of operations such that precisely the members of C are constructed. Not only does this give a computational approach to test graphs for membership in C, but it can also be exploited to prove properties of C using just arguments on the base graphs and the finitely many operations. Graph theory provides inductively defined constructions for many graph classes, including planar graphs, triangulations, k-connected graphs for k ≤ 4, regular graphs, and various intersections of these classes [4,5,31]. Most of these constructions have not been exploited computationally.For an inductively defined construction of C and a graph G ∈ C, we call a sequence of operations that is applied to a specified base graph to construct G a construction sequence of G. We will also identify a construction sequence with the sequence of

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Section: Reduction To Overlapping Intervalsmentioning

confidence: 97%

Section: Reduction To Overlapping Intervalsmentioning

confidence: 99%

Section: A Linear-time Algorithmmentioning

confidence: 99%

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Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Schmidt

2013

SIAM J. Comput.

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“…A naive approach that constructs H , contracts intervals, and runs a DFS will fail, since the overlap graph can have a quadratic number of edges. However, using a method developed by Olariu and Zomaya [23], we can compute a spanning forest of H in time linear in the number of intervals. The presentation in [23] is for the PRAM and thus needlessly complicated for our purposes.…”

Section: A Linear Time Algorithmmentioning

confidence: 99%

Certifying 3-Edge-Connectivity

2015

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We present a certifying algorithm that tests graphs for 3-edge-connectivity; the algorithm works in linear time. If the input graph is not 3-edge-connected, the algorithm returns a 2-edge-cut. If it is 3-edge-connected, it returns a construction sequence that constructs the input graph from the graph with two vertices and three parallel edges using only operations that (obviously) preserve 3-edge-connectivity. Additionally, we show how compute and certify the 3-edge-connected components and a cactus representation of the 2-cuts in linear time. For 3-vertex-connectivity, we show how to compute the 3-vertex-connected components of a 2-connected graph.Comment: 29 pages in Algorithmica, 201

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“…For the segments that are determined by nested chains, a correct order to add them to the current G c is to be determined. This is accomplished by reducing the problem to a problem on intervals based on the notion of overlap graph and a method of Olariu and Zomaya [24] for finding a spanning forest in the overlap graph. The ordering exists if and only if the spanning forest is a spanning tree.…”

Section: Introductionmentioning

confidence: 99%

Yet another optimal algorithm for 3-edge-connectivity

Tsin

2009

Journal of Discrete Algorithms

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An optimal algorithm for 3-edge-connectivity is presented. The algorithm performs only one pass over the given graph to determine a set of cut-pairs whose removal leads to the 3-edge-connected components. An additional pass determines all the 3-edge-connected components of the given graph. The algorithm is simple, easy to implement and runs in linear time and space. Experimental results show that it outperforms all the previously known linear-time algorithms for 3-edge-connectivity in determining if a given graph is 3-edge-connected and in determining cut-pairs. Its performance is also among the best in determining the 3-edge-connected components.

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A time- and cost-optimal algorithm for interlocking sets-with applications

Cited by 9 publications

References 23 publications

Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Contractions, Removals, and Certifying 3-Connectivity in Linear Time

Certifying 3-Edge-Connectivity

Yet another optimal algorithm for 3-edge-connectivity

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