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Adjacency list

About: Adjacency list is a research topic. Over the lifetime, 4419 publications have been published within this topic receiving 78449 citations.


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Journal ArticleDOI
TL;DR: A vertically integrated model that supports the representation and manipulation of points, lines, polygons, and solids without redundancy is presented and the hybrid edge, a derivative of Baumgart's winged edge and Eastman's split edge models, is presented.
Abstract: Generalizing the computational geometric support for the representation of designed artefacts over multiple different levels of symbolic abstractions (e.g. sketches, solid modelling, and drafting) is discussed. First, the need for integrating the representation and manipulation of points, lines, polygons, and solids to facilitate design of artefacts at many levels of design abstraction is established. Second, a vertically integrated model that supports the representation and manipulation of points, lines, polygons, and solids without redundancy is presented. The particular roles of the operators and the operands are discussed, along with their hierarchical integration. The utility of the model is demonstrated in performing operations that involve multiple levels of data abstraction, such as splitting solids at a polygon inscribed on their boundary. Finally, a particular implementation of the integrated model is presented. This implementation is based on the hybrid edge, a derivative of Baumgart's winged edge and Eastman's split edge models. The hybrid edge distinguishes between the role of edges as carriers of topological adjacency information and their role as carriers of directionality information. This distinction permits the hybrid edge to handle the different combinations of adjacency and directionality requirements posed by the different abstraction levels in the integrated model.

24 citations

Proceedings ArticleDOI
08 Sep 2013
TL;DR: This work shows a novel approach for computing the clustering coefficients in an undirected and unweighted graphs by exploiting the use of a vertex cover, V̂ ⊆ V, which reduces the number of times that a triangle is counted by as many as 3 times per triangle.
Abstract: Clustering coefficients, also called triangle counting, is a widely-used graph analytic for measuring the closeness in which vertices cluster together. Intuitively, clustering coefficients can be thought of as the ratio of common friends versus all possible connections a person might have in a social network. The best known time complexity for computing clustering coefficients uses adjacency list intersection and is O(V · dmax2), where dmax is the size of the largest adjacency list of all the vertices in the graph. In this work, we show a novel approach for computing the clustering coefficients in an undirected and unweighted graphs by exploiting the use of a vertex cover, V ⊆ V. This new approach reduces the number of times that a triangle is counted by as many as 3 times per triangle. The complexity of the new algorithm is O(V · hmax2 + tVC) where dmax is the size of the largest adjacency list in the vertex cover and tVC is the time needed for finding the vertex cover. Even for a simple vertex cover algorithm this can reduce the execution time 10-30% while counting the exact number of triangles (3-circuits). We extend the use of the vertex cover to support counting squares (4-circuits) and clustering coefficients for dynamic graphs.

24 citations

Book ChapterDOI
19 Aug 2013
TL;DR: This work considers minimum multicuts of superpixel affinity graphs in which all affinities between non-adjacent superpixels are negative and proposes a relaxation by Lagrangian decomposition and a constrained set of re-parameterizations for which it can optimize exactly and efficiently.
Abstract: We address the problem of segmenting an image into a previously unknown number of segments from the perspective of graph partitioning. Specifically, we consider minimum multicuts of superpixel affinity graphs in which all affinities between non-adjacent superpixels are negative. We propose a relaxation by Lagrangian decomposition and a constrained set of re-parameterizations for which we can optimize exactly and efficiently. Our contribution is to show how the planarity of the adjacency graph can be exploited if the affinity graph is non-planar. We demonstrate the effectiveness of this approach in user-assisted image segmentation and show that the solution of the relaxed problem is fast and the relaxation is tight in practice.

24 citations

Proceedings Article
Miroslav N. Velev1, Ping Gao1
22 Oct 2009
TL;DR: Novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT) by using the absolute SAT encoding of permutations, where for each of the n objects and each of its pos- sible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position.
Abstract: We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the original graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its pos- sible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation — instead of, or in addition to the exclusivity adjacency constraints — that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.

24 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the removal of any point from a connected graph G still yields a hamiltonian graph, even if the distance between two points u and v in G is at most three.
Abstract: Let G be any connected graph on 4 or more points. The graph G3 has as it s point set that of C, and two di s tinct pointE U and v are adjacent in G3 if and only if the distance betwee n u and v in G is at most three. It is shown that not only is G" hamiltonian, but the removal of any point from G" still yields a hamiltonian graph. Let G be a graph (finite, undirected , with no loops or multiple lines). A waLk of G is a finit e alternating seque nce of points and lines of G, beginning and ending with a point and where each lin e is incide nt with the points immediately preceding and followin g it. A walk in whi c h no point is repeated is called a path; the Length of a path is the number of lines in it. A graph G is connected if between e very pair of di stin ct points th er e exis ts a path, and for s uc h a graph , the distance between two points u and v is defin e d as the le ngth of the shortest path if u 0;1= v and zero if u = v. A walk with at least three points in whic h the fir st and last points are the same but all other points are di s tinc t i s called a cycle. A cycle containing all points of a graph G is called a hamiLtonian cycle of G, and G itself a hamiLtonian graph. Throughout the literature of graph theory there have been defin ed many gr a ph-valued func ­ tions / o n the class of graphs. In certain ins ta nces r es ults have been obtained to show tha t if G is connecte d a nd has s uffi ciently many points, then the graph/(G) (or its ite rates/n(G» is a hamiltoni an graph. Examples of suc h include th e line-graph func ti on L(G) and the total graph function T(G) (see [2 , 1] ,1 respectively). The Line-graph L (G) of graph G is a graph whose poi nt set can b e put in one-to-one correspond­ ence with the lin e set of G such that adjacency is preserved. The totaL graph T(G} has its point set in one-to-one correspondence with the set of points and lines of G in such a way that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. Another example which always yields a hamiltonian graph is the c ube function. In fact, if x is any line in a connected graph G with at least three points, the n the c ube of G has a h amiltonian cycle containing x. This follows from a result due to Karaganis [4] by whic h the cube of any con­ nected graph G on p( ;?: 3) points turns out to be hami ltonian-connected , i.e., between any two points the r e exists a path containing all points of G. Now if x is any line joining points u and v in G, then the addition of x to the hamiltonian path between u and v in the c ube of the graph produces a hamil­

24 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023209
2022439
2021283
2020280
2019296
2018232