Guarding in a simple polygon was motivated by art gallery problems. A guard capable of moving along a line segment in a polygon is called a mobile guard. In this paper, we discuss about two different degrees of patrol...
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Guarding in a simple polygon was motivated by art gallery problems. A guard capable of moving along a line segment in a polygon is called a mobile guard. In this paper, we discuss about two different degrees of patrol freedom of mobile guards. First, a guard diagonal is an internal diagonal that a mobile guard moving along the diagonal in a polygon and every interior point of the polygon can be seen by the mobile guard. Second, a guard chord is an internal chord that a mobile guard moving along the chord in a polygon and every interior point of the polygon can be seen by the mobile guard. In this paper, we solve the problem of finding the longest guard diagonal in O(n) time, the shortest guard diagonal in O(n alpha(n)) and the longest guard chord in O(n) time of a simple polygon P with n vertices, where alpha(n) is the inverse of Ackermann's function. (C) 2000 Elsevier Science B.V. All rights reserved.
We have been developing general user steered image segmentation strategies for routine use in applications involving a large number of data sets. In the past, we have presented three segmentation paradigms: live wire,...
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We have been developing general user steered image segmentation strategies for routine use in applications involving a large number of data sets. In the past, we have presented three segmentation paradigms: live wire, live lane, and a three-dimensional (3-D) extension of the live-wire method. In this paper, we introduce an ultra-fast live-wire method, referred to as live wire on the fly, for further reducing user's time compared to the basic live-wire method. In live wire, 3-D/four-dimensional (4-D) object boundaries are segmented in a slice-by-slice fashion. To segment a two-dimensional (2-D) boundary, the user initially picks a point on the boundary and all possible minimum-cost paths from this point to all other points in the image are computed via Dijkstra's algorithm. Subsequently a live wire is displayed in real time From the initial point to any subsequent position taken by the cursor. If the cursor is close to the desired boundary, the live wire snaps on to the boundary. The cursor is then deposited and a new live-wire segment is Found next, The entire 2-D boundary is specified via a set of live-wire segments in this fashion. A drawback of this method is that the speed of optimal path computation depends on image size. On modestly powered computers, for images of even modest size, some sluggishness appears in user interaction, which reduces the overall segmentation efficiency. In this work, we solve this problem by exploiting some known properties of graphs to avoid unnecessary minimum-cost path computation during segmentation. In live wire on the fly, when the user selects a point on the boundary the live-wire segment is computed and displayed in real time from the selected point to any subsequent position of the cursor in the image, even for large images and even on low-powered computers. Based on 492 tracing experiments from an actual medical application, we demonstrate that live wire on the fly is 1.3-31 times faster than live wire for actual segmentatio
This paper addresses the searchlight guarding problem, which is an extension of so-called graph searching/guarding problem on a weighted, undirected graph G by considering the time-slot parameter in addition to the tr...
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This paper addresses the searchlight guarding problem, which is an extension of so-called graph searching/guarding problem on a weighted, undirected graph G by considering the time-slot parameter in addition to the traditional building cost. Given a weighted, undirected graph G, suppose that there is a fugitive moving along the edges of G at an unknown speed. The task involves placing a set of searchlights at vertices to search the edges of the graph and to spot the fugitive. Assume that it costs some building cost to arrange a searchlight at some vertex. The searchlight guarding problem is to allocate a set S of searchlights at the vertices such that the total costs of the vertices for S is minimized. Furthermore, for all sets of searchlights with a minimum building cost, the one with the minimum search time will be selected, that is, the time slots needed to spot the fugitive is furthermore required to be minimized. The searchlight guarding problem has been known to be NP-hard on weighted bipartite graphs. But it is linear-time-solvable on weighted trees, weighted interval graphs, and weighted two-terminal series-parallel graphs. In this paper, the searchlight guarding problem is first proved to be NP-hard on weighted split graphs. Next, a linear time optimal algorithm to solve the problem on weighted cographs is designed. The algorithm is divided into two phases: in the first phase, a set of searchlights with a minimum guarding cost is identified and the search directions of all edges are assigned by the dynamic programming strategy. To achieve the task involved in phase one, a new graph problem, the edge-direction assignment problem on weighted complete-bipartite graphs, is defined and solved using the greedy strategy. In the second phase, the search time slots of each edge are determined. Both phases take linear time. (C) 2000 John Wiley & Sons, Inc.
We describe a method of defining new families of graphs called G-mixed, using the way of partitioning the edge set of overlap graphs. Consider a hereditary family G of graphs. An oriented graph G(V, E) is called G-mix...
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We describe a method of defining new families of graphs called G-mixed, using the way of partitioning the edge set of overlap graphs. Consider a hereditary family G of graphs. An oriented graph G(V, E) is called G-mixed if its edge set can be partitioned into two disjoint subsets E-1, E-2 such that G(V, E-1) epsilon G, G(V, E-2) is transitive and for every three vertices w, v, u if w --> v epsilon E-2 and (u, v) epsilon E-1 then (u, w) epsilon E-1;the letter G is generic and is replaced by specific names. The G-mixed graphs have a polynomial time algorithm to find maximum weight cliques, when G has such an algorithm. We define a new family of intersection graphs called interval-filament graphs which contain the polygon-circle graphs, the circle graphs, the chordal graphs and the cocomparability graphs. Let I be a family of intervals on a line L. In the plane, above L, construct to each interval i epsilon I a curve f(i) connecting its two endpoints, such that if two intervals are disjoint, their curves do not intersect;FI = {f(i) \ i epsilon I} is a family of interval filaments and its intersection graph is an interval-filament graph. We prove that a graph is an interval-filament graph iff its complement is a cointerval-mixed graph. Since cointerval graphs have a polynomial time algorithm to find maximum weight cliques, we can find maximum weight independent sets in interval-filament graphs using the algorithm for maximum weight cliques in cointerval-mixed graphs. Interval-filament graphs have also an algorithm to find maximum weight cliques. New families of intersection graphs of filaments are defined using families of circular-arcs of a Circle and families of subtrees of a tree or of a cactus. (C) 2000 Elsevier Science B.V. All rights reserved.
Given a graph G = (V, E) and two vertices s, t is an element of V, s not equal t, the Menger problem is to find a maximum number of disjoint paths connecting s and t. Depending on whether the input graph is directed o...
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Given a graph G = (V, E) and two vertices s, t is an element of V, s not equal t, the Menger problem is to find a maximum number of disjoint paths connecting s and t. Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or are disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the are disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time O(\ V \ log \ V \). Here we present a linear time, i.e., asymptotically optimal, algorithm for the are disjoint version in planar directed graphs.
The formalism of monadic second-order (MS) logic has been very successful in unifying a large number of algorithms for graphs of bounded treewidth. We extend the elegant framework of MS logic from static problems to d...
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The formalism of monadic second-order (MS) logic has been very successful in unifying a large number of algorithms for graphs of bounded treewidth. We extend the elegant framework of MS logic from static problems to dynamic problems, in which queries about MS properties of a graph of bounded treewidth are interspersed with updates of vertex and edge labels. This allows us to unify and occasionally strengthen a number of scattered previous results obtained in an ad hoc manner and to enable solutions to a wide range of additional problems to be derives automatically. As an auxiliary result of independent interest, we dynamize a data structure of Chazelle for answering queries about products of labels along paths in a tree with edges labeled by elements of a semigroup.
We consider a problem which arises in the computer-aided design of machines, motor vehicles, and other technical devices: convert the data describing a surface model of a three-dimensional workpiece from a low-informa...
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We consider a problem which arises in the computer-aided design of machines, motor vehicles, and other technical devices: convert the data describing a surface model of a three-dimensional workpiece from a low-information format to a high-information format. In the worst case, the input format only contains a list of independent descriptions of the elementary pieces, whereas the output format also contains the neighborhoods of these pieces-the topology of the model. Reconstructing the topology is a crucial step in this process. This problem is ill-posed and hence cannot be solved fully automatically. In fact, manual operations and previous format conversions make the data so "dirty" that human judgment is still required. Therefore, the aim is to approximate the topology as closely as possible to reduce the effort needed to correct the result afterward. In this paper we present a discrete approach to this problem and report the results of a computational study.
We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning forest;of a gra...
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ISBN:
(纸本)3540677151
We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning forest;of a graph with n vertices and m edges that runs in time O(tau*(m, n)) where tau* is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for tau* are tau*(m, n) = Omega(m) and tau*(m, n) = O(m . alpha(m, n)), where alpha is a certain natural inverse of Ackermann's function. Even under the assumption that tau* is super-linear, we show that if the input graph is selected from G(n,m), our algorithm runs in linear time w.h.p., regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for G(n,m) similar to the edge-exposure martingale for G(n,p).
Given a graph G = (V, E) and two vertices s, t is an element of V, s not equal t, the Menger problem is to find a maximum number of disjoint paths connecting s and t. Depending on whether the input graph is directed o...
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ISBN:
(纸本)3540633979
Given a graph G = (V, E) and two vertices s, t is an element of V, s not equal t, the Menger problem is to find a maximum number of disjoint paths connecting s and t. Depending on whether the input graph is directed or not, and what kind of disjointness criterion is demanded, this general formulation is specialized to the directed or undirected vertex, and the edge or are disjoint Menger problem, respectively. For planar graphs the edge disjoint Menger problem has been solved to optimality [W2], while the fastest algorithm for the are disjoint version is Weihe's general maximum flow algorithm for planar networks [W1], which has running time O(\ V \ log \ V \). Here we present a linear time, i.e., asymptotically optimal, algorithm for the are disjoint version in planar directed graphs.
This paper describes the LINK software system, which provides not only a graph editor and graph library, but a computing environment that employs object-oriented Scheme: to provide a flexible workbench for algorithm l...
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This paper describes the LINK software system, which provides not only a graph editor and graph library, but a computing environment that employs object-oriented Scheme: to provide a flexible workbench for algorithm learners and experimenters, Copyright (C) 2000 John Wiley & Sons, Ltd.
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