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69
Hamiltonian Triangulations for Fast Rendering
, 1994
"... Highperformance rendering engines in computer graphics are often pipelined, and their speed is bounded by the rate at which triangulation data can be sent into the machine. To reduce the data rate, it is desirable to order the triangles so that consecutive triangles share a face, meaning that only ..."
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Cited by 65 (8 self)
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Highperformance rendering engines in computer graphics are often pipelined, and their speed is bounded by the rate at which triangulation data can be sent into the machine. To reduce the data rate, it is desirable to order the triangles so that consecutive triangles share a face, meaning that only one additional vertex need be transmitted to describe each triangle. Such an ordering exists if and only if the dual graph of the triangulation contains a Hamiltonian path. In this paper, we consider several problems concerning triangulations with Hamiltonian duals. Specifically, we ffl Show that any set of n points in the plane has a Hamiltonian triangulation, and give two optimal \Theta(n log n) algorithms for constructing such a triangulation. We have implemented and tested both algorithms. ffl Consider the special case of sequential triangulations, where the Hamiltonian cycle is implied, and prove that certain nondegenerate point sets in the plane do not admit a sequential triangulati...
The median problems for breakpoints are NPcomplete
 Elec. Colloq. on Comput. Complexity
, 1998
"... The breakpoint distance between two npermutations is the number of pairs that appear consecutively in one but not in the other. In the median problem for breakpoints one is given a set of permutations and has to construct a permutation that minimizes the sum of breakpoint distances to all the origi ..."
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Cited by 63 (1 self)
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The breakpoint distance between two npermutations is the number of pairs that appear consecutively in one but not in the other. In the median problem for breakpoints one is given a set of permutations and has to construct a permutation that minimizes the sum of breakpoint distances to all the original ones. Recently, the problem was suggested as a model for determining the evolutionary history of several species based on their gene orders. We show that the problem is already NPhard for three permutations, and that this result holds both for signed and for unsigned permutations.
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 55 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Graph Sandwich Problems
, 1994
"... The graph sandwich problem for property \Pi is defined as follows: Given two graphs G ) such that E ` E , is there a graph G = (V; E) such that E which satisfies property \Pi? Such problems generalize recognition problems and arise in various applications. Concentrating mainly o ..."
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Cited by 49 (8 self)
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The graph sandwich problem for property \Pi is defined as follows: Given two graphs G ) such that E ` E , is there a graph G = (V; E) such that E which satisfies property \Pi? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NPcompleteness of others. We describe
Computing Simple Circuits from a Set of Line Segments . . .
, 1987
"... Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that d ..."
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Cited by 26 (1 self)
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Given a collection of line segments in the plane we would like to connect the segments by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NPcomplete. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NPcomplete.
Hardness and approximation results for black hole search in arbitrary graphs
 In Proc. 12th Coll. on Structural Information and Communication complexity (SIROCCO’05
, 2005
"... Abstract. A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node, without leaving any trace. We consider the task of locating a black hole in a (partially) synchronous arbitrary network, assuming an upper bound on the ti ..."
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Cited by 19 (5 self)
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Abstract. A black hole is a highly harmful stationary process residing in a node of a network and destroying all mobile agents visiting the node, without leaving any trace. We consider the task of locating a black hole in a (partially) synchronous arbitrary network, assuming an upper bound on the time of any edge traversal by an agent. For a given graph and a given starting node we are interested in finding the fastest possible Black Hole Search by two agents (the minimum number of agents capable to identify a black hole). We prove that this problem is NPhard in arbitrary graphs, thus solving an open problem stated in [2]. We also give a 7/2approximation algorithm, thus improving on the 4approximation scheme observed in [2]. Our approach is to explore the given input graph via some spanning tree. Even if it represents a very natural technique, we prove that this approach cannot achieve an approximation ratio better than 3/2.
SingleStrip Triangulation of Manifolds with Arbitrary Topology
, 2004
"... Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for cre ..."
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Cited by 16 (5 self)
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Triangle strips have been widely used for efficient rendering. It is NPcomplete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50 % in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a singlestrip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.
Flowshop scheduling with limited temporary storage
 Journal of the ACM
, 1980
"... We examine the problem of scheduling 2machine flowshops in order to minimize makespan, using a limited amount of intermediate storage buffers. Although there are efficient algorithms for the extreme cases of zero and infinite buffer capacities, we show that all the intermediate (finite capacity) ca ..."
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Cited by 16 (0 self)
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We examine the problem of scheduling 2machine flowshops in order to minimize makespan, using a limited amount of intermediate storage buffers. Although there are efficient algorithms for the extreme cases of zero and infinite buffer capacities, we show that all the intermediate (finite capacity) cases are NPcomplete. We prove exact bounds for the relative improvement of execution times when a given buffer capacity is used. We also analyze an efficient heuristic for solving the 1buffer problem, showing that it has a 3/2 worstcase performance. Furthermore, we show that the "nowait " (i.e., zero buffer) flowshop scheduling problem with 4 machines is NPcomplete. This partly settles a wellknown open question, although the 3machine case is left open here. *Research supported by NSF Grant MCS7701192 +Research supported by NSF/RANN grant APR7612036
Multitrack Interval Graphs
, 1995
"... . A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The trac ..."
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Cited by 11 (2 self)
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. A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for Km;n is determined by proving that the arboricity of Km;n equals its "caterpillar arboricity". Recognition of graphs with track number 2 is shown to be NPcomplete. 1. Introduction Combinatorial properties of interval systems were investigated as early as the 1930's by Tibor Gallai. His two beautiful unpublished remarks are now phrased as saying that interval graphs and their complements are perfect graphs. In 1968, Gallai suggested considering more general set systems consisting of unions of d intervals, one each from d parallel lines. Such set systems have been called separated dintervals, since the d parallel lines can be viewed as d disjoint host intervals on a single li...