International Journal of Computational Geometry & Applications, 2010
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension (d = 2, 3) under the discrete Fréchet distance. Given n polygonal chains C in d-dimension (d = 2, 3), each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (C). Our main results are summarized as follows.
A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet... more A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the problem of simplifying 3D polygonal chains under the discrete Fréchet distance. We present efficient polynomial time algorithms for simplifying a single chain, including the first near-linear O(nlogn) time exact algorithm for the continuous min-# fitting problem. Our algorithms generalize to any fixed dimension d > 3. Motivated by the ridge-based model simplification we also consider simplifying a pair of chains simultaneously and we show that one version of the general problem is NP-complete.
We present new approximation algorithms for the NP-hard problems of labeling points with maximum-... more We present new approximation algorithms for the NP-hard problems of labeling points with maximum-size uniform circles and circle pairs (MLUC and MLUCP). Our algorithms build on the important concept of maximal feasible region and new algorithmic techniques. We obtain several results: a (2.98 + )-approximation for MLUC, improving previous factor 3.0+ ; a (1.491+ )-approximation for MLUCP, improving previous factor 1.5; and the first non-trivial lower bound 1.0349 for both MLUC and MLUCP, improving previous lower bound 1 + O(1/n).
Designing and analysing efficient algorithms is important in practical applications, but it is al... more Designing and analysing efficient algorithms is important in practical applications, but it is also fun and frequently instructive, even for simple problems with no immediate applications. In this self-contained paper we try to convey some of fun of algorithm design and analysis. Hopefully, the reader will find the discussion instructive as well. We focus our attention on a single problem
ABSTRACT This paper presents space-efficient algorithms for some basic tasks (or problems) on a b... more ABSTRACT This paper presents space-efficient algorithms for some basic tasks (or problems) on a binary image of n pixels, assuming that an input binary image is stored in a read-only array with random-access. Although efficient algorithms are available for those tasks if O(n) work space (of O(nlogn) bits) is available, we aim to propose efficient algorithms using only limited work space, i.e., O(1) or O(n) space. Tasks to be considered are (1) CCC to count the number of connected components, (2) MERR to report the minimum enclosing rectangle of every connected component, and (3) LCCR to report a largest connected component. We show that we can solve each of CCC, MERR, and LCCR in O(nlogn) time using only O(1) space. If we can use O(n) work space, we can solve them in O(n),O(n), and O(n+mlogm) time, respectively, where m is the number of pixels in the largest connected component.
In 1926, Jarník introduced the problem of drawing a convex ngon with vertices having integer coor... more In 1926, Jarník introduced the problem of drawing a convex ngon with vertices having integer coordinates. He constructed such a drawing in the grid [1, c · n 3/2 ] 2 for some constant c > 0, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n)-time algorithm to construct such a point set.
In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids ... more In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids generated by any contour interpolation method. Given two arbitrary parallel contour slices with n vertices in 3D, let α be the smallest angle in the constrained Delaunay triangulation of the corresponding 2D contour overlay, we present a contour interpolation method which reconstructs a
In this work we study several problems regarding balanced bipartitions of 3-colored sets of point... more In this work we study several problems regarding balanced bipartitions of 3-colored sets of points, and of 3-colored sets of lines. We say that a bipartition of a colored set is balanced if each of its two parts contains the same quantity of elements of each color. First we consider sets of lines and study the existence of segments that intersect the arrangement in a balanced set; we also study the dual problem, that is, we consider point sets in the plane and study bipartitions induced by double wedges. Then we consider point sets on a closed Jordan curve and study bipartitions induced by arcs. Last, we consider point sets in the plane lattice and study bipartitions induced by L-lines. This is an extended abstract of a presentation given at EuroCG 2013. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.
Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-kno... more Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.
ABSTRACT A set of vertical bars planted on given points of a horizontal line defines a fence comp... more ABSTRACT A set of vertical bars planted on given points of a horizontal line defines a fence composed of the quadrilaterals bounded by successive bars. A set of bars in the plane, each having one endpoint at the origin, defines an umbrella composed of the triangles bounded by successive bars. Given a collection of bars, we study how to use them to build the fence or the umbrella of maximum total area. We present optimal algorithms for these constructions. The problems introduced in this paper are related to the Geometric Knapsack problems (Arkin et al. in Algorithmica 10:399–427, 1993) and the Rearrangement Inequality (Wayne in Scripta Math 12(2):164–169, 1946).
We present a simple O(m+n 6/ε 12) time (1+ε)-approximation algorithm for finding a minimum-cost s... more We present a simple O(m+n 6/ε 12) time (1+ε)-approximation algorithm for finding a minimum-cost sequence of lines to cut a convex n-gon out of a convex m-gon.
A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is ... more A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is a set of non-crossing straight line segments with endpoints in S. Given a set of red points and a set of blue points in the plane, the red/blue spanning tree problem is to find a geometric spanning tree
In this paper we investigate the problem of locating a mobile facility at (or near) the center of... more In this paper we investigate the problem of locating a mobile facility at (or near) the center of a set of clients that move independently, continuously, and with bounded velocity. It is shown that the Euclidean 1-center of the clients may move with arbitrarily high velocity relative to the maximum client velocity. This motivates the search for strategies for moving a facility so as to closely approximate the Euclidean 1-center while guaranteeing low (relative) velocity.
In this paper, we present a new model for RNA multiple sequence structural alignment based on the... more In this paper, we present a new model for RNA multiple sequence structural alignment based on the longest common subsequence. We consider both the off-line and on-line cases. For the off-line case, i.e., when the longest common subsequence is given as a linear graph with n vertices, we first present a polynomial O(n 2 ) time algorithm to compute its maximum nested loop. We then consider a slightly different problem-the Maximum Loop Chain problem and present an algorithm which runs in O(n 5 ) time. For the on-line case, i.e., given m RNA sequences of lengths n, compute the longest common subsequence of them such that this subsequence either induces a maximum nested loop or the maximum number of matches, we present efficient algorithms using dynamic programming when m is small.
Given a finite set of points S in R d , consider visiting the points in S with a polygonal path w... more Given a finite set of points S in R d , consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S and let G d n be a n × · · · × n grid in Z d . Kranakis et al. showed that L(G 2 n ) = 2n − 1 and
... 3. Pankaj K. Agarwal , Cecilia M. Procopiuc, Exact and approximation algorithms for clusterin... more ... 3. Pankaj K. Agarwal , Cecilia M. Procopiuc, Exact and approximation algorithms for clustering, Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, p.658-667, January 25-27, 1998, San Francisco, California, United States. ...
International Journal of Computational Geometry & Applications, 2010
Polygonal chains are fundamental objects in many applications like pattern recognition and protei... more Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension (d = 2, 3) under the discrete Fréchet distance. Given n polygonal chains C in d-dimension (d = 2, 3), each with at most k vertices, we prove fundamental properties of such a Voronoi diagram VD F (C). Our main results are summarized as follows.
A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet... more A well-known measure to characterize the similarity of two polygonal chains is the famous Fréchet distance. In this paper, for the first time, we consider the problem of simplifying 3D polygonal chains under the discrete Fréchet distance. We present efficient polynomial time algorithms for simplifying a single chain, including the first near-linear O(nlogn) time exact algorithm for the continuous min-# fitting problem. Our algorithms generalize to any fixed dimension d > 3. Motivated by the ridge-based model simplification we also consider simplifying a pair of chains simultaneously and we show that one version of the general problem is NP-complete.
We present new approximation algorithms for the NP-hard problems of labeling points with maximum-... more We present new approximation algorithms for the NP-hard problems of labeling points with maximum-size uniform circles and circle pairs (MLUC and MLUCP). Our algorithms build on the important concept of maximal feasible region and new algorithmic techniques. We obtain several results: a (2.98 + )-approximation for MLUC, improving previous factor 3.0+ ; a (1.491+ )-approximation for MLUCP, improving previous factor 1.5; and the first non-trivial lower bound 1.0349 for both MLUC and MLUCP, improving previous lower bound 1 + O(1/n).
Designing and analysing efficient algorithms is important in practical applications, but it is al... more Designing and analysing efficient algorithms is important in practical applications, but it is also fun and frequently instructive, even for simple problems with no immediate applications. In this self-contained paper we try to convey some of fun of algorithm design and analysis. Hopefully, the reader will find the discussion instructive as well. We focus our attention on a single problem
ABSTRACT This paper presents space-efficient algorithms for some basic tasks (or problems) on a b... more ABSTRACT This paper presents space-efficient algorithms for some basic tasks (or problems) on a binary image of n pixels, assuming that an input binary image is stored in a read-only array with random-access. Although efficient algorithms are available for those tasks if O(n) work space (of O(nlogn) bits) is available, we aim to propose efficient algorithms using only limited work space, i.e., O(1) or O(n) space. Tasks to be considered are (1) CCC to count the number of connected components, (2) MERR to report the minimum enclosing rectangle of every connected component, and (3) LCCR to report a largest connected component. We show that we can solve each of CCC, MERR, and LCCR in O(nlogn) time using only O(1) space. If we can use O(n) work space, we can solve them in O(n),O(n), and O(n+mlogm) time, respectively, where m is the number of pixels in the largest connected component.
In 1926, Jarník introduced the problem of drawing a convex ngon with vertices having integer coor... more In 1926, Jarník introduced the problem of drawing a convex ngon with vertices having integer coordinates. He constructed such a drawing in the grid [1, c · n 3/2 ] 2 for some constant c > 0, and showed that this grid size is optimal up to a constant factor. We consider the analogous problem for drawing the double circle, and prove that it can be done within the same grid size. Moreover, we give an O(n)-time algorithm to construct such a point set.
In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids ... more In this paper, we present the first nontrivial theoretical bound on the quality of the 3D solids generated by any contour interpolation method. Given two arbitrary parallel contour slices with n vertices in 3D, let α be the smallest angle in the constrained Delaunay triangulation of the corresponding 2D contour overlay, we present a contour interpolation method which reconstructs a
In this work we study several problems regarding balanced bipartitions of 3-colored sets of point... more In this work we study several problems regarding balanced bipartitions of 3-colored sets of points, and of 3-colored sets of lines. We say that a bipartition of a colored set is balanced if each of its two parts contains the same quantity of elements of each color. First we consider sets of lines and study the existence of segments that intersect the arrangement in a balanced set; we also study the dual problem, that is, we consider point sets in the plane and study bipartitions induced by double wedges. Then we consider point sets on a closed Jordan curve and study bipartitions induced by arcs. Last, we consider point sets in the plane lattice and study bipartitions induced by L-lines. This is an extended abstract of a presentation given at EuroCG 2013. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.
Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-kno... more Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.
ABSTRACT A set of vertical bars planted on given points of a horizontal line defines a fence comp... more ABSTRACT A set of vertical bars planted on given points of a horizontal line defines a fence composed of the quadrilaterals bounded by successive bars. A set of bars in the plane, each having one endpoint at the origin, defines an umbrella composed of the triangles bounded by successive bars. Given a collection of bars, we study how to use them to build the fence or the umbrella of maximum total area. We present optimal algorithms for these constructions. The problems introduced in this paper are related to the Geometric Knapsack problems (Arkin et al. in Algorithmica 10:399–427, 1993) and the Rearrangement Inequality (Wayne in Scripta Math 12(2):164–169, 1946).
We present a simple O(m+n 6/ε 12) time (1+ε)-approximation algorithm for finding a minimum-cost s... more We present a simple O(m+n 6/ε 12) time (1+ε)-approximation algorithm for finding a minimum-cost sequence of lines to cut a convex n-gon out of a convex m-gon.
A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is ... more A geometric spanning tree of a point set S is a tree whose vertex set is S and whose edge set is a set of non-crossing straight line segments with endpoints in S. Given a set of red points and a set of blue points in the plane, the red/blue spanning tree problem is to find a geometric spanning tree
In this paper we investigate the problem of locating a mobile facility at (or near) the center of... more In this paper we investigate the problem of locating a mobile facility at (or near) the center of a set of clients that move independently, continuously, and with bounded velocity. It is shown that the Euclidean 1-center of the clients may move with arbitrarily high velocity relative to the maximum client velocity. This motivates the search for strategies for moving a facility so as to closely approximate the Euclidean 1-center while guaranteeing low (relative) velocity.
In this paper, we present a new model for RNA multiple sequence structural alignment based on the... more In this paper, we present a new model for RNA multiple sequence structural alignment based on the longest common subsequence. We consider both the off-line and on-line cases. For the off-line case, i.e., when the longest common subsequence is given as a linear graph with n vertices, we first present a polynomial O(n 2 ) time algorithm to compute its maximum nested loop. We then consider a slightly different problem-the Maximum Loop Chain problem and present an algorithm which runs in O(n 5 ) time. For the on-line case, i.e., given m RNA sequences of lengths n, compute the longest common subsequence of them such that this subsequence either induces a maximum nested loop or the maximum number of matches, we present efficient algorithms using dynamic programming when m is small.
Given a finite set of points S in R d , consider visiting the points in S with a polygonal path w... more Given a finite set of points S in R d , consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S and let G d n be a n × · · · × n grid in Z d . Kranakis et al. showed that L(G 2 n ) = 2n − 1 and
... 3. Pankaj K. Agarwal , Cecilia M. Procopiuc, Exact and approximation algorithms for clusterin... more ... 3. Pankaj K. Agarwal , Cecilia M. Procopiuc, Exact and approximation algorithms for clustering, Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms, p.658-667, January 25-27, 1998, San Francisco, California, United States. ...
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