Papers by Prosenjit Gupta
Computational Geometry: Theory and Applications, 1999
A parting line for a convex polyhedron, P, is a closed curve on the surface of P. It de nes the t... more A parting line for a convex polyhedron, P, is a closed curve on the surface of P. It de nes the two pieces of P for which mold-halves must be made. An undercut-free parting line is one which does not create recesses or projections in P and thus allows easy de-molding of P. Computing an undercut-free parting line that is as at as possible is an important problem in mold design. In this paper, an O(n 2 )-time algorithm is presented to compute such a line, according to a prescribed atness criterion, where n is the number of vertices in P.
Information Processing Letters, 1999
Let S be a set of convex polygons in the plane with a total of n vertices, where a polygon consis... more Let S be a set of convex polygons in the plane with a total of n vertices, where a polygon consists of the boundary as well as the interior. E cient algorithms are presented for the problem of reporting output-sensitively (resp. counting) the I pairs of polygons that intersect. The algorithm for the reporting (resp. counting) problem runs in time O(n 4=3+ +I) (resp. O(n 4=3+ )), where > 0 is an arbitrarily small constant. This result is based on an interesting characterization of the intersection of two convex polygons in terms of the intersection of certain trapezoids from their trapezoidal decomposition. Also given is an alternative solution to the reporting problem, which runs in O(n 4=3 log O(1) n + I ) time, and is based on characterizing the intersection of two convex polygons via the intersection of their upper and lower chains and their leftmost vertices. The problems are interesting and challenging because the output size, I , can be much smaller than the total number of intersections between the boundaries of the polygons and because the number of polygons and their sizes can depend on n.
We consider the problem of reporting the pairwise enclosures among a set of n axes-parallel recta... more We consider the problem of reporting the pairwise enclosures among a set of n axes-parallel rectangles in IR 2 , which is equivalent to reporting dominance pairs in a set of n points in IR 4 . For more than ten years, it has been an open problem whether these problems can be solved faster than in O(n log 2 n + k) time, where k denotes the number of reported pairs. First, we give a divide-and-conquer algorithm that matches the O(n) space and O(n log 2 n + k) time bounds of the algorithm of Lee and Preparata LP82], but is simpler. Then we give another algorithm that uses O(n) space and runs in O(n log n log log n+k log log n) time. For the special case where the rectangles have at most di erent aspect ratios, we give an algorithm that runs in O( n log n + k) time and uses O(n) space. Problem 1.1 Given a set R of n axes-parallel rectangles in the plane, report all pairs (R 0 ; R) of rectangles such that R encloses R 0 .
Efficient algorithms for generalized intersection searching on non-iso-oriented objects
ABSTRACT
Computational Geometry: Generalized Intersection Searching
... BibTeX. @INPROCEEDINGS{Gupta05computationalgeometry:, author = {Prosenjit Gupta and Ravi Jana... more ... BibTeX. @INPROCEEDINGS{Gupta05computationalgeometry:, author = {Prosenjit Gupta and Ravi Janardan and Michiel Smid}, title = {Computational geometry: Generalized intersection searching}, booktitle ... 222, Making data structures persistent - Driscoll, Sarnak, et al. - 1986. ...
Information Processing Letters, 1997
In a generalized searching problem, a set S of n colored geometric objects has to be stored in a ... more In a generalized searching problem, a set S of n colored geometric objects has to be stored in a data structure, such that for any given query object q, the distinct colors of the objects of S intersected by q can be reported e ciently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp. fat triangles) with a fat triangle (resp. point). For both problems, a data structure is obtained having size O(n 1+ ) and query time O((log n) 2 + C). Here, C denotes the number of colors reported by the query, and is an arbitrarily small positive constant.
Information Processing Letters, 1996
In a generalized searching problem, a set S of n colored geometric objects has to be stored in a ... more In a generalized searching problem, a set S of n colored geometric objects has to be stored in a data structure, such that for any given query object q, the distinct colors of the objects of S intersected by q can be reported e ciently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp. fat triangles) with a fat triangle (resp. point). For both problems, a data structure is obtained having size O(n 1+ ) and query time O((log n) 2 + C). Here, C denotes the number of colors reported by the query, and is an arbitrarily small positive constant.
Computational Geometry: Theory and Applications, 1999
E cient geometric algorithms are given for the two-dimensional versions of optimization problems ... more E cient geometric algorithms are given for the two-dimensional versions of optimization problems arising in layered manufacturing, where a polygonal object is built by slicing its CAD model and manufacturing the slices successively. The problems considered are minimizing (i) the contact-length between the supports and the manufactured object, (ii) the area of the support structures used, and (iii) the area of the so-called trapped regions|factors that a ect the cost and quality of the process.
Computational Geometry: Theory and Applications, 1996
Computational Geometry: Theory and Applications, 1996
Consider a set of geometric objects, such as points, line segments, or axesparallel hyperrectangl... more Consider a set of geometric objects, such as points, line segments, or axesparallel hyperrectangles in IR d , that move with constant but possibly different velocities along linear trajectories. Efficient algorithms are presented for several problems defined on such objects, such as determining whether any two objects ever collide and computing the minimum inter-point separation or minimum diameter that ever occurs. In particular, two open problems from the literature are solved: Deciding in o(n 2 ) time if there is a collision in a set of n moving points in IR 2 , where the points move at constant but possibly different velocities, and the analogous problem for detecting a red-blue collision between sets of red and blue moving points. The strategy used involves reducing the given problem on moving objects to a different problem on a set of static objects, and then solving the latter problem using techniques based on sweeping, orthogonal range searching, simplex composition, and parametric search.
We consider a generalization of geometric range searching, with the goal of generating an informa... more We consider a generalization of geometric range searching, with the goal of generating an informative "summary" of the objects contained in a query range via the application of a suitable aggregation function on these objects. We provide some of the first results for functions such as closest pair, diameter, and width that measure the extent (or "spread") of the retrieved set. We discuss a subset of our results, including closest pair queries on point-sets in the plane and on random pointsets in R d (d ≥ 2) and guaranteed-quality approximations for diameter and width queries in the plane, all for axes-parallel query rectangles.
In a range-aggegate query problem we wish to preprocess a set S of geometric objects such that gi... more In a range-aggegate query problem we wish to preprocess a set S of geometric objects such that given a query orthogonal range q, a certain intersection or proximity query on the objects of S intersected by q can be answered efficiently. Although range-aggregate queries have been widely investigated in the past for aggregation functions like average, count, min, max, sum etc. there is little work on proximity problems. In this paper, we solve two problems. We first consider the problem of determining if any pair of points in a query orthogonal rectangle are within a constant λ of each other and give a solution that takes O(n log 2+ n) space and O(log 2 n) query time. Subsequently, we solve the problem of finding the closest pair in a query orthogonal rectangle which takes O(n log 3 n) space and O(log 3 n) query time.
Efficient external memory segment intersection for processing very large VLSI layouts
One fundamental problem that arises in VLSI layout analysis and verification is the segment inter... more One fundamental problem that arises in VLSI layout analysis and verification is the segment intersection problem: given a set of segments in the plane, find all pairwise intersections. This problem has been widely studied in the Computational Geometry. One problem with processing large VLSI layouts is that the data to be processed may be far too massive to fit in main memory. When dealing with data sets of sizes exceeding main memory, communication between the fast internal memory and the slow external memory is often the performance bottleneck and algorithms and data structures designed under the assumption of a single level of memory may be meaningless. External-memory algorithms try to optimize performance by taking into account disk accesses. One can certainly use the standard main memory algorithms for data that reside on disk but their performance is often considerably below the optimum because there is no control over how the operating system performs disk accesses. On demand thrashing can be high thus resulting in an increase in response time. Although a lot of research has been done in the recent past on efficient external-memory algorithms and data structures, such work in the area of VLSI computer-aided design is limited. We have designed and implemented a practical external-memory algorithm for reporting all intersecting pairs amongst a set of orthogonal segments. The key to our success is that we take advantage of the fact that real data sets from VLSI applications tend to obey the so-called "square-root" rule, i.e. in a set of TV line segments, the expected number of line segments intersecting a horizontal or vertical scanline in a VLSI layout is O(radic N), a fact ignored by known external-memory algorithms. Another factor that is crucial to our success is that other algorithms stores the data structures in external memory requiring I/O to access them. We reduce such disk accesses by using a clever storage scheme. Our algorithm outperforms not only a - - standard in-memory algorithm but also an existing external-memory algorithm for segment intersection reporting
Uploads
Papers by Prosenjit Gupta