Anil Maheshwari

Optimal parallel algorithms for rectilinear link-distance problems

AbstractWe provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). LetP be a trapezoided rectilinear simple polygon withn vertices. InO(logn) time usingO(n/logn) processors we can optimally compute:1.Minimum réctilinear link paths, or shortest paths in theL1 metric from any point inP to all vertices ofP.2.Minimum rectilinear link paths from any segment insideP to all vertices ofP.3.The rectilinear window (histogram) partition ofP.4.Both covering radii and vertex intervals for any diagonal ofP.5.A data structure to support rectilinear link-distance queries between any two points inP (queries can be answered optimally inO(logn) time by uniprocessor).nOur solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which usedO(n logn) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon.

An algorithm for recognizing palm polygons

A polygonP is said to be apalm polygon if there exists a pointx∈P such that the Euclidean shortest path fromx to any pointy∈P makes only left turns or only right turns. The set of all such pointsx is called thepalm kernel. In this paper we propose an O(E) time algorithm for recognizing a palm polygonP, whereE is the size of the visibility graph ofP. The algorithm recognizes the given polygonP as a palm polygon by computing the palm kernel ofP. If the palm kernel is not empty,P is a palm polygon.

Parallel Algorithms for All Minimum Link Paths and Link Center Problems

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R equal the minimum number of segments or links that are needed to construct a path in R between the points. The minimum link path problem is to compute a path consisting of minimum number of links between two points in R, when R is the inside of a simple polygon P of size ns. Recently Chandru et al. [1] proposed a parallel algorithm for computing minimum link path between two points inside P and it runs in O(log n log log n) time using O(n) processors. Here we show that minimum link paths from a point to all vertices of P can be computed in O(log2n log log n) time using O(n) processors. Using this result we propose a parallel algorithm for computing the link center of P. The link center of P is the set of points x inside P such that the link distance from x to any other point in P is minimized. The algorithm runs in O(log2n log log n) time using O(n2) processors. We also show that a triangle in the approximate link center can be computed in O(log3n log log n) time using O(n) processors. The complexity results of this paper are with respect to the CREW-PRAM model of computation.

An optimal algorithm for computing a minimum nested nonconvex polygon

Given a polygon Q of m vertices inside another polygon P of n vertices, a polygon K is called nested between P and Q when it circumscribes Q and is inscribed in P. It can be seen that there are many nested polygons between P and Q. We are interested in constructing one such nested polygon that has the minimum number of vertices.

Multi-List Ranking: Complexity and Applications

A natural combinatorial generalization of the convex layer problem, termed multi-list ranking, is introduced. It is proved to be P-complete in the general case. When the number of lists or layer size are bounded by s(n), multi-list ranking is shown to be log-space hard for the class of problems solvable simultaneously in polynomial time and space s(n). On the other hand, simultaneous polynomial-time and O(s(n) log n)-space solutions in the above cases are provided. Also, NC algorithms for multilist ranking when the number of lists or layer size are constantly bounded are given. In result, the first NC solutions (SC solutions, respectively) for the convex layer problem where the number of orientations or the layer size are constantly bounded (poly-log bounded, respectively) are derived.

Parallel algorithms for rectilinear link distance problems

The authors provide optimal parallel solutions to several fundamental link distance problems set in trapezoided rectilinear polygons. All parallel algorithms are deterministic, run in logarithmic time, have an optimal time-processor product and are designed to run on EREW PRAM. The authors develop techniques (e.g. rectilinear window partition) for solving link distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. They employ these parallel techniques for example to compute the link diameter, link center, and central diagonal of a rectilinear polygon. Their results also imply an optimal linear-time sequential algorithm for constructing a data structure to support rectilinear link distance queries between points.<<ETX>>

Multilist layering: complexity and applications

Abstract A natural combinatorial generalization of the convex layer problem, termed multilist layering , is introduced. It is observed to be P-complete in the general case. When the number of lists or layer size are bounded by s ( n ), multilist layering is shown to be logspace-hard for the class of problems solvable simultaneously in polynomial time and space s ( n ). On the other hand, simultaneous polynomial-time and O ( s ( n ) log n )-space solutions in the above cases are provided. Thus a natural, almost complete problem for Steves classes SC 1 ,SC 2 ,/4. is in particular obtained. Also, NC algorithms for multilist layering when the number of lists or the layer size is bounded by a constant are given. As a result, the first NC solutions (SC solutions, respectively) for the convex layer problem where the number of orientations or the layer size are bonded by a constant (polylog bounded, respectively) are derived.

Computing the Shortest Path Tree in a Weak Visibility Polygon

In this paper we propose two linear time algorithms for computing the shortest path tree rooted at any vertex of a weak visibility polygon. The first algorithm computes the shortest path tree in a polygon weakly visible from a given internal segment. The second algorithm computes the shortest path tree in a weak visibility polygon without the knowledge of a visibility segment. In both algorithms we use the convexity property of shortest paths in weak visibility polygons established in [4,11].

Optimal Shooting: Characterizations and Applications

We have introduced the archers problem and shown that its solution leads to the intersting class of stage graphs which we characterized to be permutation graphs. The characterization which leads to the solution for the archers problem allowed for the development of improved algorithms for matching in permutation graphs, for a class of two-processor scheduling problems, and for several geometric problems. We answer the natural question of how the archers problem generalizes to multiple stages and to three-dimensions. In two dimensions we establish upper and lower bounds on the number of stages required to represent graphs. In three dimensions we give characterization results and establish the NP-completeness of the recognition problem already for triangular stages.

An optimal parallel algorithm for computing furthest neighbors in a tree

A vertex y is said to be a furthest neighbor of a vertex x in a tree if the weight of the path from x to y is greater than or equal to the weight of the path from x to any other vertex in the tree. We propose a parallel algorithm for computing a furthest neighbor of each vertex in a tree of size n with positive (real-valued) edge weights. The algorithm runs in O(logn) time and O(n) space using O(n/logn) processors on an exclusive-read, exclusive-write parallel RAM. We show that all furthest neighbors of all vertices can also be computed within the same resource bounds. The algorithms are based on an interesting relationship between the diameter of a tree and the furthest neighbor of a vertex.