Sudebkumar Prasant Pal

A convex hull algorithm for discs, and applications

Abstract We show that the convex hull of a set of discs can be determined in Θ( n log n ) time. The algorithm is straightforward and simple to implement. We then show that the convex hull can be used to efficiently solve various problems for a set of discs. This includes O( n log n ) algorithms for computing the diameter and the minimum spanning circle, computing the furthest site Voronoi diagram, computing the stabbing region, and establishing the region of intersection of the discs.

Resolving horizontal constraints and minimizing net wire length for multi-layer channel routing

The channel routing problem in VLSI design is to route a specified interconnection among modules in as small an area as possible. Hashimoto and Stevens (1971) proposed an algorithm for solving the two-layer channel routing problem in the absence of vertical constraints. In this paper, we analyze this algorithm in two different ways. In the first analysis, we show that a graph-theoretic realization, algorithm MCC1, runs in O(m + n + e) time, where m is the size of the channel specification of n nets, and e is the size of the complement of the horizontal constraint graph. In the second analysis, algorithm MCC2, we show that a time complexity of O(m + n log n) can be achieved. Algorithms MCC1 and MCC2 guarantee optimum routing solutions under the multi-layer V/sub i+1/H/sub i/ (i/spl ges/1) routing model, where the horizontal and vertical layers of interconnect alternate. Finally, we consider the problem of minimizing the total net wire length in the V/sub i+1/H/sub i/ (i/spl ges/1) routing model. Given a channel specification and a partition of the set of nets (where the nets within each part of the partition are non-overlapping), we propose an O(m + d/sub max/log d/sub max/) time algorithm for minimizing the total net wire length, subject to the condition that nets from a part of the partition are assigned to the same track. All our solutions use the minimum number of via holes.<<ETX>>

A general graph theoretic framework for multi-layer channel routing

In this paper we propose a general framework for viewing a class of heuristics for track assignment in channel routing from a purely graph theoretic angle. Within this framework we propose algorithms for computing routing solutions using optimal or near optimal number of tracks for several well-known benchmark channels in the two-layer VH. Three-layer HVH, and multi-layer V/sub i/H/sub i/ and V/sub i/H/sub i+1/ routing models. Within the same framework we also design an algorithm for minimizing the total wire length in the two-layer VH and three-layer HVH routing models.

Visibility with One Reflection

Abstract. We extend the concept of the polygon visible from a source point S in a simple polygon by considering visibility with two types of reflection, specular and diffuse. In specular reflection a light ray reflects from an edge of the polygon according to the rule: the angle of incidence equals the angle of reflection. In diffuse reflection a light ray reflects from an edge of the polygon in all inward directions. Several geometric and combinatorial properties of visibility polygons under these two types of reflection are described, when at most one reflection is permitted. We show that the visibility polygon Vs(S) under specular reflection may be nonsimple, while the visibility polygon Vd(S) under diffuse reflection is always simple. We present a Θ(n2) worst-case bound on the combinatorial complexity of both Vs(S) and Vd(S) and describe simple O(n2 log2 n) time algorithms for constructing the sets.

Characterizing and recognizing weak visibility polygons

A polygon is said to be a weak visibility polygon if every point of the polygon is visible from some point of an internal segment. In this paper we derive properties of shortest paths in weak visibility polygons and present a characterization of weak visibility polygons in terms of shortest paths between vertices. These properties lead to the following efficient algorithms: (i) an O(E) time algorithm for determining whether a simple polygon P is a weak visibility polygon and for computing a visibility chord if it exist, where E is the size of the visibility graph of P and (ii) an O(n2) time algorithm for computing the maximum hidden vertex set in an n-sided polygon weakly visible from a convex edge.

Visibility with multiple reflections

Abstract. We show that the region lit by a point light source inside a simple n -gon after at most k reflections off the boundary has combinatorial complexity O(n2k) , for any k≥ 1 . A lower bound of Ω ((n/k-Θ(1))2k) is also established which matches the upper bound for any fixed k . A simple near-optimal algorithm for computing the illuminated region is presented, which runs in O(n2k log n) time and O(n2k) space for k>1 , and in O(n2 log2 n) time and O(n2) space for k=1 .

Visibility with multiple diffuse reflections

Abstract The combinatorial complexity of the region lit up from a point light source inside a simple n-gon with perfectly reflecting edges, after at most k specular reflections was established as O(n2k) (Aronov et al., 1996). A lower bound of Ω(( n k ) 2k ) was also established, matching the upper bound for any fixed k. In a real situation, surfaces may not be perfect mirrors; indeed most surfaces may be non-shiny or rough, causing diffuse reflection, rather than specular reflection. In contrast to the result of (Aronov et al., 1996), the combinatorial complexity of the region lit up from a point inside a simple n-gon after k diffuse reflections is established here to be O (n 2⌜ (k+1) 2 ⌝+1 ) .

NP-Completeness of Multi-Layer No-Dogleg Channel Routing and an Efficient Heuristic

In this paper we show that given a channel specification of density d,,,, the problem of determinin whether there is a routing solution using [dmar/2f tracks an the four-layer no-dogleg VHVH Manhattan routing model is NP-complete. A similar result is derived also for the six-layer VHVHVH routing model, where we wish to determine whether there is a routing solution using [d,,,/3] tracks. We also propose an efficient polynomial time heuristic for computing a multi-layer routing solution for a given channel specification. Our heuristic computes optimal multi-layer solutions for several benchmark problems.

A combinatorial approach for studying local operations and classical communication transformations of multipartite states

We develop graph theoretic methods for analyzing maximally entangled pure states distributed between a number of different parties. We introduce a technique called bicolored merging, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication. We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among n agents. Further, we discuss a new graph theoretic metric on a class of multipartite states, and its implications.

Fair leader election by randomized voting

Leader election is a fundamental problem in distributed computing where a unique node from a set of nodes declares itself as the leader (a distinguished state). We propose a notion of fairness in the leader election problem; a leader election algorithm is said to be fair if each node has the same probability of getting elected as leader. We show that existing deterministic algorithms based on comparisons of identifiers fail to achieve fairness. We demonstrate how fairness can be achieved through randomization and propose new leader election algorithms based on randomized voting. We separate the task of fair leader election on unidirectional rings into two subtasks: attrition and solitude detection following [9]. We show that tight interleaving of these two procedures as in [7], results in fair leader election on asynchronous, anonymous, unidirectional rings using expected O(n log n) bits of communication, in expected O(log n) rounds. (Here n is the number of nodes in the ring.) This matches the performance of the optimal algorithm of Abrahamson et al. [1]. Similar algorithms are presented for electing a fair leader in other models.