Andranik Mirzaian

Lagrangian relaxation for the star‐star concentrator location problem: Approximation algorithm and bounds

The star-star concentrator location problem (SSCLP), which is a network layout problem, is considered. SSCLP is formulated as an integer linear programming problem. The Lagrangian relaxation (LR) method is used to obtain suboptimal solutions (upper bounds) and lower bounds. Three different LRs are used for SSCLP. The resulting Lagrangian dual problems are shown to be equivalent to some linear programming problems. An approximation algorithm is suggested for SSCLP that produces both a feasible solution (upper bound) and a lower bound. It is shown that if z and zare the lower and upper bounds found, then z/z ≤ k, where k is the concentrator capacity. Some computational examples with up to 50 terminals and 20 potential concentrator sites are considered. All the network designs obtained are shown to be within 2.8% of optimal.

River routing in VLSI

Abstract A common framework for solving several VLSI river routing problems is developed. The main result of this paper is an O(n) time algorithm for the optimum offset problem. This improves upon the best previously known O(n log n ) time bound. A new reduction technique called halving is used to achieve this result. A variety of other applications of the halving technique are also mentioned. Algorithms for the minimum area, minimum longest wire length , and minimum total wire length problems are also given that take O ( n 2 ) time.

A linear reordering algorithm for parallel pivoting of chordal graphs

This paper provides an efficient algorithm for generating an ordering suitable for the parallel elimination of nodes in chordal graphs. The time complexity of the reordering algorithm is shown to be linear in the size of the chordal graph. The basic parallel pivoting strategy is originally by Jess and Kees [IEEE Trans. Comput., C-31 (1982), pp. 231–239]. The relevance of the reordering to parallel factorization of sparse matrices (not necessarily chordal) is also discussed.

Hamiltonian triangulations and circumscribing polygons of disjoint line segments

Abstract Let Σ={S1,…,Sn} be a finite set of disjoint line segments in the plane. We conjecture that its visibility graph, Vis(Σ), is hamiltonian. In fact, we make the stronger conjecture that Vis(Σ) has a hamiltonian cycle whose embedded version is a simple polygon (i.e., its boundary edges are noncrossing visibility segments). We call such a simple polygon a spanning polygon of Σ. Existence of a spanning polygon of Σ is equivalent to the existence of a hamiltonian triangulation of Σ. A spanning polygon P is said to be a circumscribing polygon of Σ, if it has the additional property that no segment in Σ lies in the exterior of P. We prove circumscribing polygons exist for the special case when Σ is extremally situated, i.e., when each segment Si touches the convex hull boundary of Σ. Furthermore, for this special case we give an algorithm that constructs a circumscribing polygon in O(n log n) time and this is optimal.

Selection in X + Y and matrices with sorted rows and columns☆

Abstract Let A be an n×n matrix of reals with sorted rows and columns and k an integer, 1 ⩽ k ⩽ n 2 . We present an O(n) time algorithm for selecting the k th smallest element of A. If X and Y are sorted n-vectors of reals, then Cartesian sum X + Y is such a matrix as A. One application of selection in X + Y can be found in statistics. The algorithm presented here is based on a new divide-and-conquer technique, which can be applied to similar order related problems as well. Due to the fact that the algorithm has a relatively small constant time factor, this result is of practical as well as theoretical interest.

Landmark selection strategies for path execution

A mobile robot often executes a planned path by measuring its position relative to visible landmarks at known positions and then using this information to estimate its own absolute position. The minimum number of landmarks k required for self-location depends on the types of measurements the robot can perform, such as visual angles using a video camera on a pan-tilt unit, distances using a laser range finder, or absolute orientation using a compass. The problem we address is how to find which k landmarks the robot should detect and track over which segments of a path, so that the cost of sensing (detection and tracking) is minimized. We assume that there are more landmarks visible from each point of the robots path than the minimum necessary, and that their positions are known relative to the path. We present several formulations of this problem in graph-theoretic terms, with different amounts of flexibility, generality and complexity, which can be solved by known graph-theoretic algorithms. We present uniform-cost algorithms (all landmarks have equal cost of detection and tracking), and weighted-cost algorithms (each landmark has different cost). The complexity of these algorithms is low-order polynomial in the number of landmarks k that must be simultaneously tracked at each point of the robots path. We also present an algorithm that can incorporate not only the sensing cost for each landmark, but the suitability of the landmark configurations relative to the robot. The resulting complexity is, in this case, exponential in the number of landmarks k tracked at each point of the robot path. The algorithm is still practical, since k is typically a fixed small integer.

A minimum separation algorithm for river routing with bounded number of jogs

The single-layer rectilinear river routing model with no restriction on the number of jogs per wire is considered. In particular, the author studies the model in which there is a fixed constant upper bound J on the number of jogs each wire can have. The author proposes an optimal O(n) time algorithm for the feasibility problem. This leads to an O(n log n) time algorithm for the minimum separation problem. Both algorithms take O(n) space are quite practical. This is a significant improvement over T.C. Tuan and S.L. Hakimis (1987) minimum separation algorithm, which is designed for J=2 and takes O(n/sup 3/) time and O(n/sup 2/) space.<<ETX>>

Robot Map Verification of a Graph World

In the map verification problem, a robot is given a (possibly incorrect) map M of the world G with its position and orientation indicated on the map. The task is to find out whether this map, for the given robot position and orientation in the map, is correct for the world G. We consider the world model with a graph G = (VG; EG) in which, for each vertex, edges incident to the vertex are ordered cyclically around that vertex. This holds similarly for the map M = (VM; EM). The robot can traverse edges and enumerate edges incident on the current vertex, but it cannot distinguish vertices and edges from each other. To solve the verification problem, the robot uses a portable edge marker, that it can put down and pick up as needed. The robot can recognize the edge marker when it encounters it in G. By reducing the verification problem to an exploration problem, verification can be completed in O(|VG| × |EG|) edge traversals (the mechanical cost) with the help of a single vertex marker which can be dropped and picked up at vertices of the graph world [DJMW1,DSMW2]. In this paper, we show a strategy that verifies a map in O(|VM|) edge traversals only, using a single edge marker, when M is a plane embedded graph, even though G may not be (e.g., G may contain overpasses, tunnels, etc.).

A halving technique for the longest stuttering subsequence problem

Abstract This paper presents an optimal linear-time algorithm to solve the longest stuttering subsequence problem . A suitable variation of the recently developed halving method is used to achieve this result. This demonstrates yet another application of the halving method and gives an indication that the method can be considered as an algorithmic paradigm .