On the complexity of some arborescences finding problems on a multihop radio network

An arborescence of a multihop radio network is a directed spanning tree (with rootx) such that the edges are directed away from the root. Based upon an arborescence,x canbroadcast a message to other nodes according to the directed edges of the spanning tree. The minimum transmission power arborescence problem is to find an arborescence such that the message can be broadcasted to other nodes by using a minimal amount of transmission power. The minimum delay arborescence problem is to find an arborescence such that a message can be broadcasted to other nodes by using a minimal number of broadcast transmission. In this paper we show that both these problems areNP-complete. The reductions are from the maximum leaf spanning tree problem.Areverse arborescence is similar to an arborescence except that the edges are directed toward the root. Based upon a reverse arborescence, the root node cancollect information from other nodes. In this paper we also show that the reverse minimum transmission power arborescence problem can be solved with the same computational complexity as that of finding a minimum cost spanning tree, and the reverse minimum delay arborescence problem can be solved with the same computational complexity as that of finding a spanning tree.     

An optimal time algorithm for finding a maximum weight independent set in a tree

The maximum weight independent set problem for a general graph is NP-hard. But for some special classes of graphs, polynomial time algorithms do exist for solving it. Based on the divide-and-conquer strategy, Pawagi has presented anO(|V|log|V|) time algorithm for solving this problem on a tree. In this paper, we propose anO(|V|) time algorithm to improve Pawagi's result. The proposed algorithm is based on the dynamic programming strategy and is time optimal within a constant factor.     

AnO(n 2) algorithm for finding the compact sets of a graph

A special clustering problem is discussed in this paper, called the compact set problem. The goal of the problem is to find all compact sets in a complete, weighted, undirected graph withn vertices. A subsetC of vertices is called a compact set if 1<|C|<n and the maximum weight among all edges inC is smaller than the minimum weight among all edges connecting one vertex inC and the other vertex not inC. An algorithm with complexityO(n ²) is given for the problem improving the previous results.This research was partially supported by the National Science Council of the Republic of China under Grant NSC81-0408-E-216-502.     

Intersection queries in sets of disks

In this paper we develop some new data structures for storing a set of disks that can answer different types of intersection queries efficiency. If the disks are non-intersecting we obtain a linear size data structure that can report allk disks intersecting a query line segment in timeO(n ⁺ +k), wheren is the number of disks,=log₂(1+5)–1 0.695, and is an arbitrarily small positive constant. If the segment is a full line, the query time becomesO(n +k). For intersecting disks we obtain anO(n logn) size data structure that can answer an intersection query in timeO(n ^2/3 log² n+k). We also present a linear size data structure for ray shooting queries, whose query time isO(n ).The research of the first two authors was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). The work of the third author was supported byDimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648.     

An unfeasible matching problem

LetG be a bipartite graph with natural edge weights, and letW be a function from the set of vertices ofG into natural numbers. AW-matching ofG is a subset of the set of edges ofG such that for each vertexv the total weight of edges in the subset incident tov does not exceedW(v). Letm be a natural number. We show that the problem of deciding whether there is aW-matching inG whose total weight is not less thanm is NP-complete even ifG is bipartite and its edge weights as well as theW(v)-constraints are constantly bounded.     

Approximating maximum independent sets by excluding subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.A preliminary version of this paper appeared in [9].Supported in part by National Science Foundation Grant CCR-8902522 and PYI Award CCR-9057488.Research done at Rutgers University. Supported in part by Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) fellowship.     

Quotient tree partitioning of undirected graphs

The partitioning of the vertices of an undirected graph, in a way that makes its quotient graph a tree, mirrors a way of permuting a square symmetric matrix to allow its factoring with little fill-in. We analyze the complexity of finding the best partitioning and show that it is NP-complete. We also give a new and simpler implementation of an algorithm that finds a maximal quotient tree.     

Finding squares and rectangles in sets of points

A number of problems concerning sets of points in the plane are studied, e.g. establishing whether it contains a subset of size 4, which are the vertices of a square or rectangle. Both the problems of finding axis-parallel squares and rectangles, and arbitrarily oriented squares and rectangles are studied. Efficient algorithms are obtained for all of them. Furthermore, we investigate the generalizations tod-dimensional space, where the problem is to find hyperrectangles and hypercubes. Also, upper and lower bounds are given for combinatorial problems on the maxium number of subsets of size 4, of which the points are the vertices of a square or rectangle. Then we state an equivalence between the problem of finding rectangles, and the problem of findingK _{2, 2} subgraphs in bipartite graphs. Thus we immediately have an efficient algorithm for this graph problem.This work was partially supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM). Work of the second author was also supported by the Dutch Organisation for Scientific Research (N.W.O.).     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).     

A linear time and space algorithm for finding isomorphic subtrees of a binary tree

This paper shows that it is possible to find all isomorphic subtrees of a binary tree in linear time and space.     

Optimal algorithms for total exchange without buffering on the hypercube

Two methods are given for constructing total exchange algorithms for hypercubic processor networks. This is done by means of bit sequences with special properties. The algorithms are optimal with respect to a given time model, need no intermediate message buffering and are local in the sense that every processor executes basically the same program.     

A linear time algorithm for the maximum matching problem on cographs

In this paper, we develop a sequential algorithm for the maximum matching problem on cographs. The input is a parse tree of some cograph. The time complexity of our algorithm is linear.Supported by National Science Council of Taiwan, R.O.C. under Grant NSC81-0415-E-005-03.     

On crossing numbers of hypercubes and cube connected cycles

We prove tight bounds for crossing numbers of hypercube and cube connected cycles (CCC) graphs.The research of both authors was supported by Alexander von Humboldt Foundation, Bonn, Germany.     

A note on the largest empty rectangle problem

We consider the following problem: given a rectangle containingn points, find the largest perimeter subrectangle whose sides are parallel to those of the original rectangle, whose aspect ratio is below a given bound, and which does not contain any of the given points. Chazelle, Drysdale and Lee have studied a variant of this problem with areas as the quantity to be maximized. They gave anO(nlog³ n) algorithm for that problem. We adopt a similar divide-and-conquer approach and are able to use the simpler properties of the perimeter measure to obtain anO(nlog² n) algorithm for our problem.The work of the first author was supported by the Academy of Finland and that of the second by the Natural Sciences and Engineering Research Council of Canada Grant No. A-5692.     

On systolic array methods for band matrix factorizations

A systolic array for band matrixQR factorization is presented and its connection with various other arrays explained. It is shown how the computations should be performed on banded rectangular matrices which appear in several numerical problems.     

Maximum weight independent set in trees

Computing a maximum independent set, weighted or unweighted, isNP-hard for general as well as planar graphs. However, polynomial time algorithms do exist for solving this problem on special classes of graphs. In this paper we present an efficient algorithm for computing a maximum weight independent set in trees. A divide and conquer approach based on centroid decomposition of trees is used to compute a maximum weight independent set withinO(n logn) time, wheren is the number of vertices in the tree. We introduce a notion of analternating tree which is crucial in obtaining a new independent set from the previous one.     

On systolic arrays for updating the Cholesky factorization

We have considered the systolic implementation of several methods for updating the Cholesky factorization. For positive rank-k changes there are simple one-pass arrays that implement algorithms based on elimination and plane rotations. In the case of negative rank-one changes, we do not feel that the standard algorithm [2] has a practical implementation. We have introduced a new algorithm for the case of a negative rank-k change and provided an attractive two-pass systolic implementation.     

A generalized hypergreedy algorithm for weighted perfect matching

We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph whose weights satisfy the triangle inequality. With a modified version of this algorithm we obtain a logn-approximate perfect matching heuristic for points in the Euclidean plane, inO(n log² n) time.This research was supported in part by the DIMACS Grant No. NSF-STC88-09648.This research was supported in part by the NSF under Grant No. CCR 88-07518.     

An efficient implicit data structure for relation testing and searching in partially ordered sets

In this paper, we introduce an implicit data structure which represents a forest-structured partial order to efficiently perform, with respect to time and space, the following operations: 1) testing the relation among two elements (checking whether the two elements are related) and 2) given an elementu, searching for its father. The first operation can be performed in constant time, while the second one requires polylog time (logarithmic in the case of bounded degree). The data structure represents the order relation by referring only to internal nodes of the forest, thus achieving in many cases a significant saving in space occupation. Finally, the algorithm is shown to be optimal in a restricted computation model by deriving a lower bound on the space complexity within such a model.Work partially performed in the framework of Esprit BRA project 3075 Algorithms and Complexity and of Italian MURST 40% project Algoritmi e Strutture di Calcolo and partially supported by the National Research Council (CNR) Project Sistemi Informatici e Calcolo Parallelo.     

The analysis of a nested dissection algorithm

Summary Nested dissection is an algorithm invented by Alan George for preserving sparsity in Gaussian elimination on symmetric positive definite matrices. Nested dissection can be viewed as a recursive divide-and-conquer algorithm on an undirected graph; it usesseparators in the graph, which are small sets of vertices whose removal divides the graph approximately in half. George and Liu gave an implementation of nested dissection that used a heuristic to find separators. Lipton and Tarjan gave an algorithm to findn ^1/2-separators in planar graphs and two-dimensional finite element graphs, and Lipton, Rose, and Tarjan used these separators in a modified version of nested dissection, guaranteeing bounds ofO (n logn) on fill andO(n ^3/2) on operation count. We analyze the combination of the original George-Liu nested dissection algorithm and the Lipton-Tarjan planar separator algorithm. This combination is interesting because it is easier to implement than the Lipton-Rose-Tarjan version, especially in the framework of existïng sparse matrix software. Using some topological graph theory, we proveO(n logn) fill andO(n ^3/2) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree withn ^1/2-separators. For planar and finite element graphs, the leading constant factor is smaller than that in the Lipton-Rose-Tarjan analysis. We also construct a class of graphs withn ^1/2-separators for which our algorithm does not achieve anO(n logn) bound on fill.The work of this author was supported in part by the Hertz Foundation under a graduate fellowship and by the National Science Foundation under Grant MCS 82-02948The work of this author was supported in part by the National Science Foundation under Grant MCS 78-26858 and by the Office of Naval Research under Contract N00014-76-C-0688