Partitioning trees: Matching,domination, and maximum diameter

A matching and a dominating set in a graph G are related in that they determine diameter-bounded subtree partitions of G. For a maximum matching and a minimum dominating set, the associate partitions have the fewest numbers of trees. The problem of determining a minimum dominating set in an arbitrary graph G is known to be NP-complete. In this paper we present a linear algorithm for partitioning an arbitrary tree into a minimum number of subtrees, each having a diameter at mostk, for a givenk.Research supported in part by the National Science Foundation under Grant ENG 7902960.Research supported in part by the National Science Foundation under Grant STI 7902960.     

Combinatorial Algorithms for Data Migration to Minimize Average Completion Time

The data migration problem is to compute an efficient plan for moving data stored on devices in a network from one configuration to another. It is modeled by a transfer graph, where vertices represent the storage devices, and edges represent data transfers required between pairs of devices. Each vertex has a non-negative weight, and each edge has a processing time. A vertex completes when all the edges incident on it complete; the constraint is that two edges incident on the same vertex cannot be processed simultaneously. The objective is to minimize the sum of weighted completion times of all vertices. Kim (J. Algorithms 55, 42–57, 2005) gave an LP-rounding 3-approximation algorithm when edges have unit processing times. We give a more efficient primal-dual algorithm that achieves the same approximation guarantee. When edges have arbitrary processing times we give a primal-dual 5.83-approximation algorithm. We also study a variant of the open shop scheduling problem. This is a special case of the data migration problem in which the transfer graph is bipartite and the objective is to minimize the sum of completion times of edges. We present a simple algorithm that achieves an approximation ratio of , thus improving the 1.796-approximation given by Gandhi et al. (ACM Trans. Algorithms 2(1), 116–129, 2006). We show that the analysis of our algorithm is almost tight. A preliminary version of the paper appeared in the Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006. Research of R. Gandhi partially supported by Rutgers University Research Council Grant. Research of J. Mestre done at the University of Maryland; supported by NSF Awards CCR-0113192 and CCF-0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

An optimal algorithm for shortest paths on weighted interval and circular-arc graphs,with applications

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcsand the sorted list of the interval endpoints. Our algorithm solves this problem optimally inO(n) time, wheren is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is anO(qn + n logn)-time algorithm for the minimum-weight circle-cover problem, whereq is the minimum number of arcs crossing any point on the circle; then logn term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were0(n logn) for this shortest paths problem, andO(qn logn) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (logn)-factor improvements in other problems on such graphs.The research of M. J. Atallah was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the Air Force Office of Scientific Research under Contract AFOSR-90-0107, and by the National Science Foundation under Grant CCR-9202807. D. Z. Chen's research was supported in part by the Leonardo Fibonacci Institute, Trento, Italy. The research of D. T. Lee was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the National Science Foundation, and the Office of Naval Research under Grants CCR-8901815, CCR-9309743, and N00014-93-1-0272.     

Maintaining bridge-connected and biconnected components on-line

We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(m(m,n)), where is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.Research at Princeton University supported in part by National Science Foundation Grant DCR-86-05962 and Office of Naval Research Contract N00014-91-J-1463.This work was partially done while the author was at the Department of Computer Science, Princeton University, Princeton, NJ 08544, USA.     

An efficient parallel algorithm for shortest paths in planar layered digraphs

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs.We show that these graphs admit special kinds of separators calledone- way separators which allow the paths in the graph to cross it only once. We use these separators to give divide- and -conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in ann-node planar layered digraph in timeO(log² n) usingn/logn processors. We then use results of Aggarwal and Park [1] and Atallah [4] to improve the time bound toO(log² n) in the CREW model andO(logn log logn) in the CREW model. The processor bounds still remain asn/logn for the CREW model andn/log logn for the CRCW model.Support for the first and third authors was provided in part by a National Science Foundation Presidential Young Investigator Award CCR-9047466 with matching funds from IBM, by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052, ARPA, Order 8225. Support for the second author was provided in part by NSF Research Grant CCR-9007851, by Army Research Office Grant DAAL03-91-G-0035, and by the Office of Naval Research and the Advanced Research Projects Agency under Contract N00014-91-J-4052 and ARPA Order 8225.     

Maximumk-covering of weighted transitive graphs with applications

Consider a weighted transitive graph, where each vertex is assigned a positive weight. Given a positive integerk, the maximumk-covering problem is to findk disjoint cliques covering a set of vertices with maximum total weight. An 0(kn ²)-time algorithm to solve the problem in a transitive graph is proposed, wheren is the number of vertices. Based on the proposed algorithm the weighted version of a number of problems in VLSI layout (e.g.,k-layer topological via minimization), computational geometry (e.g., maximum multidimensionalk-chain), graph theory (e.g., maximumk-independent set in interval graphs), and sequence manipulation (e.g., maximum increasingk-subsequence) can be solved inO(kn ²), wheren is the input size.This Work was supported in part by the National Science Foundation under Grant MIP-8709074 and MIP-8921540.     

Algorithms for bichromatic line-segment problems and polyhedral terrains

We consider a variety of problems on the interaction between two sets of line segments in two and three dimensions. These problems range from counting the number of intersecting pairs between m blue segments andn red segments in the plane (assuming that two line segments are disjoint if they have the same color) to finding the smallest vertical distance between two nonintersecting polyhedral terrains in three-dimensional space. We solve these problems efficiently by using a variant of the segment tree. For the three-dimensional problems we also apply a variety of recent combinatorial and algorithmic techniques involving arrangements of lines in three-dimensional space, as developed in a companion paper.Work on this paper by the first author has been supported in part by the National Science Foundation under Grant CCR-9002352. Work by the second author was supported in part by the National Science Foundation under Grant CCR-8714565. The fourth author has been supported in part by the Office of Naval Research under Grant N0014-87-K-0129, by the National Science Foundation under Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a grant from the US-Israeli Binational Science Foundation.     

Distributed verification of minimum spanning trees

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own state and label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper, we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (as long as W > (log n)^1+ε for some fixed ε > 0). Both our bounds improve previously known bounds for the problem. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings. A preliminary version of this work was presented in ACM PODC 2006. A. Korman was supported in part at the Technion by an Aly Kaufman fellowship. S. Kutten was supported in part by a grant from the Israeli Ministry for Science and Technology.     

A better performance guarantee for approximate graph coloring

Approximate graph coloring takes as input a graph and returns a legal coloring which is not necessarily optimal. We improve the performance guarantee, or worst-case ratio between the number of colors used and the minimum number of colors possible, toO(n(log logn)³/(logn)³), anO(logn/log logn) factor better than the previous best-known result.The work of the first author was supported by Air Force Grant AFOSR-86-0078 and NSF PYI Grant 8657527-CCR. The work of the second author was supported by a National Science Foundation Graduate Fellowship.     

A parallel graph partitioning algorithm for a message-passing multiprocessor

We develop a parallel algorithm for partitioning the vertices of a graph intop2 sets in such a way that few edges connect vertices in different sets. The algorithm is intended for a message-passing multiprocessor system, such as the hypercube, and is based on the Kernighan-Lin algorithm for finding small edge separators on a single processor.⁽¹⁾ We use this parallel partitioning algorithm to find orderings for factoring large sparse symmetric positive definite matrices. These orderings not only reduce fill, but also result in good processor utilization and low communication overhead during the factorization. We provide a complexity analysis of the algorithm, as well as some numerical results from an Intel hypercube and a hypercube simulator.Publication of this report was partially supported by the National Science Foundation under Grant DCR-8451385 and by AT&T Bell Laboratories through their Ph.D scholarship program.     

Space bounds for a game on graphs

We study a one-person game played by placing pebbles, according to certain rules, on the vertices of a directed graph. In [3] it was shown that for each graph withn vertices and maximum in-degreed, there is a pebbling strategy which requires at mostc(d) n/logn pebbles. Here we show that this bound is tight to within a constant factor. We also analyze a variety of pebbling algorithms, including one which achieves the 0(n/logn) bound.Research partially supported by DAAD (German Academic Exchange Service) Grant No. 430/402/563/5 (W.J.P.)Research partially supported by National Science Foundation grant MCS75-22870 and Office of Naval Research contract N0014-76-C-0688 (R.E.T. and J.R.C.).Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Searching among intervals and compact routing tables

Shortest paths in weighted directed graphs are considered within the context of compact routing tables. Strategies are given for organizing compact routing tables so that extracting a requested shortest path will takeo(k logn) time, wherek is the number of edges in the path andn is the number of vertices in the graph. The first strategy takesO (k+logn) time to extract a requested shortest path. A second strategy takes (k) time on average, assuming alln(n–1) shortest paths are equally likely to be requested. Both strategies introduce techniques for storing collections of disjoint intervals over the integers from 1 ton, so that identifying the interval within which a given integer falls can be performed quickly.This research was supported in part by the National Science Foundation under Grants CCR-9001241 and CCR-9322501 and by the Office of Naval Research under Contract N00014-86-K-0689.     

On finding optimal and near-optimal lineal spanning trees

The search for good lineal, or depth-first, spanning trees is an important aspect in the implementation of a wide assortment of graph algorithms. We consider the complexity of findingoptimal lineal spanning trees under various notions of optimality. In particular, we show that several natural problems, such as constructing a shortest or a tallest lineal tree, are NP-hard. We also address the issue of polynomial-time, near-optimization strategies for these difficult problems, showing that efficient absolute approximation algorithms cannot exist unlessP = NP.This author's research was supported in part by the Sandia University Research Program and by the National Science Foundation under Grant M IP-8603879.This author's research was supported in part by the National Science Foundation under Grants ECS-8403859 and MIP-8603879.     

Shortest paths in euclidean graphs

We analyze a simple method for finding shortest paths inEuclidean graphs (where vertices are points in a Euclidean space and edge weights are Euclidean distances between points). For many graph models, the average running time of the algorithm to find the shortest path between a specified pair of vertices in a graph withV vertices andE edges is shown to beO(V) as compared withO(E +V logV) required by the classical algorithm due to Dijkstra.Support for the first author was provided in part by NSF Grant MCS-83-08806. Support for the second author was provided in part by NSF Grants MCS-81-05324 and DCR-84-03613, an NSF Presidential Young Investigator Award, an IBM research contract, and an IBM Faculty Development Award. Support for this research was also provided in part by an ONR and DARPA under Contract N00014-83-K-0146 and ARPA Order No. 4786. Equipment support was provided by NSF Grant MCS-81-218106.     

Finding and counting given length cycles

We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results. This work was supported in part by The Basic Research Foundation administrated by The Israel Academy of Sciences and Humanities.     

On the Parameterized Complexity of Layered Graph Drawing

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. Research initiated at the International Workshop on Fixed Parameter Tractability in Graph Drawing, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 9–16, 2001, organized by S. Whitesides. Research of Canada-based authors is supported by NSERC; research of Quebec-based authors is also supported by a grant from FCAR. Research of D.R. Wood completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.     

When is a controllerH ∞-optimal?

This paper examines conditions under which a given single input, single output, linear time invariant control system isH ^∞-optimal with respect to weighted combinations of its sensitivity function and its complementary sensitivity function. The specific weighting functions considered are defined in terms of the given plant and nominal controller. This paper shows that a large class of practical controllers areH ^∞-optimal, including typical stable controllers. This research was supported in part by the National Science Foundation under Grant No. ECS-8451519, grants from Honeywell, 3M, Sperry, and E. F. Johnson Company, ONR Research Grant N00014-82-C-0157, and AFOSR Research Grant F49620-86-C-0001.     

Edge-disjoint spanning trees and depth-first search

Summary This paper presents an algorithm for finding two edge-disjoint spanning trees rooted at a fixed vertex of a directed graph. The algorithm uses depthfirst search and an efficient method for computing disjoint set unions. It requires O (e(e, n)) time and O(e) space to analyze a graph with n vertices and e edges, where (e, n) is a very slowly growing function related to a functional inverse of Ackermann's function.This work was partially supported by the National Science Foundation Grant GJ-3S604X, and by a Miller Research Fellowship, at the University of California, Berkeley; and by the National Science Foundation, Grant MCS 72-03752 A03, at Stanford University.     

An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs

Given a positive integer k and a complete graph with non-negative edge weights satisfying the triangle inequality, the remote-clique problem is to find a subset of k vertices having a maximum-weight induced subgraph. A greedy algorithm for the problem has been shown to have an approximation ratio of 4, but this analysis was not shown to be tight. In this paper, we use the technique of factor-revealing linear programs to show that the greedy algorithm actually achieves an approximation ratio of 2, which is tight. This research was supported in part by the National Science Foundation under grant CISE-EI-0305954 and was performed while the first author was at Washington University in St. Louis. A preliminary version of this paper appears in RANDOM-APPROX ’06, volume 4110 of Lecture Notes in Computer Science, pp. 49–60, 2006.     

Orienting polygonal parts without sensors

In manufacturing it is often necessary to orient parts prior to packing or assembly. We say that a planar part ispolygonal if its convex hull is a polygon. We consider the following problem: given a list ofn vertices describing a polygonal part whose initial orientation is unknown, find the shortest sequence of mechanical gripper actions that is guaranteed to orient the part up to symmetry in its convex hull. We show that such a sequence exists for any polygonal part by giving anO[n ² logn) algorithm for finding the sequence. Since the gripper actions do not require feedback, this result implies that any polygonal part can be orientedwithout sensors.This report describes research conducted in part while the author was a graduate student supported by NSF Grant DMC-8520475 and NASA-Ames Grant NCC 2-463 at the School of Computer Science at Carnegie Mellon University. The author is currently supported by a grant from the Faculty Research Initiation Fund at the University of Southern California.