Showing papers on "Longest path problem published in 1992"

A potential field approach to path planning

Abstract: A path-planning algorithm for the classical mover's problem in three dimensions using a potential field representation of obstacles is presented. A potential function similar to the electrostatic potential is assigned to each obstacle, and the topological structure of the free space is derived in the form of minimum potential valleys. Path planning is done at two levels. First, a global planner selects a robot's path from the minimum potential valleys and its orientations along the path that minimize a heuristic estimate of the path length and the chance of collision. Then, a local planner modifies the path and orientations to derive the final collision-free path and orientations. If the local planner fails, a new path and orientations are selected by the global planner and subsequently examined by the local planner. This process is continued until a solution is found or there are no paths left to be examined. The algorithm solves a much wider class of problems than other heuristic algorithms and at the same time runs much faster than exact algorithms (typically 5 to 30 min on a Sun 3/260). >

The complexity of multiway cuts (extended abstract)

Abstract: In the Multiway Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2–2/k of the optimal cut weight.

Laying out graphs using queues

Abstract: The problem of laying out the edges of a graph using queues is studied. In a k-queue layout, vertices of the graph are placed in some linear order and each edge is assigned to exactly one of the k queues so that the edges assigned to each queue obey a first-in/first-out discipline. This layout problem abstracts a design problem of fault-tolerant processor arrays, a problem of sorting with parallel queues, and a problem of scheduling parallel processors. A number of basic results about queue layouts of graphs are established, and these results are contrasted with their analogues for stack layouts of graphs (the book-embedding problem). The 1-queue graphs (they are almost leveled-planar graphs) are characterized. It is proved that the problem of recognizing 1-queue graphs is NP-complete. Queue layouts for some specific classes of graphs are given. Relationships between the queuenumber of a graph and its bandwidth and separator size are presented. An apparent tradeoff between the queuewidth and the number of...

Chip-Firing Games on Directed Graphs

Abstract: We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects. Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable. Finally, we show how the basic properties of the “probabilistic abacus” can be derived from our results.

On the all-pairs-shortest-path problem

Abstract: The following algorithm solves the distance version of the all-pairs-shortest-path problem for undirected, unweighed n-vertex graphs in time O(&f(rJ) log n), where M(n) denotes the time necessary to multiply two n x n matrices of small integers (which is currently known to be o(n2376)): Input: n x n O-1 matrix A, the adjacency matrix of undirected, connected graph G’ Output: n x n integer matrix D, with dij the length of a shortest path joining vertices i and j in G function APD(A : n x n O-1 matrix) : n x n integer matrix let Z=A. A let B be an n x n O-1 matrix, where bij = 1 iff i # j and (aij = 1 or ~tj > O) if bij = 1 for all i # j then return n x n matrix D = 2B – A let T = APD(B) let X=T. A { 2tij if Zij ~ tij . degree(j) return n x n matrix D, where dij = Zt ij – 1 if Xij < tij . degree(j) We also address the problem of actually finding a shortest path between each pair of vertices and present a randomized algorithm that matches APD() in its simplicity and in its expected running time. 1. Computing All Dktances In the following let G be an undirected, unweighed, connected graph with vertex set {1, 2, ..., n} and adjacency matrix A, and let dij denote the number of edges on a shortest path joining vertices i and j in G. In this section we show that the function APD() computes all dij correctly within the claimed time bound. Claim 1 Let Z = A oA. There is a path of length 2 in G between vertices i and j iff ~ij >0. Proofi There is a length 2 path joining i and j iff there is a vertex k adjacent to both i and j, which is exactly ‘he Cme ‘f ‘ij = ~l0❑ “Supported by NSF Presidential Young Investigator Award CCR-9058440. Email address: seidel@cs.berkeley.edu Permission to oopy without fee all or part of this material is granted providad that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice ie given that copying is by permission of the Association for Computing Machinery. To copy otharwisa, or to republish, raquirae a fee and/or spacific permission. 24th ANNUAL ACM STOC 5/92/VICTORIA, B.C., CANADA @1992 ACM ()+9791-51 2.7/92/0004/074~...$J .50 Let G’ be the simple undirected n-vertex graph obtained from G by connecting every two vertices i and j by an edge iff there is a path of length 1 or 2 between i and j in G. Note that the O-1 matrix B computed in the algorithm is the adjacency matrix of G’. G’ is the complete graph iff G has diameter at most 2, and in that csse dij = 2 if aij = O and dij = 1 if aij = 1, Thus the algorithm is correct for graphs of diameter at most 2. Let tijdenote the length of a shortest path joining i and j in G’. Claim 2 For any pair i, j of vertices, dij even implies d=$J 2tij , and dij odd implies dij = 2tij – 1. Proof: Observe that if for a pair i, j of vertices dij = 2s and i = io, il, ..., i2S-l,i2S = j is a shortest path in G, then i = io, iz, iq, ..., iz~-z, izs = j is a shortest path between i and j in G’ and has length s. Similarly, if dij = 2s–1 andi= io, il, . . . . i2~-3, i28-2, i2S-1 = j is a shortest path inG, then i = io, i2, iq, . . .,izS_.4,iz*_z,iz8_l = j is a shortest path between i and j in G1 and has length s. E!l

Rotator graphs: an efficient topology for point-to-point multiprocessor networks

Abstract: Rotator graphs, a set of directed permutation graphs, are proposed as an alternative to star and pancake graphs. Rotator graphs are defined in a way similar to the recently proposed Faber-Moore graphs. They have smaller diameter, n-1 in a graph with n factorial vertices, than either the star or pancake graphs or the k-ary n-cubes. A simple optimal routing algorithm is presented for rotator graphs. The n-rotator graphs are defined as a subset of all rotator graphs. The distribution of distances of vertices in the n-rotator graphs is presented, and the average distance between vertices is found. The n-rotator graphs are shown to be optimally fault tolerant and maximally one-step fault diagnosable. The n-rotator graphs are shown to be Hamiltonian, and an algorithm for finding a Hamiltonian circuit in the graphs is given. >

Recognizing P 4 -sparse graphs in linear time

Abstract: A graph G is $P_4 $-sparse if no set of five vertices in G induces more than one chordless path of length three $P_4 $-sparse graphs generalize both the class of cographs and the class of $P_4 $-reducible graphs One remarkable feature of $P_4 $-sparse graphs is that they admit a tree representation unique up to isomorphism It has been shown that this tree representation can be obtained in polynomial time This paper gives a linear time algorithm to recognize $P_4 $-sparse graphs and shows how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation in linear time

Finding Hamiltonian paths in cocomparability graphs using the bump number algorithm

Abstract: Hamiltonian Path/Cycle are well known NP-complete problems on general graphs, but their complexity status for permutation graphs has been an open question in algorithmic graph theory for many years. In this paper, we prove that theHamiltonian Path problem is solvable in polynomial time even for the larger class of cocomparability graphs. Our result is based on a nice relationship between Hamiltonian paths and the bump number of partial orders. As another consequence we get a new interpretation of the bump number in terms of path partitions, leading to polynomial time solutions of theHamiltonian Path/Cycle Completion problems in cocomparability graphs.

Shortest path problems with node failures

Abstract: Consider the problem of finding the shortest paths from a node source s to a node sink t in a complete network. On any given instance of the problem, only a subset of the intermediate nodes can be used to go from s to t, the subset being chosen according to a given probability law. We wish to find an a priori path from s to t such that, on any given instance of the problem, the sequence of nodes defining the path is preserved but only the permissible nodes are traversed, the others being skipped. The problem of finding an a priori path of minimum expected length is defined as the Probabilistic Shortest Path Problem (PSPP). Note that if the network is not originally complete, the PSPP methodology can still be used if we first add each missing edge, together with a deterministic length (being defined by an alternative path using nodes that have no probability of failure). In this paper, after discussing potential applications of the PSPP, we study the complexity of this class of problems. We first show that the problem is, in general, NP-hard and then we develop polynomial time procedures for special cases of it. We also consider the complexity of a related problem: the Probabilistic Minimum Spanning Tree Problem (PMSTP). Finally, we provide a discussion of the implications of the results.

Level graphs and approximate shortest path algorithms

Abstract: Shortest path algorithms for graphs have been widely studied and are of great practical utility. For the case of a general graph, Dijkstra's algorithm is known to be optimal. However, in many practical instances, there is a “level” structure which may be imposed on the underlying graph. Utilizing these levels, this paper demonstrates that the time complexity of shortest path generation may be greatly reduced. A new graph structure, the level graph, together with a simple uniformed heuristic, LGS, for searching that structure is introduced, which allows for rapid generation of approximate shortest paths. LGS is studied both analytically and via simulation. It is shown that the length of the path produced by LGS converges rapidly to that of the actual shortest path as the distance between the points increases.

Solving min‐max shortest‐path problems on a network

Abstract: In this article we consider the problem of determining a path between two nodes in a network that minimizes the maximum of r path length values associated with it. This problem has a direct application in scheduling. It also has indirect applications in a class of routing problems and when considering multiobjective shortest-path problems. We present a label-correcting procedure for this problem. We also develop two pruning techniques, which, when incorporated in the label-correcting algorithm, recognize and discard many paths that are not part of the optimal path. Our computational results indicate that these techniques are able to speed up the label-correcting procedure by many orders of magnitude for hard problem instances, thereby enabling them to be solved in a reasonable time. © 1992 John Wiley & Sons, Inc.

Computing a shortest k-link path in a polygon

Abstract: The authors consider the problem of finding a shortest polygonal path from s to t within a simple polygon P, subject to the restriction that the path have at most k links (edges). They give an algorithm to compute a k-link path with length at most (1 + epsilon ) times the length of a shortest k-link path, for any error tolerance epsilon >0. The algorithm runs in time O(n/sup 3/k/sup 3/ log (Hk/ epsilon /sup 1/k/)), where N is the largest integer coordinate among the n vertices of P. They also study the more general problem of approximating shortest k-link paths in polygons with holes. In this case, they give an algorithm that returns a path with at most 2k links and length at most that of a shortest k-link path; the running time is O(kE/sup 2/), where E is the number of edges in the visibility graph. Finally, they study the bicriteria path problem in which the two criteria are link length and 'total turn' (the integral of mod Delta theta mod along a path). They obtain in an exact polynomial-time algorithm for polygons with holes. >

The traveling salesman problem in graphs with some excluded minors

Abstract: Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.

An optimal algorithm to solve the all‐pair shortest path problem on interval graphs

Abstract: We present an O(n2) time-optimal algorithm for solving the unweighted all-pair shortest path problem on interval graphs, an important subclass of perfect graphs. An interesting structure called the neighborhood tree is studied and used in the algorithm. This tree is formed by identifying the successive neighborhoods of the vertex labeled last in the graph according to the IG-ordering.

A polynomial time algorithm for the computation of the iteration-period bound in recursive data flow graphs

Abstract: Rate-optimal scheduling of iterative data-flow graphs requires the computation of the iteration period bound. According to the formal definition, the total computational delay in each directed loop in the graph has to be calculated in order to determine that bound. As the number of loops cannot be expressed as a polynomial function of the number of modes in the graph, this definition cannot be the basis of an efficient algorithm. A polynomial-time algorithm for the computation of the iteration period bound based on longest path matrices and their multiplications is presented. >

Triangulating 3-colored graphs

Abstract: The problem of determining whether a vertex-colored graph can be triangulated without introducing edges between vertices of the same color is what is of interest here. This problem is known to be polynomially equivalent to a fundamental problem in numerical taxonomy called the perfect phylogeny problem, which is concerned with the inference of evolutionary history. This problem is also related to the problem of recognizing partial k-trees, a class of graphs that has received much attention recently. The problem in its general form is NP-complete and can be solved in $O( n^{k + 1} )$ time, where n is the number of vertices and k the number of colors. In this paper, a linear time algorithm for the case of 3-colored graphs is presented.

Reasoning MPE to multiply connected belief networks using message passing

Abstract: Finding the l Most Probable Explanations (MPE) of a given evidence, Se, in a Bayesian belief network is a process to identify and order a set of composite hypotheses, HiS, of which the posterior probabilities are the l largest; i.e., Pr(H1|Se) ≥ Pr(H2|Se) ≥ ... ≥ Pr(Hl|Se). A composite hypothesis is defined as an instantiation of all the non-evidence variables in the network. It could be shown that finding all the probable explanations is a NP-hard problem. Previously, only the first two best explanations (i.e., l = 2) in a singly connected Bayesian network could be efficiently derived without restrictions on network topologies and probability distributions. This paper presents an efficient algorithm for finding l (≥ 2) MPE in singly-connected networks and the extension of this algorithm for multiply-connected networks. This algorithm is based on a message passing scheme and has a time complexity O(lkn) for singly-connected networks; where l is the number of MPE to be derived, k the length of the longest path in a network, and n the maximum number of node states - defined as the product of the size of the conditional probability table of a node and the number of the incoming/outgoing arcs of the node.

On bends and lengths of rectilinear paths: a graph-theoretic approach

Abstract: We consider the problem of finding a rectilinear path between two designated points in the presence of rectilinear obstacles subject to various optimization functions in terms of the number of bends and the total length of the path. Specifically we are interested in finding a minimum bend shortest path, a shortest minimum bend path or a least-cost path where the cost is defined as a function of both the length and the number of bends of the path. We provide a unified approach by constructing a path-preserving graph. guaranteed to preserve all these three kinds of paths and give an O(K+e log e) algorithm to find them, where e is the total number of obstacle edges, and K is the number of intersections between tracks from extreme point and other tracks (defined in the text). K is bounded by O(et), where t is the number of extreme edges. In particular, if the obstacles are rectilinearly convex, then K is O(ne), where n is the number of obstacles. Extensions are made to find a shortest path with a bounded number of bends and a minimum-bend path with a bounded length. When a source point and obstacles are pre-given, queries for the assorted paths from the source to given points can be handled in O(log e+k) time after O(K+e log e) preprocessing, where k is the size of the goal path. The trans-dichotomous algorithm of Fredman and Willard8 and the running time for these problems are also discussed.

I/O-efficiency of shortest path algorithms: an analysis

Abstract: To establish the behavior of algorithms in a paging environment, the author analyzes the input/output (I/O) efficiency of several representative shortest path algorithms. These algorithms include single-course, multisource, and all pairs ones. The results are also applicable for other path problems such as longest paths, most reliable paths, and bill of materials. The author introduces the notation and a model of a paging environment. The I/O efficiencies of the selected single-source, all pairs, and multisource algorithms are analyzed and discussed. >

Shortest path queries in rectilinear worlds

Abstract: In this paper, a data structure is given for two and higher dimensional shortest path queries. For a set of n axis-parallel rectangles in the plane, or boxes in d-space, and a fixed target, it is possible with this structure to find a shortest rectilinear path avoiding all rectangles or boxes from any point to this target. Alternatively, it is possible to find the length of the path. The metric considered is a generalization of the L1-metric and the link metric, where the length of a path is its L1-length plus some (fixed) constant times the number of turns on the path. The data structure has size O((n log n)d−1), and a query takes O(logd−1 n) time (plus the output size if the path must be reported). As a byproduct, a relatively simple solution to the single shot problem is obtained; the shortest path between two given points can be computed in time O(ndlog n) for d≥3, and in time O(n2) in the plane.

On the parallel computation of the algebraic path problem

Abstract: The algebraic path problem is a general description of a class of problems, including some important graph problems such as transitive closure, all pairs shortest paths, minimum spanning tree, etc. In this work, the algebraic path problem is solved on a processor array with a reconfigurable bus system. The proposed algorithms are based on repeated matrix multiplications. The multiplication of two n*n matrices takes O(log n) time in the worst case, but, for some special cases, O(1) time is possible. It is shown that three instances of the algebraic path problem, transitive closure, all pairs shortest paths, and minimum spanning tree, can be solved in O(log n) time, which is as fast as on the CRCW PRAM. >

Efficient Algorithms for Solving Systems of Linear Equations and Path Problems

Abstract: Efficient algorithms are presented for solving systems of linear equations defined on and for solving path problems [11] for treewidth k graphs [20] and for α-near-planar graphs [22]. These algorithms include the following: 1. O(nk2) and O(n3/2) time algorithms for solving a system of linear equations and for solving the single source shortest path problem, 2. O(n2k) and O(n2log n) time algorithms for computing A−1 where A is an n×n matrix over a field or for computing A* where A is an n×n matrix over a closed semiring, and 3. O(n2k) and O(n2log n) time algorithms for the all pairs shortest path problems.

Analysis of free schedule in periodic graphs

Abstract: In this paper we address a scheduling problem for regular algorithms which can be described using aperiodic graph. The free schedule of those regular iterative algorithms is analyzed in a cone-like index space. Since the determination of a free schedule is closely related to the longest path problem, the structure of longest paths in periodic graphs must be determined. It will be shown that free schedules also have some kind of regularity. We also present algorithms for calculating this structure and give an estimate on the computational complexity of our algorithms. Finally, it is proven that this problem is NP–complete in the dimension of the index space.

Optimal greedy algorithms for indifference graphs

Abstract: Investigates the class of indifference graphs that models the notion of indifference relation arising in social sciences and management. The authors examine algorithmic properties of indifference graphs. Recently it has been shown that indifference graphs are characterized by a very special ordering on their sets of vertices. It is shown that this linear order can be exploited in a natural way to obtain optimal greedy algorithms for a number of computational problems on indifference graphs, including finding a shortest path between two vertices, computing a maximum matching, the center, and a Hamiltonian path. >

Finite State Machine Decomposition

Abstract: Finite state machine (FSM) decomposition is concerned with the implementation of a FSM as a set of smaller interacting submachines. Such an implementation is desirable for a number of reasons. A partitioned sequential circuit usually leads to improved performance as a result of a reduction in the longest path between latch inputs and outputs. This fact is particularly true when the individual submachines are implemented as Programmable Logic Arrays (PLAs). It appears that the primary interest in using decomposition tools in industry stems from a need to improve the performance of FSM controllers, which often dictates the required duration of the system clock. FSM decomposition can be applied directly when Programmable Gate Array (PGAs) or Programmable Logic Devices (PLDs) are the target technology. Such technologies are characterized by I/O or gate-limited blocks of logic and latches into which the circuit must be mapped. In many cases, it is desirable for reasons of clock-skew minimization or simplifying the layout to distribute the control logic for a data path in such a manner that the portions of the data path and control that interact closely are placed next to each other. FSM decomposition can also be used for this purpose. Partitioning of the logic implementing the FSM could result in simplified layout constraints resulting in smaller chip area. In PLA-based FSMs, decomposition has the effect of partitioning the PLA that implements the original FSM into smaller interacting PLAs that implement the individual submachines. In such situations, an area reduction can be attributed to PLA partitioning. Finally, it is not computationally feasible for current multilevel logic minimizers (e.g. MIS-II [10]) to search all possible area minimal solutions. In some cases, an initially-decomposed FSM could correspond to a superior starting point for multilevel logic minimization.