An optimal algorithm for the period of a strongly connected digraph (original) (raw)

An algorithm for minimum cycle basis of graphs

2008

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0, 1} incidence vector is associated with each cycle and the vector space over F 2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering.

Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms

Algorithmica, 2000

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be e ciently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O( (n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n ), for any constant 0 < < 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or nding a negative cycle in linear time.

Cycle bases in graphs characterization, algorithms, complexity, and applications

Computer Science Review, 2009

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and apriori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

Recognizing properties of periodic graphs

1991

A periodic (dynamic) graph is an infinite graph with a repetitive structure and a compact representation. A periodic graph is represented by a finite directed graph, called the dependence or the static graph, with ddimensional integer vector weights associated with its edges. For every vertex in the dependence graph there corresponds a d-dimensional lattice in the periodic graph. For every edge (u , u) in the dependence graph with vector weight a , there are infinitely many edges in the periodic graph, namely, from every point on the lattice corresponding to u to the point shifted by a on the lattice corresponding to u . Periodic graphs are used, for example, to model VLSI circuits and systems of uniform recurrence relations. In this paper we give algorithms to compute weakly connected components, to test bipartiteness, and to compute a minimum average cost spanning tree for d-dimensional periodic graphs.

More efficient periodic traversal in anonymous undirected graphs

Theoretical Computer Science, 2012

We consider the problem of periodic graph exploration in which a mobile entity with constant memory, an agent, has to visit all n nodes of an input simple, connected, undirected graph in a periodic manner. Graphs are assumed to be anonymous, that is, nodes are unlabeled. While visiting a node, the agent may distinguish between the edges incident to it; for each node v, the endpoints of the edges incident to v are uniquely identified by different integer labels called port numbers. We are interested in algorithms for assigning the port numbers together with traversal algorithms for agents using these port numbers to obtain short traversal periods. Periodic graph exploration is unsolvable if the port numbers are set arbitrarily, see [1]. However, surprisingly small periods can be achieved by carefully assigning the port numbers. Dobrev et al. [4] described an algorithm for assigning port numbers and an oblivious agent (i.e., an agent with no memory) using it, such that the agent explores any graph with n nodes within the period 10n. When the agent has access to a constant number of memory bits, the optimal length of the period was proved in [7] to be no more than 3.75n − 2 (using a different assignment of the port numbers and a different traversal algorithm). In this paper, we improve both these bounds. More precisely, we show how to achieve a period length of at most (4 + 1 3)n − 4 for oblivious agents and a period length of at most 3.5n − 2 for agents with constant memory. To obtain our results, we introduce a new, fast graph decomposition technique called a three-layer partition that may also be useful for solving other graph problems in the future. Finally, we present the first non-trivial lower bound, 2.8n − 2, on the period length for the oblivious case.

Some Problems on Dynamic/Periodic Graphs

Progress in Combinatorial Optimization, 1984

A dynamic graph is a (locally finite) infinite graph G -(V, E) in which the vertex set is V -{i1,...,n and p } , where Z is the set of integers, and the edge set has the following periodic property: (, f) is an edge of E if and only if (ip+, f+l) is an edge of E. Dynamic graphs may model a wide range of periodic combinatorial optimization problems in workforce scheduling, vehicle routing, and production scheduling.

Analysis of free schedule in periodic graphs

Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures - SPAA '92, 1992

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.

A strong flow-based formulation for the shortest path problem in digraphs with negative cycles

International Transactions in Operational Research, 2009

In this paper, we are interested in the shortest path problem between two specified vertices in digraphs containing negative cycles. We study two integer linear formulations and their linear relaxations. A first formulation, close in spirit to a classical formulation of the traveling salesman problem, requires an exponential number of constraints. We study a second formulation that requires a polynomial number of constraints and, as confirmed by computational experiments, its linear relaxation is significantly sharper. From the second formulation we propose a family of linear relaxations with fewer variables than the classical linear one.

Polynomial algorithms for finding paths and cycles in quasi-transitive digraphs

2003

A digraph D is called quasi-transitive if for any triple x, y, z of distinct vertices of D such that (x, y) and (y, z) are arcs of D there is at least one arc from x to z or from z to x. A minimum path factor of a digraph D is a collection of the minimum number of pairwise vertex disjoint paths covering the vertices of D. J. Bang-Jensen and J. Huang conjectured that there exist polynomial algorithms for the Hamiltonian path and cycle problems for quasi-transitive digraphs. We solve this conjecture by describing polynomial algorithms for finding a minimum path factor and a Hamiltonian cycle (if it exists) in a quasi-transitive digraph.