We consider an optimal-reachability problem for a timed automaton with respect to a linear cost function which results in a weighted timed automaton. Our solution to this optimization problem consists of reducing it to a (parametric) shortest-path problem for a finite directed graph. The directed graph we construct is a refinement of the region automaton due to Alur and Dill. We present an expo...

A trajectory is a directed path that defines a link between two spatial locations. That path may be as simple as the Euclidean shortest distance between a start and end point, or may involve a more complex traversal through time and space to travel from start to end. Within GI analysis, trajectories are used to represent phenomena such as movement of people as migration and commuting, goods and...

An asteroidal triple in a graph G is a set of three nonadjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. Special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triples play a central role in a characterization of directed path ...

The new Directed Reverse Path Join (DRPJ) protocol efficiently implements a Greedy routing algorithm for generating a multicast tree. The DRPJ protocol minimizes the messaging overhead from probe messages and allows a joining node to find multiple paths that are not constrained to be only the shortest paths. This enables a controllable tradeoff between path length and bandwidth consumption. Usi...

When the directed graph of an n–by–n matrix A does not contain a Hamilton cycle, we exhibit a formula for detA in terms of sums of products of proper principal minors of A. The set of minors involved depends upon the zero/nonzero pattern of A.

Significant cycle time saving can be achieved in 2.5-D milling by intelligently selecting tool sequences. The problem of finding the optimal tool sequence was formulated as finding the shortest path in a single-source single-sink directed acyclic graph. The nodes in the graph represented the state of the stock after the tool named in the node was done machining and the edges represented the cos...

The symmetric group is generated by σ = (1 2 ··· n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n. Dedicated to Herb Wilf (1931 – 2012).

Let P ⊂ R2 be a planar n-point set such that each point p ∈ P has an associated radius rp > 0. The transmission graph G for P is the directed graph with vertex set P such that for any p, q ∈ P , there is an edge from p to q if and only if d(p, q) ≤ rp. Let t > 1 be a constant. A t-spanner for G is a subgraph H ⊆ G with vertex set P so that for any two vertices p, q ∈ P , we have dH(p, q) ≤ tdG(...

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k-1) trees (each of which contains a previous best path). An O( ( log ) m K n K   ) time and O(K+m) ...

Let G = (V,E) be a directed graph with two special vertices: a source s and a sink t. Every edge (u, v) has a capacity cap(u, v) ≥ 0. For convenience, we define cap(u, v) = 0 if (u, v) in not an edge. More specifically, we work with a modified set of edges {(u, v) : (u, v) ∈ E or (v, u) ∈ E} with cap(u, v) = 0 if (u, v) 6∈ E. A flow is a real-valued function on edges with three properties: • Sk...