Grid graphs are subgraphs of the infinite 2-dimensional integer grid. The Hamiltonian path problem for general grid graphs is a well-known NP-complete problem. In this paper, we present necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in L−shaped grid graphs. We also show that a Hamiltonian path between two given vertices of a L−shaped grid ...

Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the rst known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For n-vertex graphs, our algorithm can use as little as n=2 ( p logn) space while still running in polynomial time.

A neural network solution is proposed for solving path planning problems faced by mobile robots. The proposed network is a two-dimensional sheet of neurons forming a distributed representation of the robot's workspace. Lateral interconnections between neurons are "cooperative", so that the network exhibits oscillatory behaviour. These oscillations are used to generate solutions of Bellman's dyn...

In [Adl94] Adleman used biological manipulations with DNA strings to solve some instances of the Directed Hamiltonian Path Problem. Lipton [Lip94] showed how to extend this idea to solve any NP problem. We prove that exactly the problems in PNP = p2 can be solved in polynomial time using Lipton’s model. Various modifications of Lipton’s model, based on other DNA manipulations, are investigated ...

Three skeletal rearrangement channels for the norbornadiene (N*+) to the 1,3,5-cycloheptatriene (CHT*+) radical cation conversion, initialized by opening a bridgehead-methylene bond in N*+, are investigated using the quantum chemical B3LYP, MP2 and CCSD(T) methods in conjunction with the 6-311 +G(d,p) basis set. Two of the isomerizations proceed through the norcaradiene radical cation (NCD*+), ...

This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed and used to show that the quantum mechanical version of the classical transformation does not leave the measure of the path integral invariant, inst...

Let G(V,E) be a graph with vertex set V and edge set E, and let X be either V or E (not both). Let Π be a search problem with input X and solution Y , where Y ⊆ X. Let Y ′ denote a sub-solution of Π. That is, Y ′ is a solution of Π when the input is X ′, the vertex set or edge set of some minor of G. Consider the set system (X, I), where I denotes the family of all sub-solutions of Π. We prove ...

Appendix Proof of Lemma 1: The network topology implies, that any link emanating from s belongs to a single s-t path. The proof is by interchange argument: Since any path converging network is an sp graph, we use Theorem 1 to identify an optimal strategy with a unique order of trials on the network links. Assume that M (G) = (m 1 ; : : :; m n) is a complete order on paths for an optimal strateg...