In the previous chapter, we saw algorithms to find the shortest path from a source vertex s to a target vertex t in a directed graph. As it turns out, the best algorithms for this problem actually find the shortest path from s to every possible target (or from every possible source to t) by constructing a shortest path tree. The shortest path tree specifies two pieces of information for each no...

We show that, for a stationary version of Hammersley's process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly NorthEast path L(t, t) from (0, 0) to (t, t) is equal to 2E(t − X(t)) + , where X(t) is the location of a second class particle at time t. This implies that both E(t − X(t)) + and the variance of L(t,...

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. We introduce a generalization of the shortest descending path problem, called the shortest gently descending path (SGDP) problem, where a path descends, but not too steeply. The additional constraint to disallow a very steep descent makes the paths more...

Introduction. When a fiber-reinforced viscoelastic material is cooled from cure temperature to room temperature the different coefficients of thermal expansion of the matrix and fibers lead to residual stresses. To study this phenomenon in detail, we consider a thin, isotropic, thermorheologically simple, linearly viscoelastic plate reinforced by a random system of fibers lying in the plane of ...

We address the following problem: Given a simple polygon $P$ with $n$ vertices and two points $s$ and $t$ inside it, find a minimum link path between them such that a given target point $q$ is visible from at least one point on the path. The method is based on partitioning a portion of $P$ into a number of faces of equal link distance from a source point. This partitioning is essentially a shor...

The external path length of a tree T is the sum of the lengths of the paths from the root to each external node. The maximal path length diierence, , is the diierence between the length of the longest and shortest such path. We prove tight lower and upper bounds on the external path length of binary trees with N external nodes and prescribed maximal path length diierence. In particular, we give...

In this paper,we study a variant of thepath cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a k fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices ofT are all endpoints of the paths inP . The k -fixed-endpoint path cover problem is to f...

Consider a tree T on n nodes, each having a weight drawn from [1..σ]. In this paper, we design succinct data structures to encode T using nH(WT ) + o(n lg σ) bits of space, such that we can support path counting queries in O( lg σ lg lgn + 1) time, path reporting queries in O((occ+1)( lg σ lg lgn +1)) time, and path median and path selection queries in O( lg σ lg lg σ ) time, where H(WT ) is th...

This paper gives approximation algorithms for solving the following motion planning problem: Given a set of polyhedral obstacles and points s and t, find a shortest path from s to t that avoids the obstacles. The paths found by the algorithms are piecewise linear, and the length of a path is the sum of the lengths of the line segments making up the path. Approximation algorithms will be given f...

A new path-density diagnostic for atmospheric surface-to-surface transport is formulated. The path density h gives the joint probability that air whose last surface contact occurred on patch Vi at time ti will make its next surface contact with patch Vf after a residence time t 2 (t, t 1 dt) and that it can be found in dr during its surface-to-surface journey. The dependence on t allows the ave...