We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound...

There is a close relationship between harmonic functions { which have recently been proposed for path planning { and hitting probabilities for random processes. The hitting probabilities for random walks can be cast as a Dirichlet problem for harmonic functions, in much the same way as in path planning. This equivalence has implications both for uncertainty in motion planning and in the applica...

In this paper we consider the following inverse problem for the first hitting time distribution: given a Wiener process with a random initial state, probability distribution, F (t), and a linear boundary, b(t) = μt, find a distribution of the initial state such that the distribution of the first hitting time is F (t). This problem has important applications in credit risk modeling where the pro...

We develop a systematic matrix-analytic approach, based on intertwinings of Markov semigroups, for proving theorems about hitting-time distributions for finite-state Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding sample-path-by-sample-path stochastic constructions. We employ our approach to give new proofs and constructions for two th...

We examine the relationship between PageRank and several invariants occurring in the study of random walks and electrical networks. We consider a generalized version of hitting time and effective resistance with an additional parameter which controls the ‘speed’ of diffusion. We will establish their connection with PageRank . Through these connections, a combinatorial interpretation of PageRank...

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states x and moving large sets (As)s, of the hitting time of (As)s starting from x. We prove that in the case of the d-dimensional torus the maximum hitting time of moving targets is equal to the maximum hitting time of stationary targets. Neve...

Previous work has shown the effectiveness of random walk hitting times as a measure of dissimilarity in a variety of graph-based learning problems such as collaborative filtering, query suggestion or finding paraphrases. However, application of hitting times has been limited to small datasets because of computational restrictions. This paper develops a new approximation algorithm with which hit...

In this paper an investigation is carried out to classify the steady state responses of asymmetric piecewise linear vibration isolators as double hitting, single hitting, and no hitting. In each class, the analysis has been carried out using a set of coupled nonlinear algebraic equations following Natsiavas and Gonzalez [1]. Applying perturbation technique, a closed form analytic expression of ...

Consider the random walk on graphs such that, at each step, the next visited vertex is a neighbor of the current vertex, chosen with probability proportional to the inverse of the square root of its degree. On one hand, for every graph with n vertices, the maximal mean hitting time for this degree-biased random walk is asymptotically dominated by n. On the other hand, the maximal mean hitting t...