Timothy Hunter

Timothy Hunter, Ahmed El Alaoui, Alexandre Bayen

We present new algorithms for computing the log-determinant of symmetric, diagonally dominant matrices. Existing algorithms run with cubic complexity with respect to the size of the matrix in the worst case. Our algorithm computes an approximation of the log-determinant in time near-linear with respect to the number of non-zero entries and with high probability. This algorithm builds upon the utra-sparsifiers introduced by Spielman and Teng for Laplacian matrices and ultimately uses their refined versions introduced by Koutis, Miller and Peng in the context of solving linear systems. We also present simpler algorithms that compute upper and lower bounds and that may be of more immediate practical interest.

Timothy Hunter, Aude Hofleitner, Jack Reilly et al.

Most optimal routing problems focus on minimizing travel time or distance traveled. Oftentimes, a more useful objective is to maximize the probability of on-time arrival, which requires statistical distributions of travel times, rather than just mean values. We propose a method to estimate travel time distributions on large-scale road networks, using probe vehicle data collected from GPS. We present a framework that works with large input of data, and scales linearly with the size of the network. Leveraging the planar topology of the graph, the method computes efficiently the time correlations between neighboring streets. First, raw probe vehicle traces are compressed into pairs of travel times and number of stops for each traversed road segment using a `stop-and-go' algorithm developed for this work. The compressed data is then used as input for training a path travel time model, which couples a Markov model along with a Gaussian Markov random field. Finally, scalable inference algorithms are developed for obtaining path travel time distributions from the composite MM-GMRF model. We illustrate the accuracy and scalability of our model on a 505,000 road link network spanning the San Francisco Bay Area.