Classification bandits are multi-armed bandit problems whose task is to classify a given set of arms into either positive or negative class depending on whether the rate of the arms with the expected reward of at least h is not less than w for given thresholds h and w. We study a special classification bandit problem in which arms correspond to points x in d-dimensional real space with expected rewards f(x) which are generated according to a Gaussian process prior. We develop a framework algorithm for the problem using various arm selection policies and propose policies called FCB (Farthest Confidence Bound) and FTSV (Farthest Thompson Sampling Value). We show a smaller sample complexity upper bound for FCB than that for the existing algorithm of the level set estimation, in which whether f(x) is at least h or not must be decided for every arm’s x. Arm selection policies depending on an estimated rate of arms with mean rewards of at least h are also proposed and shown to improve empirical sample complexity. According to our experimental results, the rate-estimation versions of FCB and FTSV, together with that of the popular active learning policy which selects the point with the maximum variance, outperform other policies for synthetic functions, and the rate-estimation version of FTSV is also the best performer for our real-world dataset.

Consider a set P of points in the plane sorted by the x-coordinate. A point p in P is said to be a proximate point if there exists a point q on the x-axis such that p is the closest point to q over all points in P. The proximate point problem is to determine all the proximate points in P. Our main contribution is to propose optimal parallel algorithms for solving instances of size n of the proximate points problem. We begin by developing a work-time optimal algorithm running in O(log log n) time and using n/loglogn Common-CRCW processors. We then go on to show that this algorithm can be implemented to run in O(log n) time using n/logn EREW processors. In addition to being work-time optimal, our EREW algorithm turns out to also be time-optimal. Our second main contribution is to show that the proximate points problem finds interesting, and quite unexpected, applications to digital geometry and image processing. As a first application, we present a work-time optimal parallel algorithm for finding the convex hull of a set of n points in the plane sorted by x-coordinate; this algorithm runs in O(log log n) time using n/logn Common-CRCW processors. We then show that this algorithm can be implemented to run in O(log n) time using n/logn EREW processors. Next, we show that the proximate points algorithms afford us work-time optimal (resp, time-optimal) parallel algorithms for various fundamental digital geometry and image processing problems.

The problem of computing the convex hull of a set of n sorted points in the plane is one of the fundamental tasks in image processing, pattern recognition, cellular network design, and robotics, among many others. Somewhat surprisingly, in spite of a great deal of effort, the best previously known algorithm to solve this problem on a reconfigurable mesh of size /spl radic/n/spl times//spl radic/n was running in O(log2 n) time. It was open for more than ten years to obtain an algorithm for this important problem running in sublogarithmic time. Our main contribution is to provide the first breakthrough: we propose an almost optimal convex hull algorithm running in O((log log n)/sup 2/) time on a reconfigurable mesh of size /spl radic/n/spl times//spl radic/n. With slight modifications, this algorithm can be implemented to run in O((log log n)/sup 2/) time on a reconfigurable mesh of size /spl radic/n/loglogn/spl times//spl radic/n/loglogn. Clearly, the latter algorithm is work-optimal. We also show that any algorithm that computes the convex hull of a set of n sorted points on an n-processor reconfigurable mesh must take /spl Omega/(log log n) time. Our result opens the door to an entire slew of efficient convex-hull-based algorithms on reconfigurable meshes.

For 2/spl les/k/spl les/n, the k-merge problem is to merge a collection of ksorted sequences of total length n into a new sorted sequence. The k-merge problem is fundamental as it provides a common generalization of both merging and sorting. The main contribution of this work is to give simple and intuitive work-time optimal algorithms for the k-merge problem on three PRAM models, thus settling the status of the k-merge problem. We first prove that /spl Omega/(n log k) work is required to solve the k-merge problem on the PRAM models. We then show that the EREW-PRAM and both the CREW-PRAM and the CRCW require /spl Omega/(log n) time and /spl Omega/(log log n+log k) time, respectively, provided that the amount of work is bounded by O(n log k). Our first k-merge algorithm runs in /spl Theta/(log n) time and performs /spl Theta/(n log k) work on the EREW-PRAM. Finally, we design a work-time optimal CREW-PRAM k-merge algorithm that runs in /spl Theta/(log log n+log k) time and performs /spl Theta/(n log k) work. This latter algorithm is also work-time optimal on the CREW-PRAM model. Our algorithms completely settle the status of the k-merge problem on the three main PRAM models.