Approximation algorithms for the traveling repairman and speeding deliveryman problems with unit-time windows

Constant-factor, polynomial-time approximation algorithms are presented for two variations of the traveling salesman problem with time windows. In the first variation, the traveling repairman problem, the goal is to find a tour that visits the maximum possible number of locations during their time windows. In the second variation, the speeding deliveryman problem, the goal is to find a tour that uses the minimum possible speedup to visit all locations during their time windows. For both variations, the time windows are of unit length, and the distance metric is based on a weighted, undirected graph. Algorithms with improved approximation ratios are given for the case when the input is defined on a tree rather than a general graph. The algorithms are also extended to handle time windows whose lengths fall in any bounded range.

Improved Bounds for Finger Search on a RAM

Abstract. We present a new finger search tree with O(1) worst-case update time and O(log log d) expected search time with high probability in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. For the need of the analysis we model the updates by a “balls and bins ” combinatorial game that is interesting in its own right as it involves insertions and deletions of balls according to an unknown distribution.