Dijkstra's algorithm is the standard method for computing shortest paths on arbitrary graphs. However, it is slow for large graphs, taking at least linear time. It has been long known that for real world road networks, creating a hierarchy of well-chosen shortcuts allows fast distance and path computation, with exact distance queries seemingly being answered in logarithmic time. However, these methods were but heuristics until the work of Abraham et al. [JACM 2016], where they defined a graph parameter called highway dimension which is constant for real-world road networks, and showed that in graphs of constant highway dimension, a shortcut hierarchy exists that guarantees shortest distance computation takes O(log(U+| V|)) time and O(V log(U +| V|)) space, where U is the ratio of the smallest to largest edge, and |V| is the number of vertices. The problem is that they were unable to efficiently compute the hierarchy of shortcuts. Here we present a simple and efficient algorithm to compute the needed hierarchy of shortcuts in time and space O(V log(U + |V|)), as well as supporting updates in time O(log(U + |V|)).

It is shown how to enhance any data structure in the pointer model to make it confluently persistent, with efficient query and update times and limited space overhead. Updates are performed in O(log n) amortized time, and following a pointer takes O(log c log n) time where c is the in-degree of a node in the data structure. In particular, this proves that confluent persistence can be achieved at a logarithmic cost in the bounded in-degree model used widely in previous work. This is a O(n/ log n)-factor improvement over the previous known transform to make a data structure confluently persistent.

A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G . The expected number of point-line comparisons performed by this data structure, when the queries are distributed according to D , is H˜ + O (H˜ 1/2 +1) where H˜=H˜( G,D ) is a lower bound on the expected number of point-line comparisons performed by any linear decision tree for point location in G under the query distribution D . The preprocessing algorithm runs in O ( n log n ) time and produces a data structure of size O ( n ). These results are obtained by creating a Steiner triangulation of G that has near-minimum entropy.

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life.