Using Quadrilaterals to Compute the Shortest Path

We introduce a new heuristic for the A* algorithm that references a data structure of size θ(|L| + |V |), where L represents a set of strategically chosen landmark vertices and V the set of vertices in the graph. This heuristic’s benefits are permitted by a new approach for computing landmark-based lower bounds, in which each landmark stores only the distances between it and vertices within its graph partition. During search queries, a geometric inequality based on distance information for multiple landmarks is used to establish a lower bound for the search. In comparison to previous landmark-based algorithms, this process significantly reduces the amount of preprocessed distance information that needs to be stored (typically θ(|L| · |V |)) while also granting a constant computational cost for each vertex visited during a shortest path query. Further, for graphs with non-overlapping partitions, preprocessing this data structure requires time complexity equivalent to one Dijkstra’s shortest path tree computation on the graph. We characterize the behavior of this new heuristic based on a dual landmark configuration that leverages quadrilateral inequalities to identify the lower bound for shortest path. Using this approach, we demonstrate both the utility and detriments of using polygon inequalities aside from the triangle inequality to establish lower bounds for shortest path queries. While this new heuristic does not dominate previous heuristics based on triangle inequalities, the inverse is true, as well. Further, we demonstrate that an A* heuristic function does not necessarily outperform another heuristic that it dominates. In comparison to other landmark methods, the new heuristic maintains a larger average search space while commonly decreasing the number of computed arithmetic operations. The new heuristic can significantly outperform previous methods, particularly in graphs with larger path lengths. The characterization of the use of these inequalities for bounding offers insight into its applications in other theoretical spaces. ∗This research was conducted at Nova Southeastern University Graduate School of Computer and Information Sciences in partial fulfillment for the requirements for the degree of the Doctor of Philosophy (Ph.D.) in Computer Science. Tuition assistance for the program was provided by the student’s employer, Raytheon BBN Technologies. 1 June 2015 ar X iv :1 60 3. 00 96 3v 1 [ cs .D S] 3 M ar 2 01 6