  We introduce a new heuristic for the A* algorithm that references a data
structure significantly smaller than that of ALT. 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.
