On Solving Floating Point SSSP Using an Integer Priority Queue

We address the single source shortest path planning problem (SSSP) in the case of floating point edge weights. We show how any integer based Dijkstra solution that relies on a monotone integer priority queue to create a full ordering over path lengths in order to solve integer SSSP can be used as an oracle to solve floating point SSSP with positive edge weights (floating point P-SSSP). Floating point P-SSSP is of particular interest to the robotics community. This immediately yields a handful of faster runtimes for floating point P-SSSP; for example, O(m+ n log log C δ ), where C is the largest weight and δ is the minimum edge weight in the graph. It also ensures that many future advances for integer SSSP will be transferable to floating point P-SSSP. Our work relies on a result known to [Dinic, 1978] and [Tsitsiklis, 1995]; that, under the right conditions, SSSP can be solved using a partial ordering of nodes, despite the fact that full orderings are typically used in practice. Thus, priority queues that do not extract keys in a monotonically nondecreasing order can be used to solve SSSP under a special set of conditions that always hold for floating point P-SSSP. In particular, any node that has a shortest-path-length (key value) within δ of the queue’s minimum key can be extracted (instead of the node with the minimum key) from the priority queue and Dijkstra’s algorithm will still run correctly. Our contribution is to show how any monotonic integer priority queue can be transformed into a suitable δ-nonmonotonic floating point priority queue in order to produce the necessary partial ordering for floating point values. Monotonic integer keys for floating point values are obtained by dividing the floating point key values by δ (as was done by [Dinic, 1978]) and then converting the result to an intege. The loss of precision this division causes is the reason the resulting floating point queue is δ-nonmonotonic instead of monotonic, but does not break the correctness of Dijkstra’s algorithm. We also prove that the floating point version of non-negative SSSP (which allows δ = 0 in addition to δ > 0) cannot be solved without creating a full ordering of nodes (and thus requires a fully monotonic priority heap); and so our method cannot be extended to work on non-negative floating point SSSP, in general.