Near-optimal working-set heaps and dijkstra on pointer machines

A heap is a dynamic data structure that stores a set of labeled values under the following operations: pop returns the minimum value of the heap, Push(x_i) pushes a new value x_i onto the heap, and DecreaseKey(i, v) decreases the value x_i to v. A working-set heap is a heap that supports the x_i ← pop() operation in O(log Γ(x_i)) time where Γ(x_i) is the size of the working set: the number of elements that were pushed onto the heap while x_i was in the heap. The goal of working set heap design is to maintain the working set property while minimizing the overhead of the Push and DecreaseKey operations. On a word RAM, there exist working set heaps that support Push and DecreaseKey in amortized constant time. In this paper, we show via a simple construction that pointer machines, one of the most general and least-assuming computational models, support working set heaps that support Push in amortized constant time and DecreaseKey in inverse-Ackermann time. A by-product of this analysis is that Dijkstra’s shortest path algorithm can be near-universally optimal on a pointer machine - incurring only an additive O(m α(m)) overhead compared to the optimal running time for distance ordering, where m denotes the number of edges in the graph.