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Is there any approach to estimating the insertion depth of a node in a d-heap that is able to beat (node_value / heap_max) * h, where h is the heap height, and heap_max is normalized to the heap minimum?
In this particular case, it is feasible to maintain extra/historical data to support this heuristic, given that the maintenance of which is O(1) in time.
According to Inserting a new element into a heap [Doberkat, 1981] binary heaps with a uniform key distribution see an average of 1.6 UpHeap operations per insertion. The equivalent number should be fairly close for a d-heap.