Layered Heaps Beating Standard and Fibonacci Heaps in Practice

We consider the classic problem of designing heaps. Standard binary heaps run faster in practice than Fibonacci heaps but have worse time guarantees. Here we present a new type of heap that runs faster in practice than both standard binary and Fibonacci heaps, but has asymptotic insert times arbitrarily better than O(logn), namely O((logn)) for arbitrary positive integer m. Our heap is defined recursively and maximum run time speed up occurs when a recursion depth of 1 is used, i.e. a heap of heaps. 1 Layered Heaps We will define M -layered heaps for arbitrary integer M ≥ 1. For M = 1 the M -ary heap is a standard binary heap stored in an array. For M ≥ 2, a kM -ary heap is used, with kM = 2 kM−1∗(M−1)/M . Then by inductive hypothesis, insert operations on the children heaps will be (logn)(M−1/M)∗(1/(M−1)) = (logn) . Furthermore the height of the M ary heap will also be (logn) . So insert operations take O((logn) ) time for any M we want. Pop/delete functions take standard O(logn) time, because we may need to do a children heap operation which takes (logn)(M−1)/M time a total of (logn) times, i.e. the height of the M -ary layered heap. The operations and running times for them are explained in the following pseudocode: INSERT INTO M-ARY HEAP A: Before swapping, start by placing the element at end of array (position n = N). Then do the following: • Set k = 2 (M−1)/M • While n > 0: – If A[n] > A[n/k] then swap their values – Else insert A[n] into children (M − 1)-ary layered heap that contains position n in the array. (Recursive) Then BREAK. 1 ar X iv :1 51 0. 03 36 7v 1 [ cs .D S] 1 2 O ct 2 01 5