External Memory Soft Heap, and Hard Heap, a Meldable Priority Queue

Abstract

An external memory version of soft heap that we call “External Memory Soft Heap” (EMSH) is presented. It supports Insert, Findmin, Deletemin and Meld operations and as in soft heap, it guarantees that the number of corrupt elements in it is never more than εN, where N is the total number of items inserted in it, and ε is a parameter of it called the error-rate. For $N = O(B m^{M/2(B + \sqrt{m})})$, the amortised I/O complexity of an Insert is $O(\frac{1}{B} \log_{m}\frac{1}{\epsilon})$, where M is the size of the main memory, B is the size of a disk block and m = M/B. Findmin, Deletemin and Meld all have non-positive amortised I/O complexities.

When we choose an error rate ε < 1/N, EMSH stays devoid of corrupt nodes, and thus becomes a meldable priority queue that we call “hard heap”. The amortised I/O complexity of an Insert, in this case, is $O(\frac{1}{B} \log_{m}\frac{N}{B})$, over a sequence of operations involving N Inserts. Findmin, Deletemin and Meld all have non-positive amortised I/O complexities. If the inserted keys are all unique, a Delete (by key) operation can also be performed at an amortised I/O complexity of $O(\frac{1}{B} \log_{m}\frac{N}{B})$. A balancing operation performed at appropriate intervals on a hard heap ensures that the number of I/Os performed by a sequence of S operations on it is $O(\frac{S}{B}+\frac{1}{B}\sum_{i = 1}^{S} \log_{m}\frac{N_i}{B})$, where N _i is the number of elements in the heap before the ith operation.