Algorithms on merging two heaps

Question

As I know, there exists a binomial heap or a so called mergeable heap, which is used to merge two heaps. My question is, instead of merging these heaps into one heap dynamically, if I copy these two heaps into one big array and then perform a heap building procedure, would that be a good approach or not?

Because I don't know how to create one heap using two heaps using by just heap operation. Please tell me if it is not a good way, or if you can, please give me some link, where a binomial heap with merge operation is implemented.

score 1 · Answer 1 · answered Mar 17 '12 at 20:28

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If you think about it, creating one heap by throwing away all the info embedded in the ordering of the other heaps can't possibly be optimal. Worst case, you should add all the items in heap 2 to heap 1, and that will be just half the work of creating a brand new heap from scratch.

But in fact, you can do way better than that. Merging two well-formed heaps involves little more than finding the insertion point for one of the roots in the other heap's tree, and inserting it at that point. No further work is necessary, and you've done no more than ln N work! See here for the detailed algorithm.

answered Mar 17 '12 at 20:28

Ernest Friedman-Hill

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yes but because i has never written merging two heap together,it is a problem for me to implement it – Mar 17 '12 at 20:32
There's very clear pseudocode on the Wikipedia page -- try it! – Ernest Friedman-Hill Mar 17 '12 at 20:42
pseudo code is clear,but how can i translate it into c++?i understood it by pseudo code only – Mar 17 '12 at 20:56
Obviously nobody can write the actual code for you, as we don't have your heap implementation to look at. – Ernest Friedman-Hill Mar 17 '12 at 20:58
i can create two heap using standard procedures,and could you help me in merging? – Mar 17 '12 at 21:02

amit · Answer 2 · 2012-03-17T20:42:39.870

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It will solve the problem, and it will give you a correct heap - but it will not be efficient.

Creating a [binary] heap of n elements from scratch is O(n), while merging 2 existing binomial heaps is O(logn).

edited Mar 17 '12 at 20:42

answered Mar 17 '12 at 20:28

amit

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Unless you need to copy the data (e.g. because you use an array storage), at which point you cannot go below `O(n)` operations (only less comparisons). – Has QUIT--Anony-Mousse Sep 21 '12 at 10:42

dfeuer · Answer 3 · 2012-07-14T03:38:20.450

Brodal queues and Brodal-Okasaki queues (bootstrapped skew binomial heaps) give the best worst-case asymptotic bounds for mergeable heaps, supporting O(1) insert, merge, and findMin, and O(log n) deleteMin. Brodal queues are ephemeral, and support efficient delete and decreaseKey. Brodal-Okasaki queues are confluently persistent (in fact purely functional), but don't support delete or decreaseKey. Unfortunately, Brodal and Okasaki say both these implementations are inefficient in practice, and Brodal considers his queues too complicated to be practical in any case.

Fibonacci heaps give similar amortized (but not worst-case) bounds, and are likely more efficient and practical in an amortized context. Pairing heaps are another good option: according to Wikipedia, their exact bounds are unknown, but they perform very well in practice.

score 0 · Answer 4 · edited Mar 24 '12 at 18:33

The process of merging 2 binomial heaps is pretty much similar to the merge operation in merge sort. If not knowing the merge - heap procedure is the problem, following steps might help.

repeat steps 1 through 4 until one of the heaps is empty

If the heads (which are binomial trees) of the 2 heaps are of same degree then you assign the head of the heap with greater key as the child of the child of head of heap with smaller key. Consequentely the degree of the head of the latter heap will be increased by 1 and make the head of the former heap the next element of its current head and go to step 2 else if they are of different degree, then go to step 4
If the head and the next binomial tree in the latter heap in step 1 are of same degree, then go to step 3 else go to step 1
Combine the head and its next element in the heap, in the same manner as you did in step 1 and assign the new combined Binomial tree as head and go to step 2.
See which of the 2 heaps have head with lower degree. Assign head of this heap as the head of other heap and delete it from the heap where it was initially present

Algorithms on merging two heaps

4 Answers4