Question

We have an n-node binary heap which contains n distinct items (smallest item at the root). For a k<=n, find a O(klogk) time algorithm to select kth smallest element from the heap.

O(klogn) is obvious, but couldn’t figure out a O(klogk) one. Maybe we can use a second heap, not sure.

Answer 1

Well, your intuition was right that we need extra data structure to achieve O(klogk) because if we simply perform operations on the original heap, the term logn will remain in the resulting complexity.

Guessing from the targeted complexity O(klogk), I feel like creating and maintaining a heap of size k to help me achieve the goal. As you may be aware, building a heap of size k in top-down fashion takes O(klogk), which really reminds me of our goal.

The following is my try (not necessarily elegant or efficient) in an attempt to attain O(klogk):

1. We create a new min heap, initializing its root to be the root of the original heap.

2. We update the new min heap by deleting the current root and inserting the two children of the current root in the original heap. We repeat this process k times.

3. The resulting heap will consist of k nodes, the root of which is the kth smallest element in the original heap.

Notes: Nodes in the new heap should store indexes of their corresponding nodes in the original heap, rather than the node values themselves. In each iteration of step 2, we really add a net of one more node into the new heap (one deleted, two inserted), k iterations of which will result in our new heap of size k. During the ith iteration, the node to be deleted is the ith smallest element in the original heap.

Time Complexity: in each iteration, it takes O(3logk) time to delete one element from and insert two into the new heap. After k iterations, it is O(3klogk) = O(klogk).

Hope this solution inspires you a bit.