Finding an appropriate data structure

Question

Hello,

I have N keys.

I need to find a data structure which i can do with the following operations :

1. building it in O(N)

2. finding min in O(1)

3. deleting the median in O(logn)

4. finding the n/2+7-th biggest number

I thought about using a minimum heap (building is O(n),minimum is O(1) - root).

however, I'm having hard time finding a way to do 3 and 4.

I think the median suppose to be on of the leaves, but that's as far as i reached.

Answer 1

When you say building in O(n), do you mean that addition has to be O(n), or that you have to build a collection of elements in O(n) such that addition has to be O(1)?

You could augment pretty much any data structure with an extra reference to retrieve the minimal element in constant time.

For #3, it sounds like you need to be able to find the median in O(lg n) and delete in O(1), or vice versa.

For #4, you didn't specify the time complexity.

To other posters - this is marked as homework. Please give hints rather than posting the answer.

Answer 2

A popular question asked in Data Structures 1 exams/hws/tutorials. I'll try to give you some hints, if they don't suffice, comment, and I'll give you more hints.

1. Remember that you don't have to use just one data structure, you can use several data structures.
2. Recall the definition of a median: n/2 of the numbers are larger, and n/2 of the numbers are smaller
3. What data structures do you know that are built in O(n), and complex operations on them are O(logn) or less? - Reread the tutorials slides on these data structures.
4. It might be easier for you to solve 1+3 seperately from 1+2, and then think about merging them.

Answer 3

Simple sorted Array would solve the problem for #2 #3 and #4. But the construction of it would take O(n*n*). However, there are no restrictions put on space complexity. I am thinking hard to use Hashing concept during the construction of the data structure which would bring down the order to O(n).

Hope this helps. Will get back if I find a better solution

Answer 4

Here is one more hint.

Suppose I said you need to find the 7th smallest element in a min-heap, can you do it?

Since it has been long enough, for closure, I will add the solution I was thinking of:

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You can find the median in O(N) time. Now partition the array so that Left array contains numbers <= M and right array contains numbers > M.

Keep an extra element outside the array to track the minimum.

Now build two heaps, a Max Heap on the left array and a Min Heap on the right array.

This data structure should be enough to provide deleteMedian in O(log n) time, find min in O(1) time and find n/2+7th in O(1) time.