find the th smallest

3-8. Design a data structure to support the following operations:

• insert(x,T) – Insert item x into the set T .

• delete(k,T) – Delete the kth smallest element from T.

• member(x,T) – Return true iff x ∈ T .

All operations must take O(\log n) time on an n-element set.

Solution: We use a basic binary search tree for this and add two counters which count the number of children nodes.





Here you can see how the counters in pink. Let’s look at 4. The pink 2 indicates that there are 2 nodes on the left, i.e.
smaller than 4. And the pink 3 indicates that there are 3 nodes on the right, i.e. bigger than 4. What happens if we insert an item?

We do the standard insert traversal of a binary search tree with the difference that in each node we add a one to our counter because there will be one more node if we added our new item.





You can see what happened when we added 1. Besides adding the node we increased the left counter of 2 by one and the left counter by 4 by 1 because we traveled this way. So there’s no problem on inserting new nodes with

complexity.

The next method is member which is basically search in a binary search tree which also works with a complexity of


The last one is delete(k, T) which removes the kth smallest item. Finally we can use our counters. The first counter indicates if our kth smallest item is on the left, the item or on the right.

Example 1: We want the 3th smallest item. We start at 4 and see that there are 3 items on the left, i.e. the 3th smallest item in its left children. Next we are at 2 and see that there are 1 on the left and 1 on the right, therefore we have to go right (1 left
+ item itself + 1 right item = 3). Now we have arrived 3 and there are no other child items therefore 3 is our 3th smallest item.

Example 2: We want the 4th smallest item. We start at 4 and see that there are 3 items to the left. Therefore the 4th smallest item is 4 itself.