### lower bound on finding the k-smallest elements in sorted order

Sun 01 January 2012

Any comparison based algorithm solves the problem of finding the k-smallest elements in sorted order in $$\Omega(k\lg{n})$$ time for an n-length array.

To find a lower bound on the running time of any algorithm solving the above problem using comparisons we use the decision tree model.

What are the possible solutions an algorithm solving this problem produces? For an input array of length n, any subset of length k is a possible solution. But because the algorithm needs to produce the result in sorted order, we have to take into account every possible permutation of those k-length solutions.

So in total there are $$\binom{n}{k} k!=n(n-1)\cdot \ldots \cdot (n-k+1)$$ possible solutions.

Suppose our decision tree has l leaves, and its height is h. Every possible solution is a leaf in the tree, so $$n(n-1)\cdot \ldots \cdot (n-k+1) \leq l$$. Also, a binary tree of height h, has at most $$2^h$$ leaves.

We have the following bounds for the number of leaves l:

$$(n-k+1)^k < n(n-1)\cdot \ldots \cdot (n-k+1) \leq l \leq 2^h$$

and we're looking to bound h from below.

Taking $$\lg$$ from both sides gives us $$\lg{(n-k+1)^k} = k\lg{(n-k+1)} < h$$.

For $$k \leq n/2$$ the above inequality gives us $$h > k\lg{n}$$, so the height of the tree is $$\Omega(k\lg{n})$$.

If $$k > n/2$$, we can think of the problem as finding the n-k largest elements and sorting the rest of the elements (this works because the "rest" are in fact the smallest).

We already showed that we can solve the problem if $$k \leq n/2$$ for the smallest, but it doesn't really matter whether it's the smallest or largest. So finding the $$n-k < n/2$$ largest elements will take $$\Omega((n-k)\lg{n})$$, and sorting the k > n/2 remaining elements will take $$\Omega(k\lg{k})=\Omega(k\lg{n/2})=\Omega(k\lg{n})$$.

Overall it takes $$\Omega((n-k)\lg{n})+\Omega(k\lg{n})=\Omega(k\lg{n})$$, since $$n-k < n/2 < k$$, and we're done.