I need some help understanding how to calculate the lower bound on the time complexity of merging $m$ sorted arrays of length $n$.

The bound should be $nm \lg(m)$. I need to prove this using a decision tree.

I tried counting the number of possible permutations (which would be the number of leaves in the tree) but I got stuck. I tried the following:

When merging the 2nd array with the first, we have $\binom{2n}{n}$ possibilities to place the elements. When merging the 3rd, we have $\binom{3n}{n}$ and so on, so the total number of permutations is:

$$\binom{2n}{n} \binom{3n}{n} \dots \binom{mn}{n}$$

The height of the tree is denoted $h$.

$$h > \lg{\binom{2n}{n} \binom{3n}{n} \dots \binom{mn}{n}}$$

And from here I don't know how to prove the complexity.

My question is - did I count the permutations correctly? Is there a more simple bound? How to continue from here?

Also, ould this problem possibly be equivalent to the problem of sorting an $m$-sorted array of size $nm$? Because in this problem it is easy to prove that the bound is $nm \lg(m)$.

  • $\begingroup$ The bound for unsorted arrays is $nm \lg (nm)$, not $nm \lg (m)$. Otherwise let $m = 1$ and voila we have linear time sorting. $\endgroup$ – orlp Aug 10 '19 at 10:58
  • $\begingroup$ You're right, I meant k-sorted with $k=m$. Edited. $\endgroup$ – PhysicsPrincess Aug 10 '19 at 11:06

Stirling's formula shows that $$ \binom{N}{pN} \sim \frac{2^{H(p)N}}{\sqrt{2\pi p(1-p)N}}, $$ where $H(p)$ is the binary entropy function: $$ H(p) = p\log \frac{1}{p} + (1-p) \log \frac{1}{1-p}. $$ In particular, this shows that for $k \geq 2$, $$ \binom{kn}{n} = (1+o(1)) \frac{2^{H(1/k)kn}}{\sqrt{2\pi(1/k)(1-1/k)kn}} \geq (1+o(1)) \frac{2^{n\log k}}{\sqrt{2\pi n}}. $$ Therefore $$ \log \binom{kn}{n} \geq n\log k - O(\log n), $$ where the big O doesn't depend on $k$. Summing this for $k=2,\ldots,m$, we deduce $$ h \geq n \log m! - O(m\log n) \geq nm \log m - O(nm), $$ using Stirling's formula again.

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