Asymptotic runtime of list-to-tree function

Question

I have a merge function which takes time O(log n) to combine two trees into one, and a listToTree function which converts an initial list of elements to singleton trees and repeatedly calls merge on each successive pair of trees until only one tree remains.

Function signatures and relevant implementations are as follows:

merge :: Tree a -> Tree a -> Tree a --// O(log n) where n is size of input trees
singleton :: a -> Tree a            --// O(1)
empty :: Tree a                     --// O(1)
listToTree :: [a] -> Tree a         --// Supposedly O(n)

listToTree = listToTreeR . (map singleton)

listToTreeR :: [Tree a] -> Tree a
listToTreeR []     = empty
listToTreeR (x:[]) = x
listToTreeR xs     = listToTreeR (mergePairs xs)

mergePairs :: [Tree a] -> [Tree a]
mergePairs []       = []
mergePairs (x:[])   = [x]
mergePairs (x:y:xs) = merge x y : mergePairs xs

This is a slightly simplified version of exercise 3.3 in Purely Functional Data Structures by Chris Okasaki.

According to the exercise, I shall now show that listToTree takes O(n) time. Which I can't. :-(

There are trivially ceil(log n) recursive calls to listToTreeR, meaning ceil(log n) calls to mergePairs.

The running time of mergePairs is dependent on the length of the list, and the sizes of the trees. The length of the list is 2^h-1, and the sizes of the trees are log(n/(2^h)), where h=log n is the first recursive step, and h=1 is the last recursive step. Each call to mergePairs thus takes time (2^h-1) * log(n/(2^h))

I'm having trouble taking this analysis any further. Can anyone give me a hint in the right direction?

Answer 1

It's almost there. You already know the expression is

so the only problem is to evaluate this sum. Using log(AB) = log A + log B and log 2^N = N we have

With help of calculators, we can find that X = O(2^m) = O(n), which is expected.

(If you want to compute this yourself, search for "Geometric series", or approximate the sum using an integral.)