Source: https://stackoverflow.com/questions/16987670/dynamic-programming-why-knuths-improvement-to-optimal-binary-search-tree-on2

## Question

This is Exercise 15.5-4 of Introduction to Algorithms, 3rd edition, which is about Knuth’s improvement to the DP approach to Optimal Binary Search Tree.

The DP algorithm of Optimal Binary Search Tree is:

OPTIMAL_BST(p, q, n)
let e[1..n+1, 0..n], w[1..n+1, 0..n], and root[1..n, 1..n] be new tables
for i = 1 to n+1
e[i, i - 1] = q[i - 1];
w[i, i - 1] = q[i - 1];
for l = 1 to n
for i = 1 to n - l + 1
j = i + l - 1
e[i, j] = INFINITY
w[i, j] = w[i, j - 1] + p[j] + q[j]
for r = i to j
t = e[i, r - 1] + e[r + 1, j] + w[i, j]
if t < e[i, j]
e[i, j] = t
root[i, j] = r
return e and root


The complexity is O(n3). Knuth had observed that root[i, j - 1] <= root[i, j] <= root[i + 1, j], so Exercise 15.5-4 asks to implement an O(n2) algorithm by doing some modification to the original algorithm.

Well after some effort I have figured this out: in the innermost loop, replace the line

for r = i to j


with

for r = r[i, j - 1] to r[i + 1, j]


This has been proved by this link: Optimal binary search trees

However, I’m not sure this is really O(n2): since during each innermost loop, distance from r[i, j - 1] to r[i + 1, j] is not constant, I suspect it is still O(n3).

So my question is: can you please explain to me why the improvement to DP algorithm yields O(n2) complexity?

PS: Maybe I might have read Knuth’s paper first, but really I searched the web but found no free access to the paper.

You’re correct that the distance from r[i, j - 1] to r[i + 1, j] is not constant in the worst case, but it is constant on average, which suffices to imply a quadratic running time. The total number of iterations for l is

$S = sum_{i = 1}^{n - l + 1} (r[i + 1, j] + 1 - r[i, j - 1]), j = i + l - 1\\ = sum_{i = 1}^{n - l + 1} (r[i + 1, i + l - 1] + 1 - r[i, i + l - 2])\\ = r[n - l + 2, n] + n - l + 1 - r[1, l - 1]$ therefore the average is S / (n - l + 1), which is a constant

by simplifying the telescoping sum.

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