Divide and Conquer Summary

To use divide and conquer as an algorithm design technique, we must divide the problem into two smaller subproblems, solve each of them recursively, and then meld the two partial solutions into one solution to the full problem. Whenever the merging takes less time than solving the two subproblems, we get an efficient algorithm.

Examples -

Merge Sort

Classic example of a divide-and-conquer algorithm. It takes only linear time to merge two sorted lists of n/2 elements each of which was obtained in O(n) time.

fast Fourier transform

Strassen's matrix multiplication algorithm

Fast Exponentiation

computing the value aⁿ for some reasonably large n. These problems occur in cryptography. It takes O(lg n) multiplications to compute aⁿ.

	 function power(a, n)
		if (n=0) return (1)
		x = power (a, floor(n/2))
		if (n is even) then return (x²)
		else return (a * x²)

Binary Search

fast algorithm for searching in a sorted array S of keys. To search for key q, we compare q to the middle key S[n/2]. If q appears before S[n/2], it must reside in the top half of our set; if not, it must reside in the bottom half of our set. By recursively repeating this process on the correct half, we find the key in a total of O(lg n) comparisons.

Applications include 20 questions, finding a point of transition.

Square and Other Roots

Uses the bisection method to find the solutions to a given equation. We say that x is a root of the function f if f(x) = 0. Assume that l is a value so that f(x) < 0 and r is a value so that f(x) > 0.

	 solve (f(x), l, r)
		let m = (1+r)/2
		let val = f(m)
		if (ABS(val) < TOL) return m
		if (val < 0) solve (f(x), m, r)
		else solve (f(x), l, m)

Complexity of O(lg n). This algorithm can be made faster by interpolation on the interval to find a test point closer to the actual root.

Exercises from Chapter 3

Kelly Schultz, Kalamazoo College