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Divide and Conquer Divide and Conquer • Probably most well-known technique in Computer Science • Basis for many common algorithms • • • • Binary search Quicsort/Mergesort Tree data structures Numerous individual/specialized algorithms • Basic idea • Split problem into one or more smaller problems • Solve the smaller (simpler) problem • If needed, combine results from smaller problems to get answer • Generally reduces O(n) to O(lg n) Binary Search: the basic algorithm • Also called “Decrease and Conquer” since you have just one subproblem • Can be used any time there is a sorted range to search over • Typical case: an array of sorted values • More general case: a range of discrete values • Can be thought of as an array with 1 of each • Even more general case: a range of continuous values • General idea here is root finding Root Finding as Binary Search • Also called bisection • Idea is to find the “zero” of a function • Need “transverse” intersection of the function – transition between positive and negative • Assumes all positive on one side, all negative on the other. • The “function” could be a complex problem that is not as easy to analyze • e.g. it fails below some value or succeeds above some value. • Can perform binary search on the values • Take middle value, determine if positive or negative • Keep half the interval remaining • Should stop in some tolerance of exact answer • If you know the range you begin with and the tolerance, you know exactly how many evaluations you will need! Another example: computing powers • If you need to compute xn, could do it with n multiplications • But, it is faster to compute xn/2 * xn/2 • Need to account for the cases where n is odd: x*xn/2*xn/2 • More generally, notice that x can be things other than numbers (e.g. a matrix) and * does not have to be standard multiplication One more example: binary search a data structure • Can sometimes convert one data structure to another that is easier to search • Valuable if you will perform multiple searches • Searches can be made binary instead of linear • Example: generate lists for all paths in tree from root to leaves • Creates large set of lists • But, can search those lists in O(lg n) time, once created • See book example in 3.3.1 for generating lists
