Lecture 1: Recursion

Reduction is the single most common technique used in designing algorithms. Reducing one problem X to another problem (or set of problems) Y means to write an algorithm for X , using an algorithm or Y as a subroutine or black box. For example, the congressional apportionment algorithm described in Lecture 0 reduces the problem of apportioning Congress to the problem of maintaining a priority queue under the operations INSERT and EXTRACTMAX. Those data structure operations are black boxes; the apportionment algorithm does not depend on any specific implementation. Conversely, when we design a particular priority queue data structure, we typically neither know nor care how our data structure will be used. Whether or not the Census Bureau plans to use our code to apportion Congress is completely irrelevant to our design choices. As a general rule, when we design algorithms, we may not know—and we should not care—how the basic building blocks we use are implemented, or how your algorithm might be used as a basic building block to solve a bigger problem. A particularly powerful kind of reduction is recursion, which can be defined loosely as follows: