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How to Think About Algorithms by Jeff Edmonds

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6 Euclid’s GCD Algorithm

More-of-the-input iterative algorithms extend a solution for a smaller input instance into a larger one. We will see in Chapter 9 that recursive algorithms do this too. The following is an amazing algorithm that does this. It finds the greatest common divisor (GCD) of two integers. For example, GCD (18, 12) = 6. It was first done by Euclid, an ancient Greek. Without the use of loop invariants, you would never be able to understand what the algorithm does; with their help, it is easy.

Specifications: An input instance consists of two positive integers, a and b. The output is GCD(a, b).

The Loop Invariant: Like many loop invariants, designing this one required creativity. The algorithm maintains two variables x and y whose ...

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