US EditionDistribution of run times for Euclidean algorithm
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Distribution of run times for Euclidean algorithm
The worst case run time for the Euclidean algorithm occurs when you give the algorithm a pair of consecutive Fibonacci numbers. The algorithm takes n steps to compute the greatest common divisor of Fn+1 and Fn+2. The nth Fibonacci number is the nearest integer to φn/√5 where φ = (1 + √5)/2 is the golden ratio. You can take logs and solve for the largest n such that Fn is less than x, subtract 2, and have an upper bound on the number of steps nec…
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