20-Memoization

20-Memoization - CS106X Autumn 2010 Handout 20 October...

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CS106X Handout 20 Autumn 2010 October 11 th , 2010 Memoization Let’s review why our first recursive implementation of Fibonacci was so dreadfully slow. Here’s the code again, updated to make use of the long long data type so that much, much larger Fibonacci numbers can, in theory and given an infinite amount of time, be computed: unsigned long long Fibonacci(int n) { if (n < 2) return n; return Fibonacci(n - 1) + Fibonacci(n - 2); } The code mirrors the inductive definition as closely as one could possibly imagine, but because each call to Fibonacci, in general, gives birth to two more, the running time grows exponentially with respect to n . One key observation: the initial recursive call leads to many (many, many) repeated recursive calls. The computation of the 40 th Fibonacci number, for instance, leads to: o 1 call to Fibonacci(39) o 2 calls to Fibonacci(38) o 3 calls to Finbonacci(37) o 5 calls to Fibonacci(36) o 8 calls to Fibonacci(35) o 13 calls to Fibonacci(34) o 21 calls to Fibonacci(33) …. It’s sad that Fibonacci(33) gets calls 21 different times, because it needs to build up the answer from scratch every single time, even though the answer is always the same. The overall implementation of Fibonacci is farcically slow because it spends a laughably large fraction of its time re-computing the same results over and over again. One very common, simple, and clever technique to overcome the repeated sub-problem issue is to keep track of all previously computed results in a Map , and to always consult the Map to see if something’s already been computed before committing to the time- consuming recursion. The code that appears below is an extension of the above, save for the key addition that a
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20-Memoization - CS106X Autumn 2010 Handout 20 October...

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