Dev Notes

Software Development Resources by David Egan.

Memoization when Computing Fibonacci Sequence in C


C, algorithms
David Egan

The objective of this exercise is to compute a Fibonacci sequence up to a target number of elements, saving the sequence as an array.

The Fibonacci sequence is the sequence of numbers such that each number is the sum of the two preceding numbers, starting from 0 and 1. For example, the first 6 Fibonacci numbers are:

0, 1, 1, 2, 3, 5

The exercise contrasts two methods:

  • Iterative, using a for loop and computing values in order
  • Recursive, using dynamic programming technique (memoization) to improve efficiency

Iterative

The iterative method is straightforward - loop from zero up to the target, setting the current element to:

  • 0 if the current element is at index 0.
  • 1 if the current element is at index 1.
  • The sum of the elements at the previous two adjacent indices otherwise.
void iterativeFibonacci(int *resArray, int n)
{
	for (int i = 0; i < n; i++) {
		resArray[i] = i == 0 || i == 1 ? i : resArray[i-1] + resArray[i-2];
	}
}

Recursive, with Dynamic Programming Caching

Simple recursive calls (in a tree structure) would involve multiple repeat calls performing the same calculation.

Computing the 4th number in the Fibonacci sequence would involve calling:

  • fib(4) once
  • fib(3) once
  • fib(2) twice
  • fib(1) three times
  • fib(0) twice
fib(4)
|     \
|      \
|       \
|        \
fib(3)	  fib(2)--
|    \     \      \
|     \	    \      \
|      \     \      \	
fib(2) fib(1) fib(1) fib(0)
|    \
|     \
|      \
fib(1)	fib(0)

For example, a naive recursive implementation in C looks like this:

int fib(int n)
{
	if (n < 2) {
		return n;
	}
	return fib(n - 2) + fib(n - 1);
}

The number of function calls grows out of proportion as you calculate higher numbers in the sequence: computing the 10th number in the Fibonacci sequence calls fib() 177 times - computing the 20th number calls fib() 21891 times. Each call to the function requires a stack frame, so this approach is quite inefficient both in terms of time and space complexity.

We can do better by storing results as we go - a dynamic programming technique called memoization.

Memoization

This can be implemented by using an array to hold successive numbers in the sequence. For each recursive call, we check to see if the value has already been computed by looking in the cache. If has been previously computed, we return this value. Otherwise, we perform the computation and add this to the cache.

#include <stdio.h>
#include <stdlib.h>

#define N 10
static size_t count = 0;

void iterativeFibonacci(int *resArray, int n)
{
	for (int i = 0; i < n; i++) {
		resArray[i] = i == 0 || i == 1 ? i : resArray[i-1] + resArray[i-2];
	}
}

/**
* Build a cache representing the Fibonacci sequence.
* Note that the second parameter is an array index which allows the cache to
* be checked for the required element. 
*/
int recursiveFibonacci(int *cache, int n)
{
	count++; // For analysis only

	// If the cache holds -1 at the required index, it has not yet been computed.
	if (cache[n] == -1) {
		cache[n] = recursiveFibonacci(cache, n - 1) + recursiveFibonacci(cache, n - 2);
	}
	return cache[n];
}

void printArray(int *arr, int n) {
	for (int i = 0; i < n; i++) {
		printf(i == 0 ? "[ %d" : ", %d", arr[i]);
	}
	puts(" ]\n");
}

int main(void)
{
	int r[N];
	iterativeFibonacci(r, N);
	printArray(r, N);
	
	// Initialise an array of N elements, each element set to -1
	int cache[N] = { [0 ... N-1] = -1 };

	// Set the first two elements in the sequence, which are known
	cache[0] = 0;
	cache[1] = 1;

	// The function receives a pointer to the cache array and the index of the last element.
	recursiveFibonacci(cache, N - 1);
	printArray(cache, N);
	printf("recursiveFibonacci() called %lu times.\n", count);
	return 0;
}

Using this method, computing the 20th number in the Fibonacci sequence requires 37 calls to recursiveFibonacci(). We also have the additional benefit of having the entire sequence up to and including our target in our cache array, for very little extra cost.

References


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