A topic not covered in the handouts was that of memoization which is something you'll see again in dynamic programming. The simple version of the fibonaccifunction, for example, has a most inefficient running time - as we will see later, as we compute the nth fib. number using that simple function, the run time to do so increases exponentially. We want to do better, and we can do so by saving the results of prior computations, thus avoiding the task of recomputing them.
Consider:
int fib(int n) {
if (n <2) return n; //assuming n is non-negative
int table[n+1]; //create a table to hold the results we already have
table[0] = 0;
table[1]= 1;
for (int x=2; x <= n; ++x)
table[x]=-1;
return fib_aux(table, n);
}
int fib_aux (int T[], int n){
if (T[n] >= 0) return T[n];
return (T[n] = fib_aux(T, n-1) +
fib_aux(T, n-2) );
}
The effect of this tabularization is that we compute each term of the
sequence only one time instead of many, and the runtime efficiency
of our algorithm improves to linear time.