23A. Illustration: Finding the Largest Value in an Array

Finding the largest using a loop

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Finding the largest value in an array is a problem that can be solved naturally using a scan algorithm. The idea is to keep track of the largest value seen so far. Suppose array A is currently as follows.

Here is a plan for a loop to find the largest value. Notice that variable L, which hold the largest value seen so far, is initially set to A[0], since there is no sensible value to choose for the largest value in an empty array.

That loop plan has an invariant: L = max(A[0], … A[i]). Check that the invariant assertion is true at each line.

Here is the algorithm written in C++.

  int largest(int* A, int n)
  {
    int L = A[0];
    int i = 0;
    while(i < n)
    {
      i++;
      L = max(L, A[i]);
    }
    return L;
  }

Finding the largest using recursion

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A recursive version of largest relies on two facts.

1. largest(A, 1) = A[0]. That is, if array A has only one member, then the largest value must be the first value, A[0].

2. largest(A, n) = max(A[n−1], largest(A, n−1)). That is, find the largest value v of the first n−1 values in array A. Then the largest of the first n values in array A is clearly either A[n−1] or v.

   Check that equation for the array A shown above for values of i from 0 to 4.

Here is a C++ function that follows those two equations.

  int largest(int* A, int n)
  {
    if(n == 1)
    {
      return A[0];
    }
    else
    {
      return max(A[n-1], largest(A, n-1));
    }
  }

Exercises

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1. What if the loop version of largest initializes L = 0 rather than L = A[0]? Does the function definition still work? Why or why not? Answer