Lecture 25: Algorithm correctness

lecture video

Textbook chapters

4.4

Algorithm correctness

We’ve been studying the runtimes of algorithms. But we can correctly write down the runtime without showing anything about its correctness.

What does it mean for an algorithm to be correct?

An algorithm is correct if it returns the correct value for every allowable input.

How do we prove correctness?

Counterexample

What if an algorithm is not correct? We can prove that by giving a counterexample.

Suppose I claim this algorithm returns the max value of an array:

public static double arrayMax(double[] x) {
    double curMax = x[0];
    for (int i = 1; i < x.length - 1; i ++) {
        if (x[i] > curMax) {
            curMax = x[i];
        }
    }
    return curMax;
}

We can prove that it’s incorrect by pointing out that for the input [10, 100] it returns 10.

Loop invariants

The purpose of loop invariants is to give us a way to formalize our arguments about why an algorithm is correct.

To prove some statement $\mathcal{L}$ is correct, we define $\mathcal{L}$ in terms of a series of smaller statements $\mathcal{L}_0, \mathcal{L}_1, \mathcal{L}_k$, where:

1. The initial claim, $\mathcal{L}_0$, is true before the loop begins.
2. If $\mathcal{L}_{j-1}$ is true before iteration $j$, then $\mathcal{L}_j$ will be true after iteration $j$.
3. The final statement, $\mathcal{L}_k$, implies the desired statement $\mathcal{L}L$ to be true.

Files used in class

LoopInvariants.pdf

Additional problems

• Use a loop invariant to argue that the algorithm for array max from the previous lecture is correct. To do so, make sure that you are explicit about what the overall claim that we need to show is, what sub-claim your loop invariant is, and then why 1, 2, and 3 above are true.