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Designing loop: the loop invariant

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• Property of loops
  • An important property of loops

    In each iteration of a loop statement, the entire loop body is executed

    Implication:

    Because the loop body is executed many times, to ensure correctness, the same condition must hold at the start of each iteration !!!

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  • The loop invariant:

    In order for the loop statement to execute correctly for every iteration: Some property/relationship must be valid at the start of every iteration of the loop In fact, the statements inside the loop body must make sure this pre-condition will be satisfied at the start of the next interation ! Note: I have to be vague on what the exact condition is... Because the property is different for different loops The technical term for this property that is true at the start of every iteration of the loop is: the loop invariant

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  • Loop invariant (Wikipedia):

    Informally, a loop invariant is a statement of the conditions that should be true on entry into a loop The loop invariant must be true on every entry (i.e., for every iteration) of the loop.

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• Examples of loop invariant
  • Loop 1: find all divisors of the number n

    x = 1; while ( x <= n ) // Run x = 1, 2, ..., n { Invariant: We have found all divisors of n that are < x x = the current divisor that we will try if ( n % x == 0 ) { // x is a divisor of n System.out.println(x); // Print x (because it's a divisor) } x++; // Make sure we more to the next number !! // or else: infinite loop !!! } Invariant when loop ends: We have found all divisors of n that are <= n

    Note:

    The loop invariant must be true at the start of every iteration !!! Furthermore: the loop invariant is also true when the loop exits Because: The loop exits when the loop-continuation-condition is false. In other words, the program was ready to start the next iteration of the loop Because we are starting a new iteration, the loop invariant must then be true !!!

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  • Loop 2: find all common divisors of a and b

    min = min(a,b); a = 1; while ( a <= min ) // Run a = 1, 2, ..., min(x,y) { Invariant: We have found all common divisors of a and b that are < x x = the current common divisor that we will try if ( x % a == 0 && y % a == 0 ) { // a is a divisor of x and y System.out.println(a); // Print a (because it's a common divisor) } a++; // Make sure we move to the next number !! // or else: infinite loop !!! } Invariant when loop ends: We have found all common divisors of a and b that are <= min

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  • Loop 3: find all factors of a number x

    f = 2; while ( x > 1 ) { Invariant: x = the number that remains to be factor and we have removed all factors < f if ( x % f == 0 ) { // f is a factor of x System.out.println(f); // Print f (because it's a divisor) x = x / f; // Remove factor f from x } else { f++; // Use next number as factor } } Invariant when loop ends: The number that remains to be factored = 1 and we have removed all factors including the largest factor (I.e.: we have found all the factors)

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• How to design a loop
  • No simple answer

    There is no simple answer to this question That's may be why Knuth called his collection of books: the Art of Computer Programming

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  • Some Advice:

    Use loop invariance It can help you write the steps inside the loop body Make sure the loop invariance is useful The loop is used to solve some problem When the loop terminates, the loop invariant is also true The loop invariant must imply that the problem you want to solve with the loop, has been solved (Otherwise, your loop is useless) Keep the loop invariance as simple as possible. Simple loop invariance will result in simple loops (How to keep loop invariances simple is an art....)