Steps to develop an iterative algorithm

The code structure

begin routine
    <pre-cond>
    pre-loop code % Establish loop invariant
    loop
        <loop-invariant>
        exit when <exit-cond>
        loop code % Make progress while maintaining the loop invariant
    end loop
    post-loop code % Clean up loose ends
    <post-cond>
end routine

 1) Specifications:

Preconditions: assertions about the input instances

Postconditions: assertions about the output

<pre-cond> & Code_alg => <post-cond>

Pre- and postconditons are the contract between the implementer and the user of the coded algorithm

2) Loop invariant

Coming up with the loop invariant is the hardest part of designing an algorithm. However, from it the rest of the algorithm often follows easily.

A loop invariant is an assertion that is placed at the top of a loop and that must hold true every time the computation returns to the top of the loop.

• Eatablish the loop invariant
  • 　　<pre-cond> & code_pre-loop => <loop-invariant>
• Maintain the loop invariant
  • 　　<loop-invariant^'> & not <exit-cond> & code_loop => <loop-invariant^''>　
• Main steps
  • 　　code_loop must be defined so that it can work for any state of the strucutre for which the loop invariant is true. Worry about one step at a time, and make sure you make progress everytime the algorithm goes around the loop.

3) Ending

　　<loop-invariant> & <exit-cond> & code_post-loop => <post-cond>

 Mathematical induction

 

Different types of Iterative Algorithms

The key point of any iterative algorithm is design a measure of progress and a loop invariant.

1) More of the output

Measure of progress: The amount of the output constructed

Loop invariant: The output constructed so far is correct

Selction sort

2) More of the input

Measure of progress: The amount of the input considered

Loop invariant: Pretending that this prefix of the input is the entire input, we have a complete solution

Insertion sort

3) Narrowing the search space

Measure of progress: the size of the space in which you have narrowed the search

Loop invariant: If the thing being searched for is anywhere, then it is in this narrowed subspace

Binary search

4) Work done

Measure of progress: some other creative frunction of the work done so far

Bubble sort