Chapter 2 Getting Started

2.1 Insertion sort

insertion sort is an efficient algorithm for sorting a small number of elements.

The pseudocode

INSERTION-SORT(A)
1 for j = 2 to A.length
2     key = A[j]
3     // Insert A[j] into the sorted sequence A[1~j-1].
4	   i = j - 1
5	   while i > 0 and A[j] > key
6     		A[i+1] = A[i]
7    		i = i - 1
8     A[i+1] = key

Loop invariants

Initialization: It is true prior to the first iteration of the loop.

Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration.

Termination: When the loop terminates, the invariant gives us a useful property that helps show that the algorithm is correct.

Exercises

2.1-2 Rewrite the INSERTION-SORT procedure to sort into nonincreasing instead of nondecreasing order.

C:

for(i=1;i<n;i++)
	{
		key=a[i];
		j=i-1;
		while(j>=0&&key>a[j])
		{
			a[j+1]=a[j];
			j--;
		}
		a[j+1]=key;
	}

2.1-3 Input: A sequence of n numbers A=⟨a1,a2,…,an⟩ and a value ν. Output: And index i such that ν=A[i] or the special value NIL if ν does not appear in A. Write pseudocode for linear search, which scans through the sequence, looking for v. Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.

The pseudocode just looks like this :

SEARCH(A, v):
  for i = 1 to A.length
      if A[i] == v
          return i
  return NIL

I’m going to state the loop invariant as:

At the start of each iteration of the for loop, the subarray A[1..i−1] consists of elements that are different than ν.

Here are the three properties:

Initialization Initially the subarray is the empty array, so proving it is trivial.

Maintenance On each step, we know that A[1…i−1] does not contain ν. We compare it with A[i]. If they are the same, we return i, which is a correct result. Otherwise, we continue to the next step. We have already insured that A[A…i−1] does not contain ν and that A[i] is different from ν, so this step preserves the invariant.

Termination The loop terminates when i>A.length. Since i increases by 1 and i>A.length, we know that all the elements in A have been checked and it has been found that ν is not among them. Thus, we return NIL.