CS1110 Lab 10 Developing loops from invariants 31-March, 1 April 2009 1
This lab gives you practice with developing loops from invariants and with the array algorithms that you have to
know for prelim III. Your lab instructor and any consultants that are present will help you.
In our sample answers, we don’t draw all the pictures that are required. We suggest that if an invariant is not
drawn as a picture, you draw it as a picture before continuing to work on the problem. If you need help, ask your TA
or consultant.
When you are finished, show your TA what you have done. If you finish at least exercises 1-4, your lab
instructor will mark you down as having completed the assignment. If you don’t finish that much, finish exercises 1-
4 by the next week and show them to your TA at the beginning of the lab next week. In any case, try to do all the
exercises because that will help you understand and gain skill in solving such problems.
1. Give a formula for the number of values in the segment b[x..y-1]:
If you are not sure what the formula is, then ask, and memorize it. The formula is a useful tool, and knowing it will
help you develop loops that deal with ranges of integers.
2. Draw an array b that satisfies these conditions b[0..i] >= 5, b[i+1..j] = 6, b[j+1..] <= 5.
3. Below is a precondition and postcondition for the algorithm to find the minimum of an array b[h..k]. Below that
are four different loop invariants for the problem. Write a loop (with initialization) for each one.
Precondition: h <= k
Postcondition R: b[x] is the minimum of b[h..k]
(a) invariant P1: b[x] is the minimum of b[h..t]
(b) invariant P2: b[x] is the minimum of b[h..s–1]
(c) invariant P3: b[x] is the minimum of b[r..k]
(d) invariant P4: b[x] is the minimum of b[w+1..k]
Below are the postcondition and invariants drawn as pictures. Use either the text versions or the pictures.
4. The purpose of these algorithms is to swap the values of array b and to store a value in k so that the postcondition
given below is true. Array b is not necessarily sorted initially. Besides the postcondition, we give three different
b
x y
R: h k
b b[x] is the minimum of these
P1: h t k
b b[x] is the minimum of these
P2: h s k
b b[x] is the minimum of these
P3: h r k
b b[x] is the minimum of these
P4: h w k
b b[x] is the minimum of these
