Prove that the outer loop for this algorithm is correct, using the following steps. You may assume for simplicity that the inner while loop is correct (i.e., it inserts the element A[j] in its proper position of the sub-array A[1..j]). Given the following pre-condition, loop invariant, and post-condition: • Pre-condition: A[1..n] is an array of integers. • Loop invariant: at the start of each iteration, the sub-array A[1..j – 1] consists of the elements originally in A[1..j – 1], but in sorted order. • Post-condition: the array A[1..n] consists of the elements originally in A[1..n], but in sorted order. Prove the following: • Initialization: If the pre-condition is true, then the loop invariant is initially true (prior to the first iteration of the loop). • Maintenance: If the loop invariant is true before an iteration, it remain true before the next iteration. • Termination: The loop will eventually terminate. And when it does, the loop invariant implies the post-condition. 1: for j=2: A.length do key = A[j] i = j – 1 while i > 0 and A[i] > key do A[i + 1] = A[i] i = i – 1 A[i + 1] = key 2: 3: 4: 5: 6: 7:
Prove that the outer loop for this algorithm is correct, using the following steps. You may assume for simplicity that the inner while loop is correct (i.e., it inserts the element A[j] in its proper position of the sub-array A[1..j]). Given the following pre-condition, loop invariant, and post-condition: • Pre-condition: A[1..n] is an array of integers. • Loop invariant: at the start of each iteration, the sub-array A[1..j – 1] consists of the elements originally in A[1..j – 1], but in sorted order. • Post-condition: the array A[1..n] consists of the elements originally in A[1..n], but in sorted order. Prove the following: • Initialization: If the pre-condition is true, then the loop invariant is initially true (prior to the first iteration of the loop). • Maintenance: If the loop invariant is true before an iteration, it remain true before the next iteration. • Termination: The loop will eventually terminate. And when it does, the loop invariant implies the post-condition. 1: for j=2: A.length do key = A[j] i = j – 1 while i > 0 and A[i] > key do A[i + 1] = A[i] i = i – 1 A[i + 1] = key 2: 3: 4: 5: 6: 7:
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Consider the
![Prove that the outer loop for this algorithm is correct, using the following steps. You may assume
for simplicity that the inner while loop is correct (i.e., it inserts the element A[j] in its proper
position of the sub-array A[1..j]).
Given the following pre-condition, loop invariant, and post-condition:
• Pre-condition: A[1..n] is an array of integers.
• Loop invariant: at the start of each iteration, the sub-array A[1..j – 1] consists of the
elements originally in A[1..j – 1], but in sorted order.
• Post-condition: the array A[1..n] consists of the elements originally in A[1..n], but in sorted
order.
Prove the following:
• Initialization: If the pre-condition is true, then the loop invariant is initially true (prior to
the first iteration of the loop).
• Maintenance: If the loop invariant is true before an iteration, it remain true before the next
iteration.
• Termination: The loop will eventually terminate. And when it does, the loop invariant
implies the post-condition.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F171ff07b-7b62-4691-9b2a-ffaa1328dd79%2F55156325-97f8-4139-b1b9-3852c9eb1435%2Fycbq8of.jpeg&w=3840&q=75)
Transcribed Image Text:Prove that the outer loop for this algorithm is correct, using the following steps. You may assume
for simplicity that the inner while loop is correct (i.e., it inserts the element A[j] in its proper
position of the sub-array A[1..j]).
Given the following pre-condition, loop invariant, and post-condition:
• Pre-condition: A[1..n] is an array of integers.
• Loop invariant: at the start of each iteration, the sub-array A[1..j – 1] consists of the
elements originally in A[1..j – 1], but in sorted order.
• Post-condition: the array A[1..n] consists of the elements originally in A[1..n], but in sorted
order.
Prove the following:
• Initialization: If the pre-condition is true, then the loop invariant is initially true (prior to
the first iteration of the loop).
• Maintenance: If the loop invariant is true before an iteration, it remain true before the next
iteration.
• Termination: The loop will eventually terminate. And when it does, the loop invariant
implies the post-condition.
![1: for j=2: A.length do
key = A[j]
i = j – 1
while i > 0 and A[i] > key do
A[i + 1] = A[i]
i = i – 1
A[i + 1] = key
2:
3:
4:
5:
6:
7:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F171ff07b-7b62-4691-9b2a-ffaa1328dd79%2F55156325-97f8-4139-b1b9-3852c9eb1435%2Fsxodq8.jpeg&w=3840&q=75)
Transcribed Image Text:1: for j=2: A.length do
key = A[j]
i = j – 1
while i > 0 and A[i] > key do
A[i + 1] = A[i]
i = i – 1
A[i + 1] = key
2:
3:
4:
5:
6:
7:
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