a. Prove that count[j] equals the number of times j appears in data[1::i], for every j from 1 up to r, after each iteration of the for loop in lines 4-6. b. Prove that count[j] equals the number of values in data that are less than or equal to j after every iteration of the for loop in lines 7-9.
a. Prove that count[j] equals the number of times j appears in data[1::i], for every j from 1 up to r, after each iteration of the for loop in lines 4-6. b. Prove that count[j] equals the number of values in data that are less than or equal to j after every iteration of the for loop in lines 7-9.
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![Prove the loop invariant of the counting sort algorithm
given below, you do not have to prove the correctness of
the algorithm.
Input: data: array of n integers that are between 1 and r
Input: n: size of data
Input: r: range of data
Output: permutation of data such that
data[1] ≤ data[2] ≤... <data[n]
1 Algorithm: Counting Sort
2 count = Array (r)
3 Initialize count to 0
4 for i=1 to n do
5
6 end
count[data[i]] = count[data[i]] + 1
7 for j = 2 to r do
8
12
13
count[j] = count[j] + count j - 1]
9 end
10 output= Array(n)
11 for i=1 to n do
output [count data[i]]] = data[i]
count [data[i]]= count[data[i]] - 1
14 end
15 return output](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc081ddcc-eba2-40ae-b68f-7408fff0456c%2F6a78f5e0-54df-4c81-8cf5-d2abcb885446%2Fnugid7b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Prove the loop invariant of the counting sort algorithm
given below, you do not have to prove the correctness of
the algorithm.
Input: data: array of n integers that are between 1 and r
Input: n: size of data
Input: r: range of data
Output: permutation of data such that
data[1] ≤ data[2] ≤... <data[n]
1 Algorithm: Counting Sort
2 count = Array (r)
3 Initialize count to 0
4 for i=1 to n do
5
6 end
count[data[i]] = count[data[i]] + 1
7 for j = 2 to r do
8
12
13
count[j] = count[j] + count j - 1]
9 end
10 output= Array(n)
11 for i=1 to n do
output [count data[i]]] = data[i]
count [data[i]]= count[data[i]] - 1
14 end
15 return output
![a. Prove that count[j] equals the number of times j appears in
data[1::i], for every j from 1 up to r, after each iteration of the for
loop in lines 4-6.
b. Prove that count[j] equals the number of values in data that are
less than or equal to j after every iteration of the for loop in lines
7-9.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc081ddcc-eba2-40ae-b68f-7408fff0456c%2F6a78f5e0-54df-4c81-8cf5-d2abcb885446%2Fpcheldu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a. Prove that count[j] equals the number of times j appears in
data[1::i], for every j from 1 up to r, after each iteration of the for
loop in lines 4-6.
b. Prove that count[j] equals the number of values in data that are
less than or equal to j after every iteration of the for loop in lines
7-9.
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