Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Example 9.6.4
Counting Iterations of a Loop
How many times will the innermost loop be iterated when the algorithm segment below is
implemented and run? (Assume n is a positive integer.)
for k:= 1 te n
for j:= 1 to k
for i:= 1 to j
[Statements in the body of the inner loop,
none containing branching statements that lead
outside the loop]
next i
nextj
next k
Solution
Construct a trace table for the values of k. j, and i for which the statements in the body of the
innermost loop are executed.
2-
3-
2
2-
3-
3D
2.
Because i goes from 1 to j, it is always the case that i< j. Similarly, because j goes from 1 to k it is
always the case that j< k. To focus on the details of the table construction, consider what happens
when k 3. In this case, j takes each value 1, 2, and 3. When j=1, i can only take the value 1 (because
i<i). Whenj= 2, i takes each value 1 and 2 (again because i < j). When, = 3. i takes each value 1. 2.
and 3 (yet again because i <).
Observe that there is one iteration of the innermost loop for each column of the table, and there is
one column of the table for each triple of integers (i. (1,3, k) with 1<isjsAsn Now Example 9.6.3
showed that the number of such triples is [n (n+ 1) (n+2)|/6. Thus there are [n (n + 1) (n + 2)|/6
iterations of the innermost loop.](https://content.bartleby.com/qna-images/question/9de35394-b566-452d-a0dc-ecbc1ac42ab5/fdebebd7-4cb5-4070-b3e3-9a7e7478cbb5/fjgfxd_processed.jpeg)
Transcribed Image Text:Example 9.6.4
Counting Iterations of a Loop
How many times will the innermost loop be iterated when the algorithm segment below is
implemented and run? (Assume n is a positive integer.)
for k:= 1 te n
for j:= 1 to k
for i:= 1 to j
[Statements in the body of the inner loop,
none containing branching statements that lead
outside the loop]
next i
nextj
next k
Solution
Construct a trace table for the values of k. j, and i for which the statements in the body of the
innermost loop are executed.
2-
3-
2
2-
3-
3D
2.
Because i goes from 1 to j, it is always the case that i< j. Similarly, because j goes from 1 to k it is
always the case that j< k. To focus on the details of the table construction, consider what happens
when k 3. In this case, j takes each value 1, 2, and 3. When j=1, i can only take the value 1 (because
i<i). Whenj= 2, i takes each value 1 and 2 (again because i < j). When, = 3. i takes each value 1. 2.
and 3 (yet again because i <).
Observe that there is one iteration of the innermost loop for each column of the table, and there is
one column of the table for each triple of integers (i. (1,3, k) with 1<isjsAsn Now Example 9.6.3
showed that the number of such triples is [n (n+ 1) (n+2)|/6. Thus there are [n (n + 1) (n + 2)|/6
iterations of the innermost loop.
![How many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume n, k, j, and i are positive
integers.
for k := 1 ton
for j := k to n
for i := j to n
[Statements in the body of the inner loop,
none containing branching statements
that lead outside the loop]
next i
next j
next k
As in Example 9.6.4, sketch a trace table for the algorithm segment and notice that because i goes from j to n, it is always the case that i ? vi.
Similarly, it is always the case that j ? v k and that k ? v 1. Thus, the number of iterations of the innermost loop is the same as the number of
integer triples (i, j, k) that are related to each other in a certain way, and this is the same as the number of strings of ? v vertical bars and
? crosses, where the position of the crosses indicate which ? v integers from 1 to n are included in the triple. Thus, the answer is](https://content.bartleby.com/qna-images/question/9de35394-b566-452d-a0dc-ecbc1ac42ab5/fdebebd7-4cb5-4070-b3e3-9a7e7478cbb5/2xm0jm_processed.jpeg)
Transcribed Image Text:How many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume n, k, j, and i are positive
integers.
for k := 1 ton
for j := k to n
for i := j to n
[Statements in the body of the inner loop,
none containing branching statements
that lead outside the loop]
next i
next j
next k
As in Example 9.6.4, sketch a trace table for the algorithm segment and notice that because i goes from j to n, it is always the case that i ? vi.
Similarly, it is always the case that j ? v k and that k ? v 1. Thus, the number of iterations of the innermost loop is the same as the number of
integer triples (i, j, k) that are related to each other in a certain way, and this is the same as the number of strings of ? v vertical bars and
? crosses, where the position of the crosses indicate which ? v integers from 1 to n are included in the triple. Thus, the answer is
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