In this report, you'll work with the following algorithm. You will model it with a state machine, verify that it terminates, verify that it is partially correct, and provide a time complexity estimate for it. Input :Non-negative integers x, y Output : Non-negative integer a r:= x; s:= y; a:= 0; while s + 0 do if 2 | s then r:=r+r; s:= s/2; end else a := a+r; r:=r+r; s:= (s - 1)/2; end end return a Älgorithm 1: An algorithm that performs a simple calculation on x,y. (a) Algorithm[ seems rather complex, but it is actually performing a relatively simple operation on the input values x, y. i Walk through the algorithm using your choice of an example pair of values x, y by showing the values of r, s, a after each iteration of the loop. Understand that the example values for x,y that you used are just for this part to understand how the algorithm works. For the rest of the report, you need to think about the algorithm abstractly. ii Describe what simple calculation is being performed on the input values. amal min Matiaa tlhat tha
In this report, you'll work with the following algorithm. You will model it with a state machine, verify that it terminates, verify that it is partially correct, and provide a time complexity estimate for it. Input :Non-negative integers x, y Output : Non-negative integer a r:= x; s:= y; a:= 0; while s + 0 do if 2 | s then r:=r+r; s:= s/2; end else a := a+r; r:=r+r; s:= (s - 1)/2; end end return a Älgorithm 1: An algorithm that performs a simple calculation on x,y. (a) Algorithm[ seems rather complex, but it is actually performing a relatively simple operation on the input values x, y. i Walk through the algorithm using your choice of an example pair of values x, y by showing the values of r, s, a after each iteration of the loop. Understand that the example values for x,y that you used are just for this part to understand how the algorithm works. For the rest of the report, you need to think about the algorithm abstractly. ii Describe what simple calculation is being performed on the input values. amal min Matiaa tlhat tha
Computer Networking: A Top-Down Approach (7th Edition)
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ISBN:9780133594140
Author:James Kurose, Keith Ross
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Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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![In this report, you'll work with the following algorithm. You will model it with a state machine,
verify that it terminates, verify that it is partially correct, and provide a time complexity estimate
for it.
Input :Non-negative integers x, y
Output : Non-negative integer a
r:= x;
s:= y;
a:= 0;
while s + 0 do
if 2 | s then
r:=r+r;
s:= s/2;
end
else
a := a+r;
r:=r+r;
s:= (s - 1)/2;
end
end
return a
Algorithm 1: An algorithm that performs a simple calculation on x, y.
(a) Algorithm] seems rather complex, but it is actually performing a relatively simple operation
on the input values x, y.
i Walk through the algorithm using your choice of an example pair of values x, y by
showing the values of r,s, a after each iteration of the loop. Understand that the example
values for x,y that you used are just for this part to understand how the algorithm works.
For the rest of the report, you need to think about the algorithm abstractly.
ii Describe what simple calculation is being performed on the input values.
(b) Complete the following description for a state machine model of Algorithm]. Notice that the
start state and part of the transition function need to be completed.
• States: (r,s,a) eN
• Start State: (?,?, ?)
• Transition Function: The transitions are given by the rule & that for s > 0:
|(2r, s/2,a) if 2|s
|(?, ?, ?)
8(r,s, a) =
otherwise.
(c) Verify that Algorithm] is partially correct.
i Describe what it means for Algorithm] to be partially correct.
ii Define a predicate P on the states that you believe is a preserved invariant. (Suggestion:
Think about why the loop condition is s + 0 and look at your sequence of steps from
Part 1.)
iii Prove that P is a preserved invariant.
iv Apply the Invariant Principle to prove that Algorithm] is partially correct.
(d) Verify that Algorithm] terminates.
i Verify that the derived variable p(r,s,a)::= s is strictly decreasing.
ii Use p to prove that the algorithm terminates.
(e) Using the results from 4. (d) and a definition for the Big Oh relation, prove that O(logy)
provides a time complexity estimate for Algorithm[].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b558713-79de-49d9-9842-19ce6230df52%2F658ebc13-da26-48a8-a3a2-d3e1d15a6dcf%2Fjs80yys_processed.png&w=3840&q=75)
Transcribed Image Text:In this report, you'll work with the following algorithm. You will model it with a state machine,
verify that it terminates, verify that it is partially correct, and provide a time complexity estimate
for it.
Input :Non-negative integers x, y
Output : Non-negative integer a
r:= x;
s:= y;
a:= 0;
while s + 0 do
if 2 | s then
r:=r+r;
s:= s/2;
end
else
a := a+r;
r:=r+r;
s:= (s - 1)/2;
end
end
return a
Algorithm 1: An algorithm that performs a simple calculation on x, y.
(a) Algorithm] seems rather complex, but it is actually performing a relatively simple operation
on the input values x, y.
i Walk through the algorithm using your choice of an example pair of values x, y by
showing the values of r,s, a after each iteration of the loop. Understand that the example
values for x,y that you used are just for this part to understand how the algorithm works.
For the rest of the report, you need to think about the algorithm abstractly.
ii Describe what simple calculation is being performed on the input values.
(b) Complete the following description for a state machine model of Algorithm]. Notice that the
start state and part of the transition function need to be completed.
• States: (r,s,a) eN
• Start State: (?,?, ?)
• Transition Function: The transitions are given by the rule & that for s > 0:
|(2r, s/2,a) if 2|s
|(?, ?, ?)
8(r,s, a) =
otherwise.
(c) Verify that Algorithm] is partially correct.
i Describe what it means for Algorithm] to be partially correct.
ii Define a predicate P on the states that you believe is a preserved invariant. (Suggestion:
Think about why the loop condition is s + 0 and look at your sequence of steps from
Part 1.)
iii Prove that P is a preserved invariant.
iv Apply the Invariant Principle to prove that Algorithm] is partially correct.
(d) Verify that Algorithm] terminates.
i Verify that the derived variable p(r,s,a)::= s is strictly decreasing.
ii Use p to prove that the algorithm terminates.
(e) Using the results from 4. (d) and a definition for the Big Oh relation, prove that O(logy)
provides a time complexity estimate for Algorithm[].
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