© Leong Hon Wai, 2003-2009 LeongHW, SoC, NUS (UIT2201: Algorithms) Page 1 Animation of Algorithm Goal: To understand an algorithm by animating its execution, step-by-step. Algorithm: Array-Sum (find sum of n numbers) Observe carefully: sequential instruction conditional statements, repetitive statements,
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© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 1
Animation of Algorithm
Goal: To understand an algorithm by animating its
execution, step-by-step.
Algorithm: Array-Sum (find sum of n numbers)
Observe carefully:sequential instruction
conditional statements, repetitive statements,
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 2
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-SumLet’s animate the execution of this algorithm on the input:
Input to Algo. Array-Sum:
n = 6Array A = [2, 5, 10, 3, 12, 24]
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 3
Model of Computer: Initial State
We assume that the values ofn and the array A[1..6] hasbeen read into the computer.
We assume that the values ofn and the array A[1..6] hasbeen read into the computer.
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF ? Sum ?
k ?
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 4
Start of execution…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF ? Sum ?
k ?
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 5
executing…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
Sum_SF 0;
CPU
Sum_SF 0 Sum ?
k ?
Assignment statement;The value of 0 is stored in thestorage box called Sum_sf.
Assignment statement;The value of 0 is stored in thestorage box called Sum_sf.
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 6
executing…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k 1;
CPU
Sum_SF 0 Sum ?
k 1
Assignment statement;The value of 1 is stored in thestorage box called k.
Assignment statement;The value of 1 is stored in thestorage box called k.
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 7
executing beginning of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 0 Sum ?
k 1
Conditional: (k <= n)Reads value of k and n;Check outcome of (k <= n),Condition is True, execute loop;Value of k, n are unchanged.
Conditional: (k <= n)Reads value of k and n;Check outcome of (k <= n),Condition is True, execute loop;Value of k, n are unchanged.
Is (k <= n)?
Is (1 <= 6)?True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 8
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 1)(A[k] = 2)
RHS = 0 + 2 = 2
CPU
Sum_SF 0 Sum ?
k 1
Calculate value of expression on the right-hand-side;Note: In A[k], k is used as an index (subscript) to the array A
Calculate value of expression on the right-hand-side;Note: In A[k], k is used as an index (subscript) to the array A
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 9
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 1)(A[k] = 2)
RHS = 0 + 2 = 2Sum_SF 2
CPU
Sum_SF 2 Sum ?
k 1
Calculate value of expression on the right-hand-side;Assign result to variable on left;
Calculate value of expression on the right-hand-side;Assign result to variable on left;
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 10
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k k + 1;k 1 + 1;
CPU
Sum_SF 2 Sum ?
k 1
This increments value of k;This increments value of k;
2
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 11
End of body of loop, loop back...
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 2 Sum ?
k 2
Reached end of the loop body.Must loop back and re-test the loop condition.(We will skip this in future loops)
Reached end of the loop body.Must loop back and re-test the loop condition.(We will skip this in future loops)
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 12
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 2 Sum ?
k 2
Testing the loop condition againwith new value of k.(This is call a pre-test loop,where testing is done beforeentering the loop.)
Testing the loop condition againwith new value of k.(This is call a pre-test loop,where testing is done beforeentering the loop.)
Is (k <= n)?
Is (2 <= 6)?True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 13
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 2)(A[k] = 5)
RHS = 2 + 5 = 7
CPU
Sum_SF 2 Sum ?
k 2
Notice that A[k] now gets thesecond number in the list;
Notice that A[k] now gets thesecond number in the list;
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 14
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 2)(A[k] = 5)
RHS = 2 + 5 = 7Sum_sf 7
CPU
Sum_SF 7 Sum ?
k 2
Notice that A[k] now gets thesecond number in the list;This number (5) is added to Sum_SF
Notice that A[k] now gets thesecond number in the list;This number (5) is added to Sum_SF
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 15
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k k + 1;k 2 + 1;
CPU
Sum_SF 7 Sum ?
k 2
This increments value of k;This increments value of k;
3
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 16
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 7 Sum ?
k 3
Testing the loop condition againwith new value of k.
Testing the loop condition againwith new value of k.
Is (3 <= 6)?
True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 17
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 3)(A[k] = 10)
Rhs = 7 + 10Sum_SF 17
CPU
Sum_SF 7 Sum ?
k 3
Now A[k] gets the third element Add A[3] to Sum_sf;
Now A[k] gets the third element Add A[3] to Sum_sf;
17
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 18
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k k + 1;k 3 + 1;
CPU
Sum_SF 17 Sum ?
k 3
increment k;increment k;
4
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 19
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 17 Sum ?
k 4
Testing the while loop conditionagain with new value of k.
Testing the while loop conditionagain with new value of k.
Is (4 <= 6)?
True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 20
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 4)(A[k] = 3)
Rhs = 17 + 3Sum_SF 20
CPU
Sum_SF 17 Sum ?
k 4
Add A[4] to Sum_SF;Add A[4] to Sum_SF;
20
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 21
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k 4 + 1;
CPU
Sum_SF 20 Sum ?
k 4
increment k;increment k;
5
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 22
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 20 Sum ?
k 5
Testing the while loop conditionagain with new value of k.
Testing the while loop conditionagain with new value of k.
Is (5 <= 6)?
True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 23
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 5)(A[k] = 12)
Rhs = 20 + 12Sum_SF 32
CPU
Sum_SF 20 Sum ?
k 5
Add A[5] to Sum_SF;Add A[5] to Sum_SF;
32
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 24
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k 5 + 1;
CPU
Sum_SF 32 Sum ?
k 5
increment k;increment k;
6
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 25
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 32 Sum ?
k 6
Testing the while loop conditionagain with new value of k.
Testing the while loop conditionagain with new value of k.
Is (6 <= 6)?
True
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 26
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
(k = 6)(A[k] = 24)
Rhs = 32 + 24Sum_SF 56
CPU
Sum_SF 32 Sum ?
k 6
Add A[6] to Sum_SF;Add A[6] to Sum_SF;
56
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 27
inside body of loop
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
k 6 + 1;
CPU
Sum_SF 56 Sum ?
k 6
increment k;increment k;
7
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 28
Test loop condition again…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 56 Sum ?
k 7
The loop condition fails (is false);Skip the body of loop.Go to the statement after the while loop;
The loop condition fails (is false);Skip the body of loop.Go to the statement after the while loop;
Is (7 <= 6)?
False
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 29
Jump out of loop, to next stmt…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 56 Sum 56
k 7
Copy value of Sum_SF to Sum Copy value of Sum_SF to Sum
Sum Sum_SF
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 30
Print statement…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_SF 56 Sum 56
k 7
Output of Algrithm Sum:
Sum is 56
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 31
End of Algorithm…
A[1]
A[2]
A[3]
A[4]
A[5]
A[6]
2
5
10
3
12
24
n 6
CPU
Sum_sf 56 Sum 56
k 7
Output of Algrithm Sum:
Sum is 56End of the Algorithm End of the Algorithm
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 32
Your Homework: On your own, execute algorithm Sum on this input.
Input to Algorithm Sum:
n = 4Array A = [4, 7, 3, 9, 2, 20, 10]
Array-Sum(A, n);
begin
Sum_SF 0;
k 1;
while (k <= n) do
Sum_SF Sum_SF + A[k];
k k + 1;
endwhile
Sum Sum_SF;
Print “Sum is”, Sum;
end;
Algorithm Array-Sum
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 33
Summary
Review: A. Did you understand the following concepts: 1. variables (storage boxes) 2. arrays (a collection of contiguous variables) 3. array index or subscript B. Can you follow the execution of
sequential instruction,the loop pre-test,repetitive statements?
© Leong Hon Wai, 2003-2009LeongHW, SoC, NUS
(UIT2201: Algorithms) Page 34
Self AssessmentSelf Test 1: (based on the example) 1. How many times was the loop executed? 2. How many times was the pre-loop test executed? 3. What is the value of k at the end of the algorithm?
Self Test 2: (for a general value of n) 1. How many times was the loop executed? 2. How many times was the pre-loop test executed? 3. What is the value of k at the end of the algorithm?
Self Test 3: (Modifying the algorithm slightly to …) 1. compute the average of the n numbers in array A. 2. the “sum-of-squares” of the n numbers.
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