Loop Invariant

1
Loop Invariant

• COMP 114
• Thursday March 7

2
Announcements

• Program 3

3
Loop Invariant

• An assertion
• In a loop
• Changes meaning every iteration

4
Example

• Assert.pre( true )
• i 0
• while (true)
• B i 0
• //Inv(Aj0 lt j lt iBj0)
• if (i 9) break
• i

5
Generalized Loop
assert(P) while(true) C assert(INV)
if(b) break D assert(Q)
6
Conditions

• Invariant must be true first time
• P C INV
• True on exit
• INV b gt Q
• True within loop body
• INV !b D C INV

7
Example 1

• Some entry Bk of B0...n is equal to 0
• Assign variable, sum, the sum of subarray
  B0..k-1.

Precondition for loop is assert (sum 0 i
0) // and (E j 0 lt j lt n Bj 0)

•  Desired postcondition is
• assert (Bi 0)
• // and sum (Sum j 0 lt j lt i Bj)
• // and (A j 0 lt j lt i Bj ! 0)

8
Loop

• while (true)
• // Invariant goes here
• if (Bi 0) break
• sum sum Bi
• i

9
Loop

• while (true)
• // sum (Sum j 0 lt j lt i Bj)
• // (A j 0 lt j lt i Bj ! 0)
• if (Bi 0) break
• sum sum Bi
• i

10
Dutch Flag Problem

• Dutch flag has been cut into many thin horizontal
  strips and all mixed up.
• We need to put it back together.

11
Computer Representation

• INPUT Array f0..n (n 1 strips) each of which
  has an R, W, or B.
• Sort the array so all Rs are first, followed by
  Ws, then the Bs.

R
R
R

W
W
W
W

B
B
B
12
Program (condensed to fit page)

• int x0 int y0 int zf.length-1
• while (true)
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

13
Loop Invariant

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B

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Conditions

• Invariant must be true first time
• P C INV

• True within loop body
• INV !b D C INV

• True on exit
• INV b gt Q

15
First Time Through

• int x0 int y0 int zf.length-1
• while (true)
• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1...f.length - 1 are B
• if (y z1) break 
• switch(fy)
• . . .

W
R
W
B
B
R
W
R
B
W
x
z
y
16
W
R
W
B
B
R
W
R
B
W
x
z
y

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

17
W
R
W
B
B
R
W
R
B
W
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

18
R
W
W
B
B
R
W
R
B
W
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

19
R
W
W
B
B
R
W
R
B
W
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

20
R
W
W
W
B
R
W
R
B
B
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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R
W
W
W
B
R
W
R
B
B
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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R
W
W
W
B
R
W
R
B
B
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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R
W
W
W
R
R
W
B
B
B
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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R
R
W
W
W
R
W
B
B
B
x
y
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

25
R
R
R
W
W
W
W
B
B
B
x
z
y

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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R
R
R
W
W
W
W
B
B
B
y
x
z

• // Inv f0...x-1 are R
• // fx...y-1 are W
• // fy...z are unknown color
• // fz1... f.length - 1 are B
• if (y z1) break 
• switch(fy)
• case 'W'
• y break
• case 'B'
• swap(f,y,z) z-- break
• default
• swap(f,x,y) x y

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Did That Help Understanding?

• Formal explanation
• Diagram helped too, of course

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Special Case While Statement
while(true) C assert(INV) if(b)
break D
while(!b) D
No C
Where does the assertion go?
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Possibilities
Repeat INV

• assert(INV)
• while(!b)
• D
• assert(INV)

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do-while
while(true) C assert(INV) if(b)
break D
do C while(!b)
No D
31
do-while Implementations
Before test
do C assert(INV) while(!b)
32
for Loop
P for(i low i lt high i) C Q
33
Conditions

• No looping (high lt low)
• (P i low high lt low) gt Q
• Before loop body
• P i low low lt high gt INV
• Executing loop
• INV i lt high C i INV
• At end
• INV i gt high gt Q

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for Loop Implementation

• for (int ilow inv() i lt high i)
• C

35
After Break

• Sorting

36
Have a Good Break