Invariants

FIT 1029 Algorithmic Problem Solving

Lecture 5 Invariants

Prepared by Julian Garca

COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969

WARNING This material has been reproduced and communicated to you by or on behalf of Monash University pursuant to Part VB of the Copyright Act 1968 (the Act).The material in this communication may be subject to copyright under the Act.Any further

reproduction or communication of this material by you may be the subject of copyright protection under the Act. Do not remove this notice.

Smaller examples

! Consider two glasses. One of milk andanother with the same amount of water.

! Take a spoonful of milk and mix it into the glass of water.

! Now, take a spoonful of the mixture of milk and water and put it back into the glass of milk.

milk and water problem

! Consider two glasses. One of milk andanother with the same amount of water.

! Take a spoonful of milk and mix it into the glass of water.

! Now, take a spoonful of the mixture of milk and water and put it back into the glass of milk.


Some things never change...

source: http://www.panelsonpages.com/

InvariantAn invariant is a property that does not

change.

Invariants.

Total amount of water = Total amount of milk

Amount taken out (1 spoonful)

= Amount put back in (1 spoonful)

=Amount in glass A Amount in glass B

(in the water and milk problem)

milk and water must have been equally exchanged.

Each glass ends at the same level it started.

Liquid is conserved

Selection SortInput: A list L[0, n-1] of real numbersOutput: A list sorted in ascending order

Sorts a list using selection sort

k n-1

k =0?

j index of max of L[0, k]

swap L[j] and L[k]

k k-1

Output L

2 1 5 23 5

2 1 23 55

2 12 3 55

1 2 3 552

1 2 3 552

2 3 5521

2 5 1 5 2 3

k=5

k=4

k=3

k=2

k=1

k=0

2 1 5 23 5

2 1 23 55

2 12 3 55

1 2 3 552

1 2 3 552

2 3 5521

2 5 1 5 2 3k=5

k=4

k=3

k=2

k=1

k=0

Any invariants?

L[k+1, n-1]

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

Prims algorithmInput: A graph G, simple and connectedOutput: A spanning tree of G

Finds a spanning tree of a graph G

Put the vertex in the tree

is there a vertex not in the tree?

Output tree

Choose a vertex in the Graph

Choose and edge between the tree

and a vertex not in the tree.

add the edge to the tree

yes

no



Output tree





yes

no1

2 3 45

32

5

1

4



Output tree





yes

no

Any invariants? always have a tree

Train yourself to look for invariants

Cutting the Chocolate BlockA chocolate block is

divided into squares by horizontal and vertical grooves. The object is to cut the chocolate block into individual squares.

Assume each cut is made on a single piece along a groove. How many cuts are needed?

2006-2009 Chocablog.com (http://www.chocablog.com/reviews/ms-organic-fairtrade-milk-chocolate-with-rose/)

How many pieces?

How many pieces?

1 cut 2 pieces

How many pieces?

2 cuts 3 pieces

How many pieces?

3 cuts 4 pieces

How many pieces?

4 cuts 5 pieces

How many pieces?

5 cuts 6 pieces

How many pieces?

How many pieces?

1 cut 2 pieces

How many pieces?

2 cuts 3 pieces

How many pieces?

3 cuts 4 pieces

How many pieces?

4 cuts 5 pieces

How many pieces?

5 cuts 6 pieces

Any invariants?

number of pieces = n +1number of cuts = n

Summing Sequences

Sum the numbers 1 to 100.

S = 1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

try it out

S = 1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

One way to do it...

Any invariants?

S = 1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

Another way to do it...

Any invariants?

101101101

101(50) = 5050101

50101

S = 1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100S =100 + 99 + 98 + 97+ ...+ 4 + 3 + 2 + 1+

2S = 101*100S = 101*50S = 5050

nXi=1

i =n(n+ 1)

2

Mutilated Grid

Lets go back a little bit...

32 W32 B

31 W31 B

30 W30 B

29 W29 B

28 W28 B

Any invariants?

Mutilated Grid

Mutilated Chessboard

30 W32 B

Cover Non-star Squares

Smaller problem?

One Dimensional Problem

Circle Problem

Using the Circle

Using the Circle

Many different circles - so many different tilings..

Sam Loyd, Cyclopedia of Puzzles ,Lamb Publishing Company, 1914

Slocum and Sonneveld (2006)

14-15 Puzzle

Consider the puzzle of sliding blocks below. Any block adjacent (left, right, above or below) the blank spot can be moved into the position of the blank spot. No diagonal moves are allowed. Can you find a series of moves so that all the blocks are in the order given, except that the 14 and 15 squares have exchanged places?

Image at: www.showmedistributors.com/items.php?id=games

Example Moves

1 2 3 45 6 7 89 10 11 1213 14 15

1 2 3 45 6 7 89 10 11 1213 14 15

1 2 3 45 6 7 89 10 12

13 14 1511

1 2 3 45 6 7 89 1213 14 1511

10

1 2 3 45 6 7 89 12

1314

151110

1 2 3 45 6 7 89 1213

141511

10

Swaps and

Moves 1 2 3 45 6 7 89 10 11 1213 14 15 16

1 2 3 45 6 7 89 10 11 1213 14 1516

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

15

1 move on the board=

1 permutation swap

1 2 3 45 6 7 89 10 11 1213 14 1516

1 2 3 45 6 7 89 10 1213 14 1511

16

1

2

3

4

5

6

7

8

9

10

16

12

13

14

11

15

1 2 3 45 6 7 89 10 11 1213 14 15 16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

15

1 2 3 45 6 7 89 10 11 1213 14 1516

1 2 3 45 6 7 89 10 1213 14 1511

16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16

15

1

2

3

4

5

6

7

8

9

10

16

12

13

14

11

15

1 2 3 45 6 7 89 10 11 1213 14 15 16

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

2 moves on the board=

2 permutation swaps

Moves and Swaps

! For an even number of moves there is an even number of swaps.

! For an odd number of moves there is an odd number of swaps.

Moves and Swaps

! For an even number of moves there is an even number of swaps.

! For an odd number of moves there is an odd number of swaps.

1 2 3 45 6 7 89 10 11 1213 15 14

! For the blank square to return to the starting position requires an even number of moves.

1 2 3 45 6 7 89 10 11 12

13 15 14

1 2 3 45 6 7 89 10 11 1213 14 15

! odd number of swaps. ! even number of moves

Some things never change...What have we learnt?

which is very useful.

Before Next Lecture

Attempt the following problem

A Torch Problem

Four people wish to cross a bridge at night. They need a torch to cross and they only have one torch between them. Also, since the bridge is narrow, they can only

cross at most two at a time, and can only travel at the speed of the slowest. The four people can cross in 1

minute, 2 minutes, 5 minutes and 10 minutes, respectively. Find a way for them to cross in the

minimum about of time.

Ando Hiroshige (http:www.hiroshige.org.uk)

Documents

Invariants