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Algorithmic Problem Solving
Lecture 3
River Crossing Problems
Review of last lecture
• State transition diagram of tumbler problem
• Abstraction, variables to describe problem.• Assignments, describe state changes.• Invariants, what does not change.• Make use of initial conditions.
Drawing a cube
0 1
A line 1st digit0=left, 1=right
00 10
01 11A square.2nd digit0=bottom1=top
000 100
010 110
001 101
011 111
A cube.3rd digit0=front,1=backFor example 010 = Left, top, front.Think of coordinates in 3 dimensions
The process is as follow;Draw a line – label each end 0, 1. Make a copy of this and join corresponding vertexes. Add a new coordinate 0 or 1. Repeat this process.
Connection to Tumbler Problem
• In the tumbler problem, we could represent the states of the tumblers.
• 1 = correct way up, 0 = upside down. The position being 1st, 2nd, 3rd tumbler.
• as e.g. 010 meaning (1st tumbler is upside down, 2nd tumbler is correct way up, 3rd tumbler is upside down).
• A vertex is a state and a line between states is an action or a transition.
• The whole picture is a state transition diagram.
State Transition Diagram for Tumble Problem 1
• 000 • 100
• 010 • 110
• 001 • 101
• 011 • 111
With 3 tumblers, only 4 allowable states out of 8.Which 4 depends on the initial configuration.Where is the MISSING TRANSITION? MISSING LINE?
State Transition Diagram for Tumble Problem 2
000 100
010 110
001 101
011 111
With 3 tumblers, only 4 allowable states out of 8.Which 4 depends on the initial configuration
0000 1000
0100 1100
0010 1010
0110 1110
0001 1001
0101 1101
0011 1011
0111 1111
A better representation
U = number of upside down tumblers. Parity = true if U is even, else is false.Two states ONLY. Parity.U = true or false.Two state transitions, which return you to the same state. There is no transition between states. This is one diagram, not two separate ones.
Parity.U = true
Parity.U = false
Action of inverting a pair of tumblers
Action of inverting a pair of tumblers
River Crossing Problems
• Brute force search means systematically trying all possibilities.
• Problem decomposition• Naming of items in a problem• State• State transition• State transition diagram
Farmer, Goat, Cabbage, Wolf
• A farmer wants to take a goat g, a cabbage c, and a wolf w across a river. The boat is only large enough to take the farmer and one item at a time. Also, the goat cannot be left alone with the cabbage, otherwise the goat will eat the cabbage. The wolf cannot be left along with the goat, otherwise the wolf will eat the goat.
• How can the farmer do this?• Do you understand the problem?
Questions
• Can you think of a good representation for this problem?
• How may states are there in total (ignoring the constrains concerning the goat)
• How many states are there, taking the constraints into contestation?
• Can you solve the problem?
Constraints
• An item is either on the left or the right river bank, i.e. one of two values (R, L) therefore think of Boolean algebra.
• f=g=c V g!=c• i.e. the f, g and c are all on same side or g
and c are on different banks. • f=g=w V g!=w• i.e. the f, g and w are all on same side or g
and w are on different banks.
All Possible States 1 – which states are allowed
F G C W
L L L L
L L L R
L L R L
L L R R
L R L L
L R L R
L R R L
L R R R
L L L L
L L L R
L L R L
L L R R
L R L L
L R L R
L R R L
L R R R
All Allowable States 1F G C W
L L L L allowed
L L L R allowed
L L R L allowed
L L R R allowed
L R L L allowed
L R L R NOT allowed
L R R L NOT allowed
L R R R NOT allowed
L L L L NOT allowed
L L L R NOT allowed
L L R L NOT allowed
L L R R allowed
L R L L allowed
L R L R allowed
L R R L allowed
L R R R allowed
State Transition Diagram 1
LLLL
RRLL
LRLL
RRRL RRLR
LLRL LLLR
RLRR
LLRR
RRRR
Representation (farmer, goat, cabbage, wolf)Notice the symmetry of the solution?
Brute Force
• Brute force – e.g. think about chess.• State space explosion, combinatorial
explosion• N individuals, 2 ^ N (two to the power of N)
size of state space.• This grows too fast.
Naming
• Distinct names are given to the 4 items.• There is no distinction between the wolf
and cabbage (look at the constraints). • The two solutions are symmetrical if move
wolf or cabbage first.• So why give cabbage and wolf separate
names?• This is part of the abstraction process.
Restatement of problem
• A farmer wants to take an A and two Bs across a river. The boat can only take the farmer and one item at a time. Also an A cannot be left alone with a B when the farmer is not present.
• How can the farmer do this?
Constraints 2
• An item is either on the left or the right river bank, i.e. one of two values (R, L) therefore think of Boolean algebra.
• f=A=B V A!=B• i.e. the f, A and B are all on same side or A
and B are on different banks.
All Possible States – which states are allowed
F A B B
L L L L
L L L R
L L R L
L L R R
L R L L
L R L R
L R R L
L R R R
L L L L
L L L R
L L R L
L L R R
L R L L
L R L R
L R R L
L R R R
All Allowable States 2F G C W
L L L L allowed
L L L R allowed
L L R L allowed
L L R R allowed
L R L L allowed
L R L R NOT allowed
L R R L NOT allowed
L R R R NOT allowed
L L L L NOT allowed
L L L R NOT allowed
L L R L NOT allowed
L L R R allowed
L R L L allowed
L R L R allowed
L R R L allowed
L R R R allowed
0000 1000
0100 1100
0010 1010
0110 1110
0001 1001
0101 1101
0011 1011
0111 1111
A hypercube for farmer problem
Getting all 4 items from LLLL to RRRR (i.e. from 0000 to 1111)i.e. from left, bottom, front, inside to Right, top, back outside
Jealous Couples
• Three couples (husband and wife) wish to cross a river. There is one boat which can carry two people maximum. The husbands are so jealous of each other, that none is willing to allow their wife to be with another man if they are not present too.
• How can all 3 couples get across the river?
State
• Describes where the individuals are.• Left | Boat | Right describes who is on the
left bank, in the boat or on the right bank.• 3H||3W = 3 husbands on left bank and 3
wives on right bank• What is the start/end state?
Actions
• An action is when individuals travel across the river.
• 3H|2W|1W the action of transporting two wives across the river.
• Potential ambiguity as we do not say the direction of movement.
Complete Solution
Last weeks problems
• Go over some of the problems• Are there any questions?
