new cat-mouse-cheesehayes/teaching/cs361-fall13/... · 2013. 11. 20. · cat can’t catch mouse:...

Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

Mouse Wins!

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Mouse

Wins!45Tuesday, November 19, 13

Cat-Mouse-Cheese

Cat made mistake here.Should have guarded cheese!

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Cat-Mouse-Cheese

LOLcat: I can has cheez

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Cat-Mouse-Cheese

LOLcat: I can has cheezMouse: Oh $#!%

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

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Cat-Mouse-Cheese

Cat Wins!

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Cat Wins!

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing Square

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Example: Missing SquareThis position (mouse moves next) is a draw.

Cat can’t catch mouse: mouse can keep the hole between them.

Mouse can’t get cheese, because cat can get there first.

Both sides can play forever, but eventually positions will repeat.

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Game as GraphHow can we represent this game as a graph?Think back to the robot scheduling example (solved exercise 2, chap 3). Configuration graph: each vertex was a pair of locations--one for each robot.For our graph: each vertex will be vector: (mouseLoc, catLoc, mover, cheeseLeft)

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Solving: SetUp

Idea: Classify each node as one of: MWin, CWin, Draw

Initial Set-up:1: Build the whole graph.2: If catLoc==mouseLoc, node is CWin, active3: Else if mouseLoc==cheeseLoc, node is MWin, active4: Mark rest: unknown (for now).

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Solving: “Search step”Let v be any active node. We know v is CWin or MWin. Suppose it’s a CWin.For all w such that (w,v) is a valid move for Cat: Mark w as a CWin, active.For all w such that (w,v) is a valid move for Mouse:

If, from w, EVERY valid move leads to a CWin, mark w as a CWin, and make active.

Mark v as inactive.

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Solving: “Search step”Let v be any active node. We know v is CWin or MWin. Suppose it’s a MWin.For all w such that (w,v) is a valid move for Mouse: Mark w as a MWin, active.For all w such that (w,v) is a valid move for Cat:

If, from w, EVERY valid move leads to a MWin, mark w as MWin, and make active.

Mark v as inactive.

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Solving: Wrap-upSuppose we are out of active nodes.Then loop through all nodes, and change the “unknown” ones to Draw.We are now done classifying all positions.To actually play the game, build a “play forest” as follows: whenever we activate a node w, from an already active node v, mark preferredMove[w] = v.“Retrograde Analysis”

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Running TimeClaim: This algorithm runs in linear time, if we consider the configuration graph as the input.(This is quadratic time in terms of the original game board size, or quartic time in terms of n, for an nxn board.)Proof of Claim: We make 2 passes through the nodes (initial and final), and for each active node v, we explore degree(v) edges in search step. Sum of degrees = 2 #edges.

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Correctness of the Alg.

Nodes marked as Win are clearly correct.Nodes marked as Draw can move to other Draw nodes. Always safe, as any node from which the other player could move to a Winning node would have been marked as a win for that player. So they really do lead to a Draw.

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Configuration Graphnode=state of game mover: M or C

Summary: Retrograde A

C

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

Apply Inductively!Finds all CWins

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

Apply Inductively!Finds all CWins

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

Apply Inductively!Finds all CWins

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Configuration Graphnode=state of game mover: M or C

Start with won positions, say for Cat.Augmentation rules:

(1) Cat can move there(2) Mouse has no option

Summary: Retrograde A

C

CWin

M

C

C C

C

M

M

M

M

C

...

done

Apply Inductively!Finds all CWins

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Retrograde AnalysisStart with positions where cat has won.Recursively apply rules to find all CWins.Do same for MWins.All other nodes must be Draws.

Goal: # operations done per edge is O(1).Implies: Linear Time in size of configuration graph.

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Retrograde AnalysisDetail: When applying rule 2 (when all moves for mouse lead to CWin, then that node is also a CWin): potentially doing degree2 operations.Fix: use memory. For each mouse node, keep a count of how many of its neighbors are CWins. May have to update this count degree times, but then can reuse it for each neighbor. (see whiteboard)

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Retrograde AnalysisR.A. has been used for many famous examples.

Recent proof that Rubik’s cube can be solved in 20 moves or fewer.Solution of Oware game in 2002 (900 billion positions considered).Solution of Checkers in 2007.Some chess endgames, e.g. KQR vs QR.

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