Presentation on theme: "Spiral Galaxies Puzzles are NP-complete"— Presentation transcript:
1 Spiral Galaxies Puzzles are NP-complete Erich FriedmanStetson UniversityOctober 2, 2002
2 Spiral Galaxies Puzzles puzzles consist of grid of squares and some circles.the object is to divide a puzzle into connected groups of squares that contain one circle, which must be a center of rotational symmetry.
3 P and NPP is the set of all yes/no problems which are decidable in polynomial time.NP is the set of all yes/no problems in which a “proof” for a yes answer can be checked in polynomial time.P is a subset of NP.The question whether P=NP is one of the most important open questions in computer science.
4 NP-Completeness NP-complete problems are: A problem is NP-complete if: it is in NP, andthe existence of a polynomial time algorithm to solve it implies the existence of a polynomial time algorithm for all problems in NP.NP-complete problems are:easy enough to check in polynomial timethe hardest such problems
5 Examples of NP-Completeness Examples of NP-complete problems are:3-Colorability: Can the vertices of a graph G be colored with 3 colors so that every pair of adjacent vertices has different colors?Hamiltonicity: Does a graph G have a circuit that visits each vertex exactly once?Bin Packing: Can we divide N numbers in K sets so that each set has sum less than S?Satisfiability: Are there inputs to a Boolean circuit with AND/OR/NOT gates that make the outputs TRUE?
6 Spiral Galaxies Puzzles are NP-complete The Main Result of this talk is:The question of whether or not a given SpiralGalaxies puzzle has a solution is NP-complete.To prove this, we will build arbitrary Boolean circuits in the Spiral Galaxies universe."wires" carry truth values"junctions" in wires simulate logical gatesSince Satisfiability is NP-complete, Spiral Galaxies puzzles are also NP-complete.
7 The Construction We need: wires variables that can have either truth valueway to end a wire that forces it to be TRUENOT gateAND gateOR gateway to split the signal in a wireway to allow wires to cross
8 Wires and Signalswires are rectangles of height 2 with a circle every 3 units.a wire carries the value TRUE if the solution involves 3x2 rectangles and FALSE if the solution involves alternating 5x2 and 1x2 rectangles.A TRUE signala FALSE signal
9 Variables variables are configurations with two local solutions. A TRUE variablea FALSE variable
10 Ending Wiresto force a TRUE or FALSE signal in a wire, we can end the wire at an appropriate point.forcing a TRUE signalforcing a FALSE signal
11 NOT Gatethe NOT gate is a wire that contains a pair of circles that are only 2 units away.
16 Moving Wiresto shift a wire one unit left, we use three consecutive circles each a distance of 2.5 units from the previous one.to shift a wire one unit up, we use three consecutive circles each raised .5 units from the previous one.
17 Filling in the Holesto make the puzzle rectangular, we put a circle in every grid square that is not a part of the circuit.
18 SummaryFor any given circuit, we can find a Spiral Galaxies puzzle that can be solved if and only if there is a set of inputs to the circuit that make the output TRUE.This Satisfiability problem for circuits is known to be NP-complete.The mapping we gave is polynomial.Therefore whether or not a given Spiral Galaxies puzzle has a solution is also NP-complete.
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