DCS Lecture how to solve it Patrick Prosser
- Slides: 68
DCS Lecture how to solve it Patrick Prosser
Your Challenge Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers
That’s illegal, okay? 6 5 Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers
That’s illegal, okay? 3 3 Put a different number in each circle (1 to 8) such that adjacent circles cannot take consecutive numbers
The Puzzle • Place numbers 1 through 8 on nodes – Each number appears exactly once – No connected ? nodes have consecutive numbers ? You have 4 minutes! ? ? ?
Bill Gates asks … how do we solve it? How do we solve it?
Heuristic Search Which nodes are hardest to number? ? ? Heuristic: a rule of thumb
Heuristic Search ? ?
Heuristic Search Which are the least constraining values to use? ? ?
Heuristic Search Values 1 and 8 ? ? ? 1 8 ? ? ?
Heuristic Search Values 1 and 8 ? ? ? 1 8 ? ? Symmetry means we don’t need to consider: 8 1 ?
Inference/propagation ? ? ? 1 8 ? ? ? We can now eliminate many values for other nodes Inference/propagation: reasoning
Inference/propagation {1, 2, 3, 4, 5, 6, 7, 8} ? ? ? 1 8 ? ? ?
Inference/propagation {2, 3, 4, 5, 6, 7} ? ? ? 1 8 ? ? ?
Inference/propagation {3, 4, 5, 6} ? ? ? 1 8 ? ? ?
Inference/propagation {3, 4, 5, 6} ? ? ? 1 8 ? ? {3, 4, 5, 6} By symmetry ?
Inference/propagation ? {3, 4, 5, 6} {1, 2, 3, 4, 5, 6, 7, 8} ? ? 1 8 ? ? {3, 4, 5, 6} ?
Inference/propagation ? {3, 4, 5, 6} {2, 3, 4, 5, 6, 7} ? ? 1 8 ? ? {3, 4, 5, 6} ?
Inference/propagation ? {3, 4, 5, 6} ? ? 1 8 ? ? {3, 4, 5, 6} ?
Inference/propagation ? By symmetry {3, 4, 5, 6} ? ? 1 8 ? ? {3, 4, 5, 6} ?
Inference/propagation ? {3, 4, 5, 6} ? ? 1 8 ? {2, 3, 4, 5, 6} {3, 4, 5, 6, 7} ? ? {3, 4, 5, 6}
Inference/propagation ? {3, 4, 5, 6} ? ? 1 8 ? {2, 3, 4, 5, 6} {3, 4, 5, 6, 7} ? ? {3, 4, 5, 6} Value 2 and 7 are left in just one node’s domain
Inference/propagation 7 {3, 4, 5, 6} ? ? 1 8 {2, 3, 4, 5, 6} {3, 4, 5, 6, 7} And propagate … 2 ? ? {3, 4, 5, 6}
Inference/propagation {3, 4, 5} 7 {3, 4, 5, 6} ? ? 1 8 {2, 3, 4, 5, 6} {3, 4, 5, 6, 7} ? {3, 4, 5} And propagate … 2 ? {3, 4, 5, 6}
Inference/propagation {3, 4, 5} 7 {4, 5, 6} ? ? 1 8 {2, 3, 4, 5, 6} {3, 4, 5, 6, 7} ? {3, 4, 5} And propagate … 2 ? {4, 5, 6}
Inference/propagation {3, 4, 5} 7 {4, 5, 6} ? ? 1 8 ? ? {3, 4, 5} 2 {4, 5, 6} Guess a value, but be prepared to backtrack … Backtrack?
Inference/propagation {3, 4, 5} 7 {4, 5, 6} 3 ? 1 8 ? ? {3, 4, 5} {4, 5, 6} Guess a value, but be prepared to backtrack … 2
Inference/propagation {3, 4, 5} 7 3 ? 1 8 ? ? {3, 4, 5} And propagate … {4, 5, 6} 2
Inference/propagation {5, 6} 7 3 ? 1 8 ? ? {4, 5} And propagate … {4, 5, 6} 2
Inference/propagation {5, 6} 7 3 ? 1 8 ? ? {4, 5} Guess another value … {4, 5, 6} 2
Inference/propagation 7 3 5 1 8 ? ? {4, 5} Guess another value … {4, 5, 6} 2
Inference/propagation 7 3 5 1 8 ? ? {4, 5} And propagate … {4, 5, 6} 2
Inference/propagation 7 {4} And propagate … 3 5 1 8 ? ? {4, 6} 2
Inference/propagation 7 {4} 3 5 1 8 4 ? {4, 6} One node has only a single value left … 2
Inference/propagation 7 3 5 1 8 4 6 {6} 2
Solution! 7 3 5 1 8 4 6 2
Bill Gates says … how does a computer solve it? How does a computer solve it?
A Constraint Satisfaction Problem ? ? • Variable, vi for each node • Domain of {1, …, 8} • Constraints – All values used Alldifferent(v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8) ? – No consecutive numbers for adjoining nodes |v 1 - v 2 | > 1 |v 1 - v 3 | > 1 …
How we might input the problem to a program Viewing the problem as a “graph” with 8 “vertices” and 17 “edges”
Graph Theory?
Our Problem as a Graph 8 vertices, 17 edges vertex 0 is adjacent to vertex 1 vertex 3 is adjacent to vertex 7 0 1 2 6 7 5 4 3
By the way, Bill Gates says … Computer scientists count from zero
A Java (Constraint) Program to solve our problem
Read in the name of the input file
Make a “Problem” and attach “variables” to it Note: variables represent our vertices
Constrain all variables take different values
Read in edges and constrain corresponding variables/vertices non-consecutive
Solve the problem! Using constraint propagation and backtracking search
Print out the number of solutions
Bill Gates wants to know … Why have you read in the puzzle as a file?
So that we can be more general 0 1 2 3 8 9 10 6 5 7 4
This technology is called “constraint programming”
Constraint programming • Model problem by specifying constraints on acceptable solutions – define variables and domains – post constraints on these variables • Solve model – choose algorithm • incremental assignment / backtracking search • complete assignments / stochastic search – design heuristics It is used for solving the following kinds of problems
Some sample problems that use constraint programming • Crew scheduling (airlines) • Railway timetabling • Factory/production scheduling • Vehicle routing problems • Network design problems • Design of locks and keys • Spatial layout • workforce management • …
BT workforce management
Constraints are everywhere! • No meetings before 10 am • Network traffic < 100 Gbytes/sec • PCB width < 21 cm • Salary > 45 k Euros …
A Commercial Reality • First-tier software vendors use CP technology
Bill Gates is watching … You know, we’re doing something on this!
So, how do YOU solve it?
Computing Science at Glasgow Learn to program a computer, learn a bit of discrete maths, algorithmics, learn about hardware, security and data protection, computer graphics, information management, project management, interactive systems, computer networks, operating systems, professional issues, software engineering, machine learning, bioinformatics, grid computing … and of course constraint programming!
That was a 4 th year lecture … Constraint Programming An Introduction by example with help from Toby Walsh, Chris Beck, Barbara Smith, Peter van Beek, Edward Tsang, . . .
That’s all for now folks
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