Transcript The 2007 DCS Lecture - School of Computing Science

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}
{3,4,5,6}
?
?
1
8
?
?
{3,4,5,6}
?
Inference/propagation
?
By symmetry
{3,4,5,6}
{3,4,5,6}
?
?
1
8
?
?
{3,4,5,6}
{3,4,5,6}
?
Inference/propagation
?
{3,4,5,6}
{3,4,5,6}
?
?
1
8
?
{2,3,4,5,6}
{3,4,5,6,7}
?
?
{3,4,5,6}
{3,4,5,6}
Inference/propagation
?
{3,4,5,6}
{3,4,5,6}
?
?
1
8
?
{2,3,4,5,6}
{3,4,5,6,7}
?
?
{3,4,5,6}
{3,4,5,6}
Value 2 and 7 are left in just one node’s domain
Inference/propagation
7
{3,4,5,6}
{3,4,5,6}
?
?
1
8
{2,3,4,5,6}
{3,4,5,6,7}
And propagate …
2
?
?
{3,4,5,6}
{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}
{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(v1 v2 v3 v4 v5 v6 v7 v8)
?
– No consecutive numbers for
adjoining nodes
|v1 - v2 | > 1
|v1 - v3 | > 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 10am
• Network traffic < 100
Gbytes/sec
• PCB width < 21cm
• Salary > 45k 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 4th 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