Lecture 16: Local Search
© J. Christopher Beck 2008
1
Outline



Local Search on Crystal Maze
Core Ideas
Local Search for Scheduling
© J. Christopher Beck 2008
2
Readings

P Ch C.5
© J. Christopher Beck 2008
3
And now for something
completely different …


Local search is
quite a different
set of ideas from
tree search
Be prepared to
“think different(ly)”
© J. Christopher Beck 2008
4
Crystal Maze

Place the numbers 1 through 8 in the
nodes such that:
Each number appears exactly once
– No connected
?
?
nodes have
consecutive
numbers
?
?
?

© J. Christopher Beck 2008
?
?
?
5
Local Search Idea

Randomly assign values (even if the
constraints are “broken”)


Initial state will probably be infeasible
Make “moves” to try to move toward a
solution
© J. Christopher Beck 2008
6
Random Initial Solution
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
7
Random Initial Solution
1?
2?
5?
3?
6?
4?
7?
8?
“Broken” constraint
Cost = # of broken constraints
© J. Christopher Beck 2008
8
What Should We Do Now?

Move:


Swap two numbers
Which two numbers?


Randomly pick a pair
The pair that will lead to the biggest
decrease in cost

Cost: number of broken constraints
© J. Christopher Beck 2008
9
What Should We Do Now?

Move:


Swap two numbers
Which two numbers?


Randomly pick a pair
The pair that will lead to the biggest
decrease in cost

Cost: number of broken constraints
© J. Christopher Beck 2008
10
Random Initial Solution
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
11
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
3
4
5
6
7
8
0
© J. Christopher Beck 2008
0
0
0
0
0
0
12
Random Initial Solution
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
13
Swap 1 & 2
2?
© J. Christopher Beck 2008
1?
5?
3?
6?
4?
7?
8?
14
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
4
5
6
7
8
0
0
0
0
0
0
15
Random Initial Solution
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
16
Swap 1 & 3
3?
© J. Christopher Beck 2008
2?
5?
1?
6?
4?
7?
8?
17
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
0
4
5
6
7
8
0
0
0
0
0
0
18
Random Initial Solution
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
19
Swap 1 & 4
4?
© J. Christopher Beck 2008
2?
5?
3?
6?
1?
7?
8?
20
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
0
4
-1
5
6
7
8
0
0
0
0
0
0
21
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
0
-1
0
4
-1
1
0
0
5
0
-1
0
0
0
6
-2
-2
0
0
0
0
7
-3
-1
-1
-1
1
-1
0
8
-2
-3
0
0
-1
0
0
0
22
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
0
-1
0
4
-1
1
0
0
5
0
-1
0
0
0
6
-2
-2
0
0
0
0
7
-3
-1
-1
-1
1
-1
0
8
-2
-3
0
0
-1
0
0
0
23
Current State
1?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
7?
8?
24
Swap 1 & 7: Cost 3
7?
2?
5?
3?
6?
4?
1?
8?
Only made 1 move (so far)!
© J. Christopher Beck 2008
25
New Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
0
0
© J. Christopher Beck 2008
3
0
0
0
4
0
2
0
0
5
2
0
0
0
0
6
0
1
1
1
1
0
7
3
1
1
1
2
0
0
8
0
1
-1
1
0
0
1
0
26
Current State
7?
© J. Christopher Beck 2008
2?
5?
3?
6?
4?
1?
8?
27
Swap 3 & 8: Cost 2
7?
© J. Christopher Beck 2008
2?
5?
8?
6?
4?
1?
3?
28
Swap 6 & 7: Cost 1
6?
© J. Christopher Beck 2008
2?
5?
8?
7?
4?
1?
3?
29
Moves

Initial State: Cost 6


Swap 1 & 7: Cost 3


calculate cost difference table
Swap 3 & 8: Cost 2


calculate cost difference table
calculate cost difference table
Swap 6 & 7: Cost 1

calculate cost difference table
© J. Christopher Beck 2008
30
Cost Difference Table
1
2
3
4
5
6
7
8
1
0
2
1
0
© J. Christopher Beck 2008
3
1
1
0
4
1
2
1
0
5
2
2
1
2
0
6
2
1
4
1
2
0
7
1
3
1
3
1
1
0
8
1
1
2
1
2
1
1
0
31
Now what?


There are no improving
moves to make!
So far, we have been “hillclimbing”
cost
© J. Christopher Beck 2008
moves
32
Now what?

Options:



Restart from a new random state
Take the least worse move (increase cost
by minimal amount)
Try a new style of local search
© J. Christopher Beck 2008
33
Now what?

For now, let’s just stop


In the next lecture, we will
see one way to deal with this
But if we stop, we haven’t
solved the problem

No guarantee that a local search will find
the best (or even any) solution!!
© J. Christopher Beck 2008
34
Core Ideas of Local Search




Start with some (random?) assignment
of variables
Look in “neighborhood” formed by
making a small change to the
assignment
Choose best neighbor
Repeat
© J. Christopher Beck 2008
35
To Define a Local Search …

Define





a way to find a starting solution
a neighborhood
a rule for choosing a neighbor
a cost function (if you don’t already have
one)
(This is a bit of a simplification – more
detail in the next lecture)
© J. Christopher Beck 2008
36
A Simple
Scheduling Problem
Jobs

Processing times
0
J0R0[15]  J0R1[50]
1
J1R1[50]  J1R0[50]
2
J2R0[30]  J2R1[15]
Solve with local search
© J. Christopher Beck 2008
37