1. Chapter 5. Advanced Search
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Course Outline Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence, and Searching
Knowledge Representation
Searching
Search Methodologies
Advanced Search
Genetic Algorithms (relatively new study area)
Knowledge Represenation and Automated Reasoning
Propositinoal and Predicate Logic
Inference and Resolution for Problem Solving
Rules and Expert Systems
Part III – Machine Learning
Part IV – Advanced Topics
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3. Chapter Objectives
Given a contraint satisfaction algorithm, define variables and constraints.
Use of Most-Constrained Variable and Most-Constraining Variable heuristics for the 8-queens problem and the map coloring problem.
Use of Least-Constraining Variable heuristic for the 8-queens problem and the map coloring problem.
Use of Heuristic Repair for the 8-queens problem.
...
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4. Chapter Outline Constraint Satisfaction Search
Forward Checking
Most-Constrained Variable First
Least-Constraining Variable First
Heuristic Repair
Combinatorial Optimization Problems
How to use greedy approach – Local Search
How to improve local search
Exchanging Heuristic
Iterated Local Search
Simulated Annealing
Parallel Search
Genetic Algorithms for Search
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5. Two types of search problems
Goal is known.
A solution is to find a best path from the initial node to a goal node.
Type 2
Goal is not known.
A solution is to find a goal.
Type 1
Type 2
Is a goal known?
Yes No
What is a solution?
A best path from the initial node to a goal node.
A goal node
Examples
3 missionaries and 3 cannibals,
The tower of Hanoi
Algorithm A*
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6. 1. Constraint Satisfaction Problems
Problem: Put n queens on an n × n board with no two queens on the same row, column, or diagonal.
[Q] How to solve? Generic search with heuristics?
[Q] Local search?
From an initial board,
which one do we need to move first?
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7. Example: The 8-queens problem
Combinatorial optimization problems involve assigning values to a number of variables.
A constraint satisfaction problem (CSP) is a combinatorial optimization problem with a set of constraints.
Example: The 8-queens problem
Eight queens must be placed on
a chess board in such a way that
no two queens are on the same
diagonal, row, or column
[Q] How to model the problem?
8 variables, a through h,
with domain {0, 1, 2, …, 8}
Each variable can have a value 0 to 8.
The values must satisfy the constraints. [Q] Meaning is?
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8. Example: The 8-queens problem
Combinatorial optimization problems involve assigning values to a number of variables.
A constraint satisfaction problem (CSP) is a combinatorial optimization problem with a set of constraints.
Example: The 8-queens problem
Eight queens must be placed on
a chess board in such a way that
no two queens are on the same
diagonal, row, or column
[Q] How to solve?
[Q] Can be solved using generic search?
BFS? DFS? A*?
Local search?
What kind of search tree?
Huge space? What is the complexity?
With many variables it is essential to use heuristics.
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9. 1.1 Forward Checking with Heuristics
Huge search tree?
[Q] Do we have to visit any choice (, i.e., state or node in the search,) that conflicts with constraints?
To delete, from the set of possible future choices, any that have been rendered impossible by placing the queen on that square
If placing a queen on the board results in removing all remaining squares, then backtracking immediately. (Not like hill climbing. Difference?)
[Q] How about to use heuristics to choose the next node?
Most-Constrained Variable First
Least-Constraining Variable First
Not local search, rather similar to DFS
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10. 1.2 Most-Constrained Variables
Most-Constrained Variable First heuristic:
At each stage of the search, this heuristic involves working with the variable that has the least possible number of valid choices.
In the example of the 8-queens problem,
assigning a value to 8 variables,
a through h
E.g., currently a = 1; b = 3; c = 5, then
d has 3 choices.
e has 3 choices.
f has 1 choice.
g has 3 choices.
h has 3 choices.
The next move is
to place a queen in column f, not d.
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11. Next move? a -> b -> c -> f d has 2 choices. e has 2 choices.
g has 2 choices.
h has 1 choices.
The next move is
to place a queen in column h.
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12. Next move? a -> b -> c -> f -> h d has 1 choices.
e has 2 choices.
g has 1 choices.
The next move is
to place a queen in column d.
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13. Next move? a -> b -> c -> f -> h -> d e has 1 choices.
g has 1 choices.
The next move is
to place a queen in column e.
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14. Next move? a -> b -> c -> f -> h -> d -> e
g has 0 choice.
Ooops!
Then backtracking
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15. Next move? a -> b -> c -> f -> h -> d
e has 1 choices. (Failed)
g has 1 choices.
The next move is
to place a queen in column g.
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16. Next move? a -> b -> c -> f -> h -> d ->g
e has 0 choice.
Backtracking
a -> b -> c -> f -> h -> d
=>
a -> b -> c -> f -> h
a -> b -> c -> f -> h -> g …
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17. [Q] Is it guaranteed to find a solution? [Q] How to implement?
Term project?
[Q] Is it guaranteed to find a solution?
[Q] How to implement?
[Q] How to represent the problem?
1×(n+1) array for a state
n(n+1) different states
The start state – [0, 0, …, 0]
A goal state ???
[Q] The generic search framework would not work???
The number of possible states from the start state is n × n.
DFS with the selection strategy of
the heuristic
You may use recursion.
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18. [Q] What is the next choice?
[Q] What if there are ties?
Then, most-constraining variable heuristic for breaking the tie.
To assign a value to the variable that places the greatest number of constraints on future variables.
E.g., map coloring problem with only 3 colors
[Q] What is the next choice?
[Q] Which province is most constrained?
[Q] Is it guaranteed to find a solution?
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19. 1.3 Least-Constraining Variables
Instead of the previous two heuristics – “most-constrained variable first”, and “most-constraining variable” to break ties,
“Least-Constraining Variable First” heuristic
To assign a value to a variable that leaves the greatest number of choices for other variables.
More intuitive?
Tie breaker?
Stop and backtracking
[Q] Is it guaranteed to find a solution?
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20. This heuristic makes n-queens problems with extremely large values of n, e.g., 1000, quite solvable.
Can you try with n=8?
#constraingCells(a:1) = … = #constraingCells(a:8) = 14
#constraingCells(b) = 14 …
#constraingCells(h) = 14
a = 1;
Next move?
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21. #constraingCells(a:1) = 14 #constraingCells(b) = 14 …
Term project?
#constraingCells(a:1) = 14
#constraingCells(b) = 14 …
#constraingCells(h) = 14
a = 1;
Next move?
b:3 -> 10
b:4 -> 9
b:5 -> 8
b:6 -> 7
b:7 -> 6
b:8 -> 5
c:2 -> 11
c:4 -> 9
c:5 -> 8
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22. Project idea
The n-queens problem with the “least constraining variable first” heuristic, or with the “most constrained variable first” heuristic
Class scheduling problem
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23. 1.4 Heuristic Repair
In the previous slides, general searching is used.
A heuristic method for solving CSPs:
Generate a possible solution (randomly, or using a heuristic to generate a position that is close to a solution), and then make small changes to bring it closer to satisfying constraints.
[Q] Local search?
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24. Heuristic Repair for the 8-Queens Problem
Initial state – one queen is conflicting with another.
We’ll now move that queen to the square with the fewest conflicts.
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25. Second state – now the queen on the f column is conflicting, so we’ll move it to the square with fewest conflicts.
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26. [Q] Is it guaranteed to find a solution with the heuristic repair?
Units
[Q] Is it guaranteed to find a solution with the heuristic repair?
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27. 2. Combinatorial Optimization Problems
[Wikipedia] In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. 
In many such problems, exhaustive search is not feasible. It operates on the domain of those optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution.
Some common problems involving combinatorial optimization are the traveling salesman problem ("TSP") and the minimum spanning tree problem ("MST").
Combinatorial optimization is a subset of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, mathematics, auction theory, and software engineering.
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28. [Q] How to solve combinatorial optimization problems?
Units
Let’s start discussing the type 2 problems that we do NOT know even how to test a goal. Not like n-queens problem.
[Q] How to solve combinatorial optimization problems?
From greedy approach – local search
To advanced local search algorithms
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29. 2.1 Local Search
Like heuristic repair, local search methods start from a random state, and make small changes (improvement) until a goal state is achieved.
Most local search methods are susceptible to local maxima, like hill-climbing. Especially when we do NOT know what is the optimal.
Local search methods are known as metaheuristics. Here are very important local search methods.
Simulated annealing
Genetic algorithms
Colony optimization
Neural networks
We will discuss some ideas of how to improve local search in the following slides.
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30. Cumulative improvement
Initial
state
A bit better state
[Q] How to evaluate?
Cumulative improvement
A bit better state
[Q] How to solve?
A bit better state
Susceptable to local maxima
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31. How to Improve Local Search
Units
How to Improve Local Search
[Q] Any good idea?
Keep cumulative improvement
Give more diversity while keeping stability to search better local optimal solutions
Give some random walks to get out of local optimal solutions so that better solutions can be found.
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32. 2.2 Exchanging Heuristics
A simple local search method.
Heuristic repair is an example of an exchanging heuristic.
Involves exchanging one or more variables at each step by giving them different values
Exchanging variables until the new state becomes better
Repeat this step until a solution is found
A k-exchange involves swapping the values of k variables.
Can be used to solve the traveling salesman problem.
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33. Ramdom choice -> Diversity Cumulative improvement
Units
Initial
state
A bit better state
A bit better state
Ramdom choice -> Diversity
A bit better state
A bit better state
Cumulative improvement
A bit better state
A bit better state
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34. Units
2.3 Iterated Local Search
A local search is applied repeatedly from different initial states.
Useful in cases where the search space is extremely large, and exhaustive search will not be possible.
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35. 2.4 Simulated Annealing
A method based on the way in which metal is heated and then cooled very slowly in order to make it extremely strong.
Aims at obtaining a minimum value for some function of a large number of variables.
This value is known as the energy of the system.
Based on metropolis Monte Carlo Simulation.
Simple Monte Carlo Simulation: a method of learning information about the shape of a search space.
E.g., a square partially contained within a circle.
How to identify what proportion of the square is within
the circle?
By random sampling
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36. Algorithm: A random initial state is selected
A small random change is made.
To select a new state that makes a small change to the current state
If this change lowers the system energy, it is accepted.
If it increases the energy, it may be accepted, depending on a probability called the Boltzmann acceptance criteria:
e(-dE/T) , where
T is the current temperature, and
dE is the increase in energey that has been produced by moving the previous state to the new state.
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37. To determine whether to move to a higher energy state or not,
e(-dE/T) , where
T is the current temperature, and
dE is the increase in energey that has been produced by moving the previous state to the new state.
To determine whether to move to a higher energy state or not,
if a random number within (0, 1) < the probability above, then move.
When the process starts, T is high, meaning increases in energy are relatively likely to happen.
Over successive iterations, T lowers and increases in energy become less likely. P
T (decreasing)
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38. TRU-COMP3710 Ramdom choice -> better diversity
Initial
state
Ramdom choice -> better diversity
A bit better state
A bit better state
A bit better state
A bit worse
state
A bit better state
A bit worse state
A bit better state
A bit worse state
A bit better state
Cumulative improvement
Diversity is gradually becoming less effective.
-> better stability
A bit better state
A bit better state
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39. [Q] Why is Simulated Annealing good?
Units
[Q] Why is Simulated Annealing good?
Because the energy of the system is allowed to increase by using random selection, simulated annealing may be able to escape from local minima.
Simulated annealing is a widely used local search method for solving problems with very large number of variables.
For example: scheduling problems, traveling salesman, placing VLSI (chip) components.
[Q] How to improve?
Combination of iterated local search and simulated annealing?
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40. Units
2.5 Parallel Search
Some search methods can be easily split into tasks which can be solved in parallel. -> improved diversity
Important concepts to consider are:
Divide and conquer?
Task distribution
Load balancing
Tree ordering
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41. 2.6 Genetic Algorithms A method based on biological evolution.
Create chromosomes which represent possible solutions to a problem.
The best chromosomes in each generation are bred with each other to produce a new generation.
Much more detail on this later.
A form of local search, which has
Cumulative improvement
Better diversity
Better stability
Randomness
Quick searching
Advanced form of the combination of iterated local search and simulated annealing
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42. Start with k randomly generated states (population) – 1st generation
A state is represented as a string over a finite alphabet (often a string of 0s and 1s) (encoding)
Evaluation function (fitness function). Higher values for better states.
Multiple states (also called individuals) -> diversity
(Possibly better) Next generation
A successor state is generated by combining two parent states.
Produce the next generation of states by selection according to the evaluation with fitness function, crossover, and mutation.
Selection -> cumulative improvement
Population, Crossover -> diversity
Mutation -> random walk …
. . .
. . .
. . .
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43. [Q] How to represent the left individual (, i.e., state)?
List of column values, e.g., (3, 1, 7, 5, 8, 6, 4, 6)
Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7 / 2 = 28). The larger, the better, in this example.
[Q] Fitness values of the next states?
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44. The fitness value: 23 (= 28 – 5) The fitness value: 24
The fitness value:
23 (= 28 – 5)
The fitness value: 24
The fitness value: 23
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45. Can you evaluate them?
Fitness function: number of non-attacking pairs of queens (min = 0, max = 8 × 7/2 = 28)
24/( ) = 31%
23/( ) = 29% etc
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46. Two types of search problems
Goal is known.
A solution is to find a best path from the initial node to a goal node.
Type 2
Goal is not known.
A solution is to find a goal. Sometimes we do not know how to test.
Type 1
Type 2
Is a goal known?
Yes No
What is a solution?
A best path from the initial node to a goal node.
A goal node
Examples
3 missionaries and 3 cannibals,
The tower of Hanoi
n-Queens
TSP
Algorithm A*
CSP heuristics
Genetic algorithms
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47. Project idea Units Local search for CSP heuristics for
The traveling salesman problem
CSP heuristics for
The n-queens problem
Genetic algorithm for
The class scheduling problem
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