1. Artificial Intelligence
Search: 3
Ian Gent

2. Artificial Intelligence
Search 3
Part I : Best First Search
Part II: Heuristics for 8s Puzzle
Part III: A*

3. Search Reminder Search states, Search trees
Don’t store whole search trees, just the frontier

4. Best First Search All the algorithms up to now have been hard wired
I.e. they search the tree in a fixed order
use heuristics only to choose among a small number of choices
e.g. which letter to set in SAT / whether to be A or a
Would it be a good idea to explore the frontier heuristically?
I.e. use the most promising part of the frontier?
This is Best First Search

5. Best First Search
Best First Search is still an instance of general algorithm
Need heuristic score for each search state
MERGE: merge new states in sorted order of score
I.e. list always contains most promising state first
can be efficiently done if use (e.g.) heap for list
no, heaps not done for free in Lisp, Prolog.
Search can be like depth-first, breadth-first, or in-between
list can become exponentially long

6. Search in the Eights Puzzle
The Eights puzzle is different to (e.g.) SAT
can have infinitely long branches if we don’t check for loops
bad news for depth-first,
still ok for iterative deepening
Usually no need to choose variable (e.g. letter in SAT)
there is only one piece to move (the blank)
we have a choice of places to move it to
we might want to minimise length of path
in SAT just want satisfying assignment

7. Search in the Eights Puzzle
Are the hard wired methods effective?
Breadth-first very poor except for very easy problems
Depth-first useless without loop checking
not much good with it, either
Depth-bounded -- how do we choose depth bound?
Iterative deepening ok
and we can use increment = 2 (why?)
still need good heuristics for move choice
Will Best-First be ok?

8. Search in the Eights Puzzle
How can we use Best-First for the Eights puzzle?
We need good heuristic for rating states
Ideally want to find guaranteed shortest solution
Therefore need to take account of moves so far
And some way of guaranteeing no better solution elsewhere

9. Manhattan distance heuristic
There is an easy lower bound on #moves required
Just calculate how far each piece is from its goal
add up this for each piece
sum is minimum number of moves possible
This is Manhattan distance
because pieces move according to Manhattan geometry
Manhattan is not exact
Why not?
Can use it as heuristic as estimate of distance to solution
makes sense to explore apparently nearest first

10. The Eights Puzzle Inaccuracy of Manhattan Manhattan distance = ?
optimal solution = 18

11. Using Heuristics Take the Manhattan distance as an example
In Best first, order all states in list by Manhattan
In Depth first, order only new states by Manhattan
still hope to explore most promising first
In Breadth first, similarly
Heuristics important to all search algorithms
Almost all problems solved by search solved by good heuristics
Excepting small problems like 8’s puzzle

12. Manhattan Distance in 8’s
Manhattan distance in 8’s puzzle is NOT a good heuristic
It can be misled
Suppose we have a small Manhattan distance for move A
but any solution for move A must reverse move A eventually (e.g. to allow a vital move B)
We have in reality made the solution 2 moves longer
moving piece A and then putting it back again
Heuristic thinks we are closer to a solution
Infinite loops can occur in Best First + Manhattan

13. Total distance Heuristic
Can use Manhattan as basis of excellent heuristic
The result will in fact be the A* algorithm
sorry about the name
pronounced “A star”
Total distance heuristic takes account of moves so far
Manhattan distance + moves to reach this position
This must be a lower bound on #moves from start state to goal state via the current state

14. A-ghastly name-* Actually the name is just A*
The Total distance heuristic has a guarantee
1. heuristic score is guaranteed lower bound on true path cost via the current state
2. heuristic score of solution is the true cost of solution
A* = Best First + heuristic with this guarantee
A* guarantees that first solution found is optimal
Helpful because we can stop searching immediately
otherwise must continue to find possible better solutions
e.g. in Depth First for 8s puzzle.

15. The A* Guarantee A* guarantees to find optimal solution
Proof: suppose not, and we derive a contradiction
Then there is a solution with higher cost found first
must be earlier in list than precursor of optimal solution
heuristic cost = true cost (by guarantee 2)
true cost of worse solution > true cost of optimal
true cost of optimal  heuristic cost of precursor (guar. 1)
 true cost of worse solution > heuristic cost of precursor
 precursor of optimal earlier in list than worse solution
Contradiction, w5 (which was what was wanted)

16. Branch and Bound BnB is not always bed and breakfast
Branch and bound is similarly inspired to A*
Unlike A* may not guarantee optimal solution first
As in A*, look for a bound which is guaranteed lower than the true cost
Search the branching tree in any way you like
e.g. depth first (no guarantee), best first
Cut off search if cost + bound > best solution found
If heuristic is cost + bound, search = best first
then BnB = A*

17. BnB example: TSP Consider the Travelling Salesperson Problem
Branch and Bound might use depth-first search
Cost so far is sum of costs of chosen edges
Bound might be cost of following minimum spanning tree of remaining nodes
MST: tree connected to all nodes of min cost among all such trees
all routes have to visit all remaining nodes
can’t possibly beat cost of MST
Bounds often much more sophisticated
e.g. using mathematical programming optimisations

18. Summary and Next Lecture
Best first tries to explore most promising node first
In 8s puzzle, Manhattan distance is one heuristic
Total distance is much better and has guarantees
Best First + Guarantees = A*
Branch and Bound also uses guaranteed bounds
Next Lecture: Heuristics in decision problems
so far looked at heuristics for optimisation
what about when just want any old solution, e.g. SAT
Look at heuristics in these situations