CSC344: AI for Games
Lecture 4: Informed search
Patrick Olivier
p.l.olivier@ncl.ac.uk
Uninformed search: summary
Breadth-first
Depth-first
Iterative deepening
Informed search strategies









Best-first search
Greedy best-first search
A* search
Heuristics
Local search algorithms
Hill-climbing search
Simulated annealing search
Local beam search
Genetic algorithms
Best-first search



Important: A search strategy is defined by
picking the order of node expansion
Uniform cost search uses cost so far: g(n)
Best first search uses:




evaluation function: f(n)
expand most desirable unexpanded node
implementation: order the nodes in fringe in
decreasing order of f(n)
Special cases:


greedy best-first search
A* search
Greedy best-first search



Evaluation function f(n) = h(n)
Heuristics are rules-of-thumb that are likely (but
not guaranteed) to help in problem solving
For example:


hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that
appears to be closest to goal
Greedy search: Arad  Bucharest
Straight line distance to
Bucharest
Greedy search: Arad  Bucharest
Greedy search: Arad  Bucharest
Greedy search: Arad  Bucharest
Greedy search: Arad  Bucharest
Properties of greedy search

Complete?



Time?


O(bm) a good heuristic gives dramatic improvement
Space?


In finite space if modified to detect repeated states
e.g., Iasi  Neamt  Iasi  Neamt  …
O(bm) keeps all nodes in memory
Optimal?

No
A* search


Idea: don’t just use estimate of distance to
the goal, but the cost of paths so far
Evaluation function: f(n) = g(n) + h(n)




g(n) = cost so far to reach n
h(n) = estimated cost from n to goal
f(n) is estimated cost of path through n to goal
Class exercise: Arad  Bucharest using A*
and the staight line distance as hSLD(n)
A* search: Arad  Bucharest
A* search: Arad  Bucharest
A* search: Arad  Bucharest
A* search: Arad  Bucharest
A* search: Arad  Bucharest
A* search: Arad  Bucharest
Admissible heuristics




A heuristic h(n) is admissible if for every node
n, h(n) ≤ h*(n), where h*(n) is the true cost to
reach the goal state from n.
An admissible heuristic never overestimates the
cost to reach the goal, i.e. it is optimistic
Example: hSLD(n) (never overestimates the
actual road distance)
Theorem: If h(n) is admissible, A* using treesearch is optimal
Proof of A* optimality of A* (1)
Suppose some suboptimal goal G2 has been generated and
is in the fringe. Let n be an unexpanded node in the fringe
such that n is on a shortest path to an optimal goal G.
f(G2) = g(G2)
g(G2) > g(G)
f(G) = g(G)
f(G2) > f(G)
since h(G2) = 0
since G2 is suboptimal
since h(G) = 0
from above
Proof of A* optimality of A* (2)
Suppose some suboptimal goal G2 has been generated and
is in the fringe. Let n be an unexpanded node in the fringe
such that n is on a shortest path to an optimal goal G.
f(G2) > f(G)
from above
h(n) ≤ h^*(n)
since h is admissible
g(n) + h(n) ≤ g(n) + h*(n)
f(n) ≤ f(G)
Hence f(G2) > f(n), and A* will never select G2 for expansion
Properties of A* search

Complete?


Yes – unless there are infinitely many nodes f(n) ≤ f(G)
Time?

Exponential unless the error in the heuristic grows no faster
that the logarithm of the actual path cost
h(n)  h* (n)  Olog h* (n) 
A* is an optimally efficient (no other algorithm is guaranteed to
expand less nodes for the same heuristic)

Space?


All nodes in memory (same as time complexity)
Optimal?

Yes
Heuristic functions

sample heuristics for 8-puzzle:




h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
h1(S) = ?
h2(S) = ?
Heuristic functions

sample heuristics for 8-puzzle:





h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
h1(S) = 8
h2(S) = 3+1+2+2+2+3+3+2 = 18
dominance:



h2(n) ≥ h1(n) for all n (both admissible)
h2 is better for search (closer to perfect)
less nodes need to be expanded
Example of dominance



randomly generate 8-puzzle problems
100 examples for each solution depth
contrast behaviour of heuristics & strategies
d
2
4
6
8
10
12
14
16
18
20
22
24
IDS
10
112
680
6384
47127
…
…
…
…
…
…
…
A*(h1)
6
13
20
39
93
227
539
1301
3056
7276
18094
39135
A*(h2)
6
12
18
25
39
73
113
211
363
676
1219
1641
Local search algorithms




In many optimisation problems, paths
are irrelevant; goal state the solution
State space = set of "complete"
configurations
Find configuration satisfying
constraints, e.g., n-queens: n queens
on an n ×n board with no two queens
on the same row, column, or diagonal
Use local search algorithms which keep
a single "current" state and try to
improve it
Hill-climbing search



"climbing Everest in thick fog with amnesia”
we can set up an objective function to be “best”
when large (perform hill climbing)
…or we can use the previous formulation of
heuristic and minimise the objective function
(perform gradient descent)
Local maxima/minina

Problem: depending on initial
state, can get stuck in local
maxima/minina
1/(1+H(n)) = 1/17
1/(1+H(n)) = 1/2
Local minima
Simulated annealing search

Idea: escape local maxima by allowing some
"bad" moves but gradually decrease their
frequency and range (VSLI layout, scheduling)
Local beam search




Keep track of k states rather than just one
Start with k randomly generated states
At each iteration, all the successors of all
k states are generated
If any one is a goal state, stop; else select
the k best successors from the complete
list and repeat.
Genetic algorithm search

A successor state is generated by combining
two parent states


Start with k randomly generated states
(population)


A state is represented as a string over a finite
alphabet (often a string of 0s and 1s)


Evaluation function (fitness function). Higher
values for better states.