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Contour interpretation of A*
Artificial Intelligence
CS 4633/6633 Artificial Intelligence
heuristic error = |h(n) - h*(n)|
● Complexity of search is exponential unless
the heuristic error grows logarithmically in
the path cost
● But heuristic search is usually at least
proportional to the path cost. Therefore,
search complexity is usually exponential.
T=
Goal
(
)
B B D −1
B −1
T = total number of nodes generated
D = depth of solution
B = effective branching factor
CS 4633/6633 Artificial Intelligence
Properties of heuristics
CS 4633/6633 Artificial Intelligence
f=16
the average number of successors that
emerge from any node
T = B0 + B1 + B2 + B3 + … + BD
CS 4633/6633 Artificial Intelligence
f(n’) = max(f(n), g(n’)+h(n’))
f=14
Effective branching factor
●
Given two admissible heuristics h1 and h2,
if h1(n) ≥ h2(n) for all n, then h1 dominates
h2 and creates a more efficient search
● Monotone heuristic = The f() value never
decreases along any path.
● Monotonicity can be maintained by the
pathmax equation:
f=12
CS 4633/6633 Artificial Intelligence
Complexity of A*
●
f=10
Start
Heuristic Functions and
Analysis of Search
How can we create heuristics?
●
An admissible heuristics can be created from a relaxed
(simplified) model of a problem. An optimal solution to the
relaxed problem is an admissible heuristic for the original
problem.
– number-of-tiles-out-of-place heuristic: the rules of the
8-puzzle are relaxed so that a tile can move anywhere
– Manhattan heuristic: the rules of the 8-puzzle are
relaxed so that a tile can move to any adjacent square
– minimal spanning tree heuristic for traveling salesman
problem: problem is relaxed so that a solution can be
any structure that connects all cities
CS 4633/6633 Artificial Intelligence