General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

A ∗ search: Properties
Complete Yes (unless there are infinitely many nodes n with f(n) ≤ f(0))
Time Exponential in [relative error in h × length of solution]
Space Same as time (variant of BFS)
Optimal Yes
A ∗ expands all (some/no) nodes with f(n) < h ∗ (n)
The run-time depends on how good we approximated the real cost h ∗ with h.
c○: Michael Kohlhase 488
Since the availability of admissible heuristics is so important for informed search (particularly for
A ∗ ), let us see how such heuristics can be obtained in practice. We will look at an example, and
then derive a general procedure from that.
Admissible heuristics: Example 8-puzzle
Example 608 Let h1(n) be the number of misplaced tiles in node n (h1(S) = 6)
Example 609 Let h2(n) be the total manhattan distance from desired location of each tile.
(h2(S) = 2 + 0 + 3 + 1 + 0 + 1 + 3 + 4 = 14)
Observation 610 (Typical search costs) (IDS ˆ= iterative deepening search)
Dominance
nodes explored IDS A ∗ (h1) A ∗ (h2)
d = 14 3,473,941 539 113
d = 24 too many 39,135 1,641
c○: Michael Kohlhase 489
Definition 611 Let h1 and h2 be two admissible heuristics we say that h2 dominates h1 if
h2(n) ≥ h1(n) or all n.
Theorem 612 If h2 dominates h1, then h2 is better for search than h1.
Relaxed problems
c○: Michael Kohlhase 490
Finding good admissible heuristics is an art!
Idea: Admissible heuristics can be derived from the exact solution cost of a relaxed version
of the problem.
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