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Optimal Depth-First Strategies for And-Or TreesRussell Greiner*, Ryan Hayward and Michael Molloy
University of Alberta University of Toronto *[email protected]
Yummy Yummy Sweet Sweet v [Milk [Milk & (Fruit (Fruit v Cereal)]Cereal)]Yummy! ?
What to test? Which order? …
?
?
…
A Strategy specified when to perform which tests…
S
M
C
F
++
+
- -
smcf C
S
F
+
-
nl
M
S
+
+
-
M
+
-
Given of each test,
Cost = $1Prob = 80%
Cost = $3Prob = 70%
costsuccess probability
… which strategy is best? correct minimize expected cost C[] !
Expected Cost of subtree rooted in is …
C[] = c() + Pr(+) C[+] + Pr( -) C[-]
…
For depth 1…
X1 X2 Xm
Yummy
Sweet
Fruit Cereal
Milk
i#1
i#2
…
Order s.t. )(
)Pr(
)(
)Pr(
)(
)Pr(
2
2
1
1
m
m
Xc
X
Xc
X
Xc
X
X1 X2 Xm…
Order s.t. )(
)Pr(
)(
)Pr(
)(
)Pr(
2
2
1
1
m
m
Xc
X
Xc
X
Xc
X
Why not… Depth-First Algorithm
[Simon/Kadane, AIJ, 1975]
DFADFA( AndOr tree ): strategy1. Order leaf nodes (in each “penultimate subtree”)
by P(+Xi)/c(Xi) for Or-nodes; P(-Xi)/c(Xi) for And-node 2. Compute Probability P, ExpectedCost C of this subtree3. Replace subtree with single MegaNode,
w/ prob P, cost C 4. Recur…
Yummy
Sweetp=0.3
c=1
i#1
i#2Milkp=0.8
c=1Fruit
p=0.2 c=1
Cerealp=0.7
c=1
Yummy
Sweetp=0.3
c=1
i#1
i#2Milkp=0.8
c=1
Fruitp=0.2
c=1
Cerealp=0.7
c=1
Yummy
Sweetp=0.3
c=1
i#1
Milkp=0.8
c=1
NCFp=0.76 c=1.3
Yummy
Sweetp=0.3
c=1NMCF
p=0.608 c=2.04
C before F CF
M before C+F
C
FM
S before M+C+F
C
FM
S
smcf
p=0.1824 c=2.428
Applications:1. What will Baby eat?2. Efficient medical diagnosis3. Mining for gold [Simon/Kadane, 75] “Satisficing search” 4. Competing on Game show [Garey, 73]5. Performing inference in simple expert system [Smith, 89]
ResultsResults
“Structure” Tests (in)dependent Precondition Results
Or-Tree
And-TreeIndependent Yes O(n ln n)
[Smith, 89]
Or-DAG
And-DAGIndependent No NP-hard
[Greiner, 91]
Or-Tree
And-TreeDependent No NP-hard
AndOr-Tree,
depth 2Independent Yes O(n ln n)
AndOr-Tree
(… read-once Boolean formula)
Independent No ???
POSITIVE Boolean formula
Independent No NP-hard(Exact3Cover)
Boolean formula Independent No NP-hard(SAT)
Tree (not DAG)
Independent testsArbitrary costs
Theorem:Theorem:DFA is optimal for
depth-1 trees depth-2 trees (-DNF, - CNF).
TheoremTheorem:
DFA is SUBoptimal for depth-3 trees. TheoremTheorem:
>0, DFA on unit-cost tests can be n1- worse than optimal!!
…
A1 Am… B1 Bm
… Z1 Zm…
Yummy
Sweet
FruitCereal
Milk
i#1
i#2
S
M
C
F
+ +
+
- -
smcf
C
S
F
+
-
nl
M
S
+
+
-
M
+
-
DFA returns:
C[ smcf ] = 2.428 > 2.392 = C[nl ] !!
But , non-DFA …
Note for unit-cost tests: Max possible expected cost is n. Max possible expected cost is 1.
So n times worse is worst possible!!
Why? DFA forces siblings to be considered together, so bad (low p/c) nodes can hamper good (high p/c) siblings.
Linear Strategy:A strategy is linear if it performs the tests in fixed linear order,skipping any test that will not help answer the question, given known info.
S
M
C
F
++
+
- -
smcf
C
S
F
+
-
nl
M
S
+
+
-
M+
-
Linear!
Non-Linear: Sometimes sometimes
M before S
S before M
TheoremTheorem:DFA produces a LINEAR strategy..
TheoremTheorem:>0, and-or tree whose optimal linear strategy
costs n1/3- worse than optimal!!
Precondition Model:Each intermediate node is a probabilistic test --- with its success probability and cost..
Yummy
Sweetp=0.3
c=1
i#1
Milkp=0.8
c=1
Fruitp=0.2
c=1
Cerealp=0.7
c=1
Labp=0.95
c=5.1
Laboratory test for Fruit, CerealCost to SEND TO LAB is 5.1Only 95% chance mail will succeed
TheoremTheoremDFA is optimal for
0-alternation trees 1-alternation trees
Results, wrt Preconditions…
Corollary:Corollary: DFA is SUBoptimal for depth-3 trees. >0, DFA on unit-cost tests can be
n1- worse than optimal!!
DFA DFA but … uses [Smith’89] for each “leaf subtree
Note: if internal tests have cost=0 and probability=1then DFA = DFA
Future work:Future work:
Complexity of computing strategy for ?
poly time algorithm?
Empirical studies, on real-world tasks
optimalnear-optimal
linear--
AndOr treesread-k formulae
This work was partially fundedby various grants from NSERC
