Best-First Search! Minimizing Space or Time!! RBFS! Save space, take more time!
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1 Best-First Search! Minimizing Space or Time!! RBFS! Save space, take more time! RBFS-1
2 RBFS general properties! Similar to A* algorithm developed for heuristic search! RBFS-2
3 RBFS general properties 2! Similar to A* algorithm developed for heuristic search!» Both are recursive in the same sense! RBFS-3
4 RBFS general properties 3! Similar to A* algorithm developed for heuristic search!» Both are recursive in the same sense! Difference between A* and RBFS! RBFS-4
5 RBFS general properties 3! Similar to A* algorithm developed for heuristic search!» Both are recursive in the same sense! Difference between A* and RBFS!» A* keeps in memory all of the already generated nodes! RBFS-5
6 RBFS general properties 4! Similar to A* algorithm developed for heuristic search!» Both are recursive in the same sense! Difference between A* and RBFS!» A* keeps in memory all of the already generated nodes!» RBFS only keeps the current search path and the sibling nodes along the path! RBFS-6
7 RBFS space 2!» When does RBFS suspend the search of a subtree?! RBFS-7
8 RBFS space 3!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best! RBFS-8
9 RBFS space 3!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best!» What does it do when a subtree is suspended?! RBFS-9
10 RBFS space 3!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best!» What does it do when a subtree is suspended?! > It forgets the subtree to save space! RBFS-10
11 RBFS space 4!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best!» What does it do when a subtree is suspended?! > It forgets the subtree to save space!» What is the space complexity?! RBFS-11
12 RBFS space 5!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best!» What does it do when a subtree is suspended?! > It forgets the subtree to save space!» What is the space complexity?! > Linear the depth of the search! RBFS-12
13 RBFS space 6!» When does RBFS suspend the search of a subtree?! > When it no longer looks the best!» What does it do when a subtree is suspended?! > It forgets the subtree to save space!» What is the space complexity?! > Linear the depth of the search! Same as IDA* RBFS-13
14 RBFS memory!» When RBFS suspends searching a subtree, what does it remember?! RBFS-14
15 RBFS memory 2!» When RBFS suspends searching a subtree, what does it remember?! > An updated f-value of the root of the subtree! RBFS-15
16 Updated f-values!» How does RBFS update the f-values?! RBFS-16
17 Updated f-values 2!» How does RBFS update the f-values?! > Backing up the f-values in the same way as A* does! RBFS-17
18 f-value notation! Static f-value!» f(n)! > Value returned by the evaluation function! > Always the same! RBFS-18
19 f-value notation 2! Static f-value!» f(n)! > Value returned by the evaluation function! > Always the same! Backed-up value!» F(N)! > Changes during the search! Depends upon descendants of N RBFS-19
20 F(N) definition! RBFS backs up f-values in the same way as A*!» How is F(N) defined?! RBFS-20
21 F(N) definition 2! RBFS backs up f-values in the same way as A*!» How is F(N) defined?! > If N has never been expanded?! RBFS-21
22 F(N) definition 3! RBFS backs up f-values in the same way as A*!» How is F(N) defined?! > If N has never been expanded?! F(N) = f(n) RBFS-22
23 F(N) definition 4! RBFS backs up f-values in the same way as A*!» How is F(N) defined?! > If N has never been expanded?! F(N) = f(n) > If N has been expanded?! RBFS-23
24 F(N) definition 5! RBFS backs up f-values in the same way as A*!» How is F(N) defined?! > If N has never been expanded?! F(N) = f(n)! > If N has been expanded?! f ( T ) = min ( F ( S j ) ) Where S j are the subtrees of N RBFS-24
25 Subtree exploration!» How does RBFS explore subtrees?! RBFS-25
26 Subtree exploration 2!» How does RBFS explore subtrees?! > As in A*, within a given f-bound! RBFS-26
27 Subtree exploration 3!» How does RBFS explore subtrees?! > As in A*, within a given f-bound!» How is the bound determined?! RBFS-27
28 RBFS subtree exploration 4!» How does RBFS explore subtrees?! > As in A*, within a given f-bound!» How is the bound determined?! > From the F-values of the siblings along the current search path! RBFS-28
29 Subtree exploration 5!» How does RBFS explore subtrees?! > As in A*, within a given f-bound!» How is the bound determined?! > From the F-values of the siblings along the current search path! > The smallest F-value! The closest competitor RBFS-29
30 Subtree exploration 6! Suppose N is currently the best node! RBFS-30
31 Subtree exploration 7! Suppose N is currently the best node! > N is expanded! RBFS-31
32 Subtree exploration 8! Suppose N is currently the best node! > N is expanded! > N s children are expanded! RBFS-32
33 Subtree exploration 9! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! RBFS-33
34 Subtree exploration 10! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! > F(N) > Bound! RBFS-34
35 Subtree exploration 10! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! > F(N) > Bound!» Then what happens?! RBFS-35
36 Subtree exploration 11! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! > F(N) > Bound!» Then what happens?! > Nodes below N are forgotten! RBFS-36
37 Subtree exploration 12! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! > F(N) > Bound!» Then what happens?! > Nodes below N are forgotten! > N s F-value is updated! RBFS-37
38 Subtree exploration 13! Suppose N is currently the best node! > N is expanded! > N s children are expanded!» Until when?! > F(N) > Bound!» Then what happens?! > Nodes below N are forgotten! > N s F-value is updated! > RBFS selects which node to expand next! RBFS-38
39 F-value inheritance! F-values can be inherited from a node s parents! RBFS-39
40 F-value inheritance 2! F-values can be inherited from a node s parents! Let N be a node about to be expanded! RBFS-40
41 F-value inheritance 3! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded! RBFS-41
42 F-value inheritance 4! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children! RBFS-42
43 F-value inheritance 5! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! RBFS-43
44 F-value inheritance 6! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again! RBFS-44
45 F-value inheritance 7! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again!» Compute f(n k )! RBFS-45
46 F-value inheritance 8! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again!» Compute f(n k )!» F(N k ) = max ( F(N), f(n k ) )! RBFS-46
47 F-value inheritance 9! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again!» Compute f(n k )!» F(N k ) = max ( F(N), f(n k ) )! > N k s F-value can be inherited from N! RBFS-47
48 F-value inheritance 10! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again!» Compute f(n k )!» F(N k ) = max ( F(N), f(n k ) )! > N k s F-value can be inherited from N! N k was generated earlier RBFS-48
49 F-value inheritance 11! F-values can be inherited from a node s parents! Let N be a node about to be expanded!» If F(N) > f(n) then N had already been expanded!» F(N) was determined from N s children!» Children have been removed from memory! Suppose a child N k of N is generated again!» Compute f(n k )!» F(N k ) = max ( F(N), f(n k ) )! > N k s F-value can be inherited from N! N k was generated earlier F(N k ) was F(N), otherwise F(N) would be smaller RBFS-49
50 Fig 12.2 snapshots! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 7 f = 9 A E S is expanded!! A is found to be the best child! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-50
51 Fig 12.2 snapshots 2! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 7 f = 9 A E f = 8 f = 10 B C A is expanded with bound 9!! C has F-value 10!! Stop expansion, backup F value! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-51
52 Fig 12.2 snapshots 3! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 10 f = 9 A E Forget expansion from A!! A has backed up F value 10!! E is best to expand next! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-52
53 Fig 12.2 snapshots 4! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 10 f = 9 A E F f = 11 E is expanded with bound 10!! F has F-value 11!! Stop expansion, backup F value! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-53
54 Fig 12.2 snapshots 5! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 10 f = 11 A E Forget expansion from E!! E has backed up F value 11!! A is best to expand next! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-54
55 Fig 12.2 snapshots 6! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 10 f = 11 A E f = 10 f = 10 f = 12 B C D When B and C are! regenerated, they! inherit F value 10 from! the parent! A is expanded with bound 11!! D has F-value 12!! Stop expansion, backup F value! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-55
56 Fig 12.2 snapshots 7! S = A E 2 9 = = B F 2 11 = = C G 3 11 = = D 3 T 11 = S f = 12 f = 11 A E Forget expansion from A!! A has backed up F value 12!! E is best to expand next! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-56
57 Fig 12.2 snapshots 8! S S 7 = A 2 2 E A E f = 11 8 = = = = B F 2 11 = C G 3 11 = D 3 T 11 = T F G f = 11 f = 11 f = 11 E is expanded with bound 12!! Reach goal, search ends! f(n) in mocha = g(n) in clover + h(n) in magenta RBFS-57
58 Algorithm! Function NewF (N, F(N), Bound)! If F(N) > Bound then NewF := F(N)! else if goal(n) then exit search with success! else if N has no children then NewF := infinity! else! dead end! for each child N k of N do!!if f(n) < F(N) then F(N k ) := max( F(N), Ff(N k ))!!else F(N k ) := f(n k )!!sort children N k in increasing order of F-value!!while F(N 1 ) Bound and F(N 1 ) < infinity do!! Bound1 := min(bound, F-value of N 2!! F(N 1 ) := NewF(N 1, F(N 1 ), Bound1)!! reorder nodes N 1, N 2, according to new F(N 1 )!!end!!newf := F(N 1 )! RBFS-58
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