6.01: Introduction to EECS I. Optimizing a Search

Transcription

1 6.0: Introduction to CS I Optimizing a Search May 3, 0

2 Nano-Quiz Makeups Wednesday, May 4, 6-pm, everyone can makeup/retake NQ everyone can makeup/retake two additional NQs you can makeup/retake other NQs excused by Sˆ3 If you makeup/retake a NQ, the new score will replace the old score, even if the new score is lower!

3 Last Time We developed systematic ways to automate a search. xample: Find minimum distance path between points on a rectangular grid. C F G H I

4 Search xample Find minimum distance path between points on a rectangular grid. C F G H I Represent all possible paths with a tree (shown to just length 3). C G F F H F H H Find the shortest path from to I.

5 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C

6 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C

7 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C

8 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C

9 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C

10 Order Matters Replace first node in agenda by its children: C G F F H F H H step genda 0: : : C 3: C epth First Search

11 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH

12 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH

13 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH

14 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH

15 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH

16 Order Matters Replace last node in agenda by its children: C G F F H F H H step genda 0: : : G 3: G GH also epth First Search

17 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 0: : : C 3: C G

18 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 0: : : C 3: C G

19 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 0: : : C 3: C G

20 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 0: : : C 3: C G

21 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 0: : : C 3: C G

22 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda : : C 3: C G 4: C G

23 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda : C 3: C G 4: C G 5: G C CF

24 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 3: C G 4: C G 5: G C CF 6: G C CF F H

25 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 5: G C CF 6: G C CF F H 7: G C CF F H

26 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 7: G C CF F H 8: G C CF F H F H

27 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 8: G C CF F H F H 9: C CF F H F H G GH

28 Order Matters Remove first node from agenda. dd its children to end of agenda. C G F F H F H H step genda 8: G C CF F H F H 9: C CF F H F H G GH readth First Search

29 Order Matters Replace last node by its children (depth-first search): implement with stack (last-in, first-out). Remove first node from agenda. dd its children to the end of the agenda (breadth-first search): implement with queue (first-in, first-out).

30 Today Generalize search framework uniform cost search. Improve search efficiency heuristics.

31 ction Costs Some actions can be more costly than others. Compare navigating from to I on two grids. C 5 C 5 F F G H I G H I Modify search algorithms to account for action costs Uniform Cost Search

32 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda:

33 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda:

34 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda: C C

35 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda: C G C G

36 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda: C G F C G CF

37 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda: C G F H C G CF H

38 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited genda: C G F H C G CF H

39 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited C G F H I genda: C G CF H CFI

40 readth-first with ynamic Programming First consider actions with equal costs. C F G H I Visited C G F H I genda: C G CF H CFI

41 readth-first with ynamic Programming Notice that we expand nodes in order of increasing path length. C F G H I Visited C G F H I genda: 0 C G CF H CFI 3 3 4

42 readth-first with ynamic Programming This algorithm fails if path costs are not equal. C 5 5 F G H I Visited C G F H I genda: 0 C 6 G CF H CFI 3 Nodes are not expanded in order of increasing path length.

43 Uniform Cost Search ssociate action costs with actions. numerate paths in order of their total path cost. Find the path with the smallest path cost = sum of action costs along the path. implement agenda with priority queue.

44 Priority Queue Same basic operations as stacks and queues, with two differences: items are pushed with numeric score: the cost. popping returns the item with the smallest cost.

45 Priority Queue Push with cost, pop smallest cost first. >>> pq = PQ() >>> pq.push( a, 3) >>> pq.push( b, 6) >>> pq.push( c, ) >>> pq.pop() c >>> pq.pop() a

46 Priority Queue Simple implementation using lists. class PQ: def init (self): self.data = [] def push(self, item, cost): self.data.append((cost, item)) def pop(self): (index, cost) = util.argmaxindex(self.data, lambda (c, x): -c) return self.data.pop(index)[] # just return the data item def empty(self): return len(self.data) == 0 The pop operation in this implementation can take time proportional to the number of nodes (in the worst case). [There are better algorithms!]

47 Search Node class SearchNode: def init (self, action, state, parent, actioncost): self.state = state self.action = action self.parent = parent if self.parent: self.cost = self.parent.cost + actioncost else: self.cost = actioncost def path(self): if self.parent == None: return [(self.action, self.state)] else: return self.parent.path() + [(self.action, self.state)] def inpath(self, s): if s == self.state: return True elif self.parent == None: return False else: return self.parent.inpath(s)

48 Uniform Cost Search def ucsearch(initialstate, goaltest, actions, successor): startnode = SearchNode(None, initialstate, None, 0) if goaltest(initialstate): return startnode.path() agenda = PQ() agenda.push(startnode, 0) while not agenda.empty(): parent = agenda.pop() if goaltest(parent.state): return parent.path() for a in actions: (news, cost)= successor(parent.state, a) if not parent.inpath(news): newn = SearchNode(a, news, parent, cost) agenda.push(newn, newn.cost) return None goaltest was previously performed when children pushed on agenda. Here, we must defer goaltest until all children are pushed (since a later child might have a smaller cost). The goaltest is implemented during subsequent pop.

49 ynamic Programming Principle The shortest path from X to Z that goes through Y is made up of the shortest path from X to Y and the shortest path from Y to Z. We only need to remember the shortest path from the start state to each other state! Want to remember shortest path to Y. Therefore, defer remembering Y until all of its siblings are considered (similar to issue with goaltest) i.e., remember expansions instead of visits.

50 ucsearch with ynamic Programming def ucsearch(initialstate, goaltest, actions, successor): startnode = SearchNode(None, initialstate, None, 0) if goaltest(initialstate): return startnode.path() agenda = PQ() agenda.push(startnode, 0) expanded = {} while not agenda.empty(): parent = agenda.pop() if not expanded.has_key(parent.state): expanded[parent.state] = True if goaltest(parent.state): return parent.path() for a in actions: (news, cost) = successor(parent.state, a) if not expanded.has_key(news): newn = SearchNode(a, news, parent, cost) agenda.push(newn, newn.cost) return None

51 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0

52 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0

53 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0 C 6

54 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0 C 6 G

55 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0 C 6 G F 3 H 3

56 ucsearch with ynamic Programming C 5 5 F G H I xpanded: genda: 0 C 6 G F 3 H 3

57 ucsearch with ynamic Programming C 5 5 F G H I xpanded: G genda: 0 C 6 G F 3 H 3 GH 3

58 ucsearch with ynamic Programming C 5 5 F G H I xpanded: G F genda: 0 C 6 G F 3 H 3 GH 3 FC 8 FI 4

59 ucsearch with ynamic Programming C 5 5 F G H I xpanded: G F H genda: 0 C 6 G F 3 H 3 GH 3 FC 8 FI 4 HI 4

60 ucsearch with ynamic Programming C 5 5 F G H I xpanded: G F H genda: 0 C 6 G F 3 H 3 GH 3 FC 8 FI 4 HI 4

61 ucsearch with ynamic Programming C 5 5 F G H I xpanded: G F H genda: 0 C 6 G F 3 H 3 GH 3 FC 8 FI 4 HI 4

62 Conclusion Searching spaces with unequal action costs is similar to searching spaces with equal action costs. Just substitute priority queue for queue.

63 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. Is there a way to make the search process consider not just the starting point but also the goal?

64 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: genda: 0

65 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: genda: 0 F H

66 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: genda: 0 F H C

67 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: genda: 0 F H C G

68 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F genda: 0 F H C G FC FI

69 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H genda: 0 F H C G FC FI HG HI

70 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H genda: 0 F H C G FC FI HG HI

71 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C genda: 0 F H C G FC FI HG HI

72 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C genda: 0 F H C G FC FI HG HI

73 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C G genda: 0 F H C G FC FI HG HI

74 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C G genda: 0 F H C G FC FI HG HI

75 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C G genda: 0 F H C G FC FI HG HI

76 Stumbling upon the Goal Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. xample: Start at, go to I. C F G H I xpanded: F H C G genda: 0 F H C G FC FI HG HI Too much time searching paths on wrong side of starting point!

77 Heuristics Our searches so far have radiated outward from the starting point. We only notice the goal when we stumble upon it. This results because our costs are computed for just the first part of the path: from start to state under consideration. We can add heuristics to make the search process consider not just the starting point but also the goal. Heuristic: estimate the cost of the path from the state under consideration to the goal.

78 Heuristics dd Manhattan distance to complete the path to the goal. C F G H I xpanded: genda:

79 Heuristics dd Manhattan distance to complete the path to the goal. C F G H I xpanded: genda: 4 4 F H

80 Heuristics dd Manhattan distance to complete the path to the goal. C F G H I xpanded: F genda: 4 4 F H FC 4 FI

81 Heuristics dd Manhattan distance to complete the path to the goal. C F G H I xpanded: F H genda: 4 4 F H FC 4 FI HG 4 HI

82 Heuristics dd Manhattan distance to complete the path to the goal. C F G H I xpanded: F H genda: F 4 4 H FC FI 4 HG HI 4

83 * = ucsearch with Heuristics heuristic function takes input s and returns the estimated cost from state s to the goal. def ucsearch(initialstate, goaltest, actions, successor, heuristic): startnode = SearchNode(None, initialstate, None, 0) if goaltest(initialstate): return startnode.path() agenda = PQ() agenda.push(startnode, 0) expanded = { } while not agenda.empty(): n = agenda.pop() if not expanded.has_key(n.state): expanded[n.state] = True if goaltest(n.state): return n.path() for a in actions: (news, cost) = successor(n.state, a) if not expanded.has_key(news): newn = SearchNode(a, news, n, cost) agenda.push(newn, newn.cost + heuristic(news)) return None

84 dmissible Heuristics n admissible heuristic always underestimates the actual distance. If the heuristic is larger than the actual cost from s to goal, then the best solution may be missed not acceptable! If the heuristic is smaller than the actual cost, the search space will be larger than necessary not desireable, but right answer. The ideal heuristic should be as close as possible to actual cost (without exceeding it) easy to calculate * is guaranteed to find shortest path if heuristic is admissible.

85 Check Yourself Consider three heuristic functions for the eight puzzle : a. 0 b. number of tiles out of place c. sum over tiles of Manhattan distances to their goals Let M i = # of moves in the best solution using heuristic i Let i = # of states expanded during search with heuristic i Which of the following statements is/are true?. M a = M b = M c. a = b = c 3. M a > M b > M c 4. a b c 5. the same best solution will result for all three heuristics

86 Check Yourself Heuristic a: 0 Heuristic b: number of tiles out of place Heuristic c: sum over tiles of Manhattan distances to their goals ll three heuristics are admissible each gives an optimum length solution: M a = M b = M c. heuristic c heuristic b heuristic a a b c. ifferent heuristics can consider multiple solutions with same path cost in different orders.

87 Check Yourself Consider three heuristic functions for the eight puzzle : a. 0 b. number of tiles out of place c. sum over tiles of Manhattan distances to their goals Let M i = # of moves in the best solution using heuristic i Let i = # of states expanded during search with heuristic i Which of the following statements is/are true?. M a = M b = M c. a = b = c 3. M a > M b > M c 4. a b c 5. the same best solution will result for all three heuristics

88 Check Yourself Results. Heuristic a: 0 Heuristic b: number of tiles out of place Heuristic c: sum over tiles of Manhattan distances to their goals Heuristic visited expanded moves a 77,877,475 b 6,47 6,5 c 3,048,859 Heuristics can be very effective!

89 Summary eveloped a new class of search algorithms: uniform cost. llows solution of problems with different action costs. eveloped a new class of optimizations: heuristics. Focuses search toward the goal. Nano-Quiz Makeups: Wednesday, May 4, 6-pm, everyone can makeup/retake NQ everyone can makeup/retake two additional NQs you can makeup/retake other NQs excused by Sˆ3 If you makeup/retake a NQ, the new score will replace the old score, even if the new score is lower!

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