Analysis of Uninformed Search Methods

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1 nalysis of Uninformed earch Methods rian. Williams ep 21 st, 2004 lides adapted from: Tomas Lozano Perez, Russell and Norvig IM rian Williams, Fall 04 1

2 ssignments Remember: Problem et #2: imple cheme and earch due Monday, eptember 27th, Reading: Uninformed search: more of IM h. 3 rian Williams, Fall 04 2

3 Outline Recap nalysis epth-first search readth-first search Iterative deepening rian Williams, Fall 04 3

4 omplex missions must carefully: Plan complex sequences of actions chedule tight resources Monitor and diagnose behavior Repair or reconfigure hardware. Most I problems, like these, may be formulated as state space search. rian Williams, Fall 04 4

5 Problem olving: Formulate raph and Find Paths Formulate oal Formulate Problem tates oose rain stronaut Fox rain stronaut oose Fox stronaut oose rain Fox stronaut Fox oose rain stronaut oose rain Fox Operators enerate olution stronaut oose rain Fox rain Fox stronaut rain Fox oose stronaut oose equence of states stronaut oose oose stronaut Fox rain rain Fox stronaut oose rain Fox oose rain Fox oose Fox Fox stronaut oose Fox stronaut rain stronaut stronaut rain stronaut oose rain rain oose Fox rian Williams, Fall 04 5

6 Elements of lgorithm esign escription: (last Wednesday) stylized pseudo code, sufficient to analyze and implement the algorithm Implementation (last Monday). rian Williams, Fall 04 6

7 epth First earch (F) epth-first: dd path extensions to front of Q Pick first element of Q readth First earch (F) readth-first: dd path extensions to back of Q Pick first element of Q rian Williams, Fall 04 7

8 Issue: tarting at and moving top to bottom, will depth-first search ever reach? rian Williams, Fall 04 8

9 epth-first Effort can be wasted in more mild cases Q 3 1 () 2 ( ) ( ) 3 ( ) ( ) ( ) ( ) ( ) ( )( ) 5 ( ) 6 ( )( ) visited multiple times Multiple paths to, & How much wasted effort can be incurred in the worst case? rian Williams, Fall 04 9

10 How o We void Repeat Visits? Idea: Keep track of nodes already visited. o not place visited nodes on Q. oes this maintain correctness? ny goal reachable from a node that was visited a second time would be reachable from that node the first time. oes it always improve efficiency? Visits only a subset of the original paths, such that each node appears at most once at the head of a path in Q. rian Williams, Fall 04 10

11 How o We Modify imple earch lgorithm Let Q be a list of partial paths, Let be the start node and Let be the oal node. 1. Initialize Q with partial path () as only entry; 2. If Q is empty, fail. Else, pick some partial path N from Q 3. If head(n) =, return N (goal reached!) 4. Else a) Remove N from Q b) Find all children of head(n) and create all the one-step extensions of N to each child. c) dd to Q all the extended paths; d) o to step 2. rian Williams, Fall 04 11

12 imple earch lgorithm Let Q be a list of partial paths, Let be the start node and Let be the oal node. 1. Initialize Q with partial path () as only entry; set Visited = ( ) 2. If Q is empty, fail. Else, pick some partial path N from Q 3. If head(n) =, return N (goal reached!) 4. Else a) Remove N from Q b) Find all children of head(n) not in Visited and create all the one-step extensions of N to each child. c) dd to Q all the extended paths; d) dd children of head(n) to Visited e) o to step 2. rian Williams, Fall 04 12

13 Note: Testing for the oal lgorithm stops (in step 3) when head(n) =. ould have performed test in step 6, as each extended path is added to Q. ut, performing test in step 6 will be incorrect for optimal search, discussed later. We chose step 3 to maintain uniformity with these future searches. rian Williams, Fall 04 13

14 Outline nalysis epth-first search readth-first search Iterative deepening rian Williams, Fall 04 14

15 Elements of lgorithm esign escription: (last Wednesday) stylized pseudo code, sufficient to analyze and implement the algorithm Implementation (last Monday). nalysis: (today) oundness: when a solution is returned, is it guaranteed to be correct? ompleteness: is the algorithm guaranteed to find a solution when there is one? Time complexity: how long does it take to find a solution? pace complexity: how much memory does it need to perform search? rian Williams, Fall 04 15

16 haracterizing earch lgorithms Level 0 Level 1 b = 3 d = 1 m = 2 Level 2 b = maximum branching factor, number of children d = depth of the shallowest goal node m = maximum length of any path in the state space rian Williams, Fall 04 16

17 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 17

18 ase Time for epth-first case time T is proportional to number of nodes visited Level 0 Level 1 Level 2 Level m 1 b*1 b*b... b*b m-1 T dfs = [b m + b + 1]*c dfs b * T dfs = [b m+1 + b m + b]*c dfs where cdfs is time per node olve recurrence [b 1] * T dfs = [b m+1 1]*c dfs T dfs = [b m+1 1] / [b 1] *c dfs rian Williams, Fall 04 18

19 ost Using Order Notation case time T is proportional to number of nodes visited Level 0 Level 1 Level 2 1 b*1 b*b... b*b m-1 Order Notation T = O(e) if T c * e for some constant c T dfs = [b m +1 1] / [b 1] *c dfs = O(b m+1 ) ~ O(b m ) for large b rian Williams, Fall 04 19

20 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~ b m readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 20

21 ase pace for epth-first Level 0 case space dfs is proportional to maximal length of Q Level 1 Level m rian Williams, Fall 04 21

22 ase pace for epth-first Level 0 case space dfs is proportional to maximal length of Q Level 1 Level m b-1 b-1... b If a node is queued its parent and siblings have been queued, and its parent dequeued. dfs [(b-1)*m+1] *c dfs where cdfs is space per node The children of at most one sibling is expanded at each level. dfs = [(b-1)*m+1] *c dfs dfs = O(b*m) rian Williams, Fall 04 22

23 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 23

24 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 24

25 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 25

26 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 26

27 Level 0 ase Time for readth-first case time T is proportional to number of nodes visited Level 1 Level d Level d+1 Level m... onsider case where solution is at level d: rian Williams, Fall 04 27

28 ase Time for readth-first case time T is proportional to number of nodes visited Level 0 Level 1 Level d Level d+1 Level m... onsider case where solution is at level d: 1 b... b d b d+1 -b T bfs = [b d+1 + b d + b b]*c bfs ~ O(b d+1 ) for large b rian Williams, Fall 04 28

29 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first ~b d+1 case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 29

30 Level 0 ase pace for readth-first case space dfs is proportional to maximal length of Q Level 1 Level d Level d+1 rian Williams, Fall 04 30

31 ase pace for readth-first case space dfs is proportional to maximal length of Q Level 0 Level 1 Level d Level d+1 1 b... b d b d+1 -b bfs = [b d+1 - b + 1]*c bfs = O(b d+1 ) rian Williams, Fall 04 31

32 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first ~b d+1 b d+1 case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 32

33 readth-first Finds hortest Path Level 0 Level 1 Level d Level d+1 Nodes visited earlier can t include First reached ssuming each edge is length 1, other paths to must be at least as long as first found rian Williams, Fall 04 33

34 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first ~b d+1 b d+1 Yes unit lngth case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 34

35 ost and Performance Which is better, depth-first or breadth-first? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first ~b d+1 b d+1 Yes unit lngth Yes case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 35

36 The of The Which is better, depth-first or breadth-first? ssume d = m in the worst case, and call both m. Take the conservative estimate: b m+ 1 = O(b m +1) earch Method Time pace hortest Path? uaranteed to find path? epth-first b m+1 b*m No Yes for finite graph readth-first b m+1 b m Yes unit lngth Yes case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 36

37 For best first search, which runs out first time or memory? rowth for est First earch b = 10; 10,000 nodes/sec; 1000 bytes/node epth Nodes Time Memory 2 1, seconds 1 megabyte 4 111, seconds 106 megabytes minutes 10 gigabytes hours 1 terabyte days 101 terabytes years 10 petabytes ,523 years 1 exabyte rian Williams, Fall 04 37

38 How o We et The est of oth Worlds? earch hortest uaranteed to Method Time pace Path? find path? epth-first ~b m b*m No Yes for finite graph readth-first ~b d+1 b d+1 Yes unit lngth Yes case time is proportional to number of nodes visited case space is proportional to maximal length of Q rian Williams, Fall 04 38

39 Outline nalysis Iterative deepening rian Williams, Fall 04 39

40 Iterative eepening (I) Idea: Explore tree in breadth-first order, using depth-first search. earch tree to depth 1,. Level 0 Level 1 Level 2 Level 3 rian Williams, Fall 04 40

41 Iterative eepening (I) Idea: Explore tree in breadth-first order, using depth-first search. earch tree to depth 1, then 2,. Level 0 Level 1 Level 2 Level 3 rian Williams, Fall 04 41

42 Iterative eepening (I) Idea: Explore tree in breadth-first order, using depth-first search. earch tree to depth 1, then 2, then 3. Level 0 Level 1 Level 2 Level 3 rian Williams, Fall 04 42

43 peed of Iterative eepening Level 0 Level 1 Level 2 d+1 d*b... 2*b d-1 Level d 1*b d ompare speed of F vs I: T bfs = 1+b + b b d + (b d+1 b) = O(b d+2 ) T ids = (d + 1)1 + (d)b + (d - 1)b b d = O(b d+1 ) Iterative deepening performs better than breadth-first! rian Williams, Fall 04 43

44 ummary Most problem solving tasks may be encoded as state space search. asic data structures for search are graphs and search trees. epth-first and breadth-first search may be framed, among others, as instances of a generic search strategy. ycle detection is required to achieve efficiency and completeness. omplexity analysis shows that breadth-first is preferred in terms of optimality and time, while depth-first is preferred in terms of space. Iterative deepening draws the best from depth-first and breadth-first search. rian Williams, Fall 04 44

Problem Solving as State Space Search

Problem Solving as State Space Search Problem olving as tate pace earch rian. Williams 6.070 pril 5 th, 00 lides adapted from: 6.0 Tomas Lozano Perez, Russell and Norvig IM rian Williams, pring 0 elf-iagnosing Explorers courtesy of JPL rian