Robot Programming with Lisp
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1 6. Search Algorithms Gayane Kazhoyan (Stuart Russell, Peter Norvig) Institute for University of Bremen
2 Contents Problem Definition Uninformed search strategies BFS Uniform-Cost DFS Depth-Limited Iterative Deepening Informed Search Greedy A* Heuristics Hill-climbing aka gradient ascent/descent Simulated annealing Organizational 2
3 Problem types Deterministic, fully observable = single-state problem Agent knows exactly which state it will be in. Solution is a sequence of actions Deterministic, non-observable = conformant problem Agent may have no idea where it is. Solution (if any) is a sequence of actions Nondeterministic, partially observable = contingency problem must perceive the world during execution solution is a contingent plan or a policy often replan during execution Unknown state space = exploration problem ( online ) 3
4 Example: vacuum world Single-state, start in #5. Solution? [Right, Vacuum] Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8} e.g., Right goes to {2, 4, 6, 8}. Solution? [Right, Vacuum, Left, Vacuum] Contingency, start in #5 Vacuum can dirty a clean carpet. Local sensing only at current location. Solution? [Right, if dirt then Vacuum]
5 Single-state problem formulation A problem is defined by four items: initial state operators (or successor function S(x)) e.g., Vacuum(x) clean room goal test path cost (additive) e.g., sum of distances, number of operators executed, etc. A solution is a sequence of operators leading from the initial state to a goal state 5
6 Example: The 8-puzzle Start State states? operators? goal test? path cost? Goal State 6
7 Example: The 8-puzzle Start State Goal State states: integer locations of tiles (ignore intermediate positions) operators: move blank left, right, up, down (ignore unjamming etc.) goal test: current state = goal state path cost: 1 per move 7
8 Example: vacuum world state space graph L R L R S S L R R L R R L L S S S S L R L R states? operators? goal test? path cost? S S 8
9 Example: vacuum world state space graph L R L R S S L R R L R R L L S S S S L R L R S states: integer dirt and robot locations (ignore dirt amounts) operators: Left, Right, Vacuum goal test: no dirt in current state path cost: 1 per operator S 9
10 Example: robotic assembly P R R R R R states? operators? goal test? path cost? 10
11 Example: robotic assembly P R R R R R states: real-valued coordinates of robot joint angles and parts of the object to be assembled operators: continuous motions of robot joints goal test: assembly object is complete path cost: time to execute 11
12 Search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) function General-Search( problem, strategy) returns a solution, or failure initialize the search tree using the initial state of problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return corresponding solution else expand the node and add the resulting nodes to the search tree end 12
13 General search example Oradea Neamt Zerind Iasi Sibiu Fagaras Vaslui Timisoara Rimnicu Vilcea Lugoj Pitesti Mehadia Urziceni Hirsova Bucharest Dobreta Craiova Giurgiu Eforie 13
14 General search example 14
15 General search example Zerind Sibiu Timisoara 15
16 General search example Zerind Sibiu Timisoara Oradea Fagaras Rimnicu Vilcea 16
17 General search example Zerind Sibiu Timisoara Oradea Fagaras Rimnicu Vilcea Sibiu Bucharest 17
18 Implementation of search algorithms function General-Search( problem, Queuing-Fn) returns a solution, or failure nodes Make-Queue(Make-Node(Initial-State[problem])) loop do if nodes is empty then return failure node Remove-Front(nodes) if Goal-Test[problem] applied to State(node) succeeds then return node nodes Queuing-Fn(nodes, Expand(node, Operators[problem])) end 18
19 Implementation contd: states vs. nodes A state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x) States do not have parents, children, depth, or path cost! parent State state Node children depth = 6 g = 6 The Expand function creates new nodes, filling in the various fields and using the Operators (or SuccessorFn) of the problem to create the corresponding states. 19
20 Search strategies A strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: completeness does it always find a solution if one exists? time complexity number of nodes generated/expanded space complexity maximum number of nodes in memory optimality does it always find a least-cost solution? Time and space complexity are measured in terms of b maximum branching factor of the search tree d depth of the least-cost solution m maximum depth of the state space (may be ) 20
21 Contents Problem Definition Uninformed search strategies BFS Uniform-Cost DFS Depth-Limited Iterative Deepening Informed Search Greedy A* Heuristics Hill-climbing aka gradient ascent/descent Simulated annealing Organizational 21
22 Uninformed search strategies Uninformed strategies use only the information available in the problem definition Uninformed search strategies are: Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search 22
23 Breadth-first search Expand shallowest unexpanded node Implementation: QueueingFn = put successors at end of queue (FIFO queue) 23
24 Breadth-first search... Zerind Sibiu Timisoara 24
25 Breadth-first search... Zerind Sibiu Timisoara Oradea 25
26 Breadth-first search... Zerind Sibiu Timisoara Oradea Oradea Rimnicu Fagaras Vilcea Lugoj 26
27 Properties of breadth-first search Complete? Time? Space? Optimal? 27
28 Properties of breadth-first search Complete: Yes Time: 1 + b + b 2 + b b d = O(b d ), i.e., exponential in d Space: O(b d ) (keeps every node in memory) Optimal: Yes (if cost = 1 per step); not optimal in general Space is the big problem; can easily generate nodes at N MB/sec. 28
29 Romania with step costs in km Zerind Timisoara 111 Oradea 140 Dobreta Lugoj Mehadia 120 Sibiu Craiova Fagaras Rimnicu Vilcea 97 Pitesti Neamt Bucharest Giurgiu Iasi Urziceni Vaslui Hirsova Eforie 29
30 Uniform-cost search Expand least-cost unexpanded node Implementation: QueueingFn = insert in order of increasing path cost (FIFO queue) 30
31 Uniform-cost search Zerind Sibiu Timisoara 31
32 Uniform-cost search Zerind Sibiu Timisoara Oradea 32
33 Uniform-cost search Zerind Sibiu Timisoara Oradea Lugoj 33
34 Properties of uniform-cost search Complete? Time? Space? Optimal? 34
35 Properties of uniform-cost search Complete: Yes, if step cost ɛ Time: # of nodes with g cost of optimal solution Space: # of nodes with g cost of optimal solution Optimal: Yes g(n) is the cost of the path up to node n. 35
36 Depth-first search Expand deepest unexpanded node Implementation: QueueingFn = insert successors at front of queue (LIFO) 36
37 Depth-first search... Zerind Sibiu Timisoara 37
38 Depth-first search... Zerind Sibiu Timisoara Oradea 38
39 Depth-first search... Zerind Sibiu Timisoara Oradea Zerind Sibiu Timisoara I.e., depth-first search can perform infinite cyclic excursions. Need a finite, non-cyclic search space (or repeated-state checking). 39
40 DFS on a depth-3 binary tree 40
41 DFS on a depth-3 binary tree 41
42 DFS on a depth-3 binary tree 42
43 DFS on a depth-3 binary tree 43
44 DFS on a depth-3 binary tree 44
45 DFS on a depth-3 binary tree, contd. 45
46 DFS on a depth-3 binary tree 46
47 DFS on a depth-3 binary tree 47
48 DFS on a depth-3 binary tree 48
49 Properties of depth-first search Complete? Time? Space? Optimal? 49
50 Properties of depth-first search Complete: No: fails in infinite-depth spaces, spaces with loops modify to avoid repeated states along path. Complete in finite spaces Time: O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space: O(bm), i.e., linear space! Optimal: No 50
51 Depth-limited search Depth-limited search = depth-first search with depth limit l Implementation: Nodes at depth l have no successors 51
52 Iterative deepening search function Iterative-Deepening-Search( problem) returns a solution sequence inputs: problem, a problem for depth 0 to do result Depth-Limited-Search( problem, depth) if result cutoff then return result end 52
53 Iterative deepening search l = 0 53
54 Iterative deepening search l = 1 54
55 Iterative deepening search l = 1 Zerind Sibiu Timisoara 55
56 Iterative deepening search l = 2 56
57 Iterative deepening search l = 2 Zerind Sibiu Timisoara 57
58 Iterative deepening search l = 2 Zerind Sibiu Timisoara Oradea 58
59 Iterative deepening search l = 2 Zerind Sibiu Timisoara Oradea Oradea Fagaras Rimnicu Vilcea 59
60 Iterative deepening search l = 2 Zerind Sibiu Timisoara Oradea Oradea Fagaras Rimnicu Vilcea Lugoj 60
61 Properties of iterative deepening search Complete? Time? Space? Optimal? 61
62 Properties of iterative deepening search Complete: Yes Time: (d + 1)b 0 + db 1 + (d 1)b b d = O(b d ) Space: O(bd) Optimal: Yes, if step cost = 1 Can be modified to explore uniform-cost tree. Iterative deepening search uses only linear space and not much more time than other uninformed algorithms 62
63 Contents Problem Definition Uninformed search strategies BFS Uniform-Cost DFS Depth-Limited Iterative Deepening Informed Search Greedy A* Heuristics Hill-climbing aka gradient ascent/descent Simulated annealing Organizational 63
64 Informed search Idea: use an evaluation function for each node as an estimate of desirability Expand most desirable unexpanded node Implementation: QueueingFn = insert successors in decreasing order of desirability Informed search algorithms are: greedy search A search 64
65 Romania with straight line distances in km Oradea 71 Neamt 87 Zerind Iasi Sibiu 99 Fagaras Vaslui Timisoara Rimnicu Vilcea Lugoj 97 Pitesti Hirsova Mehadia Urziceni Bucharest 120 Dobreta 90 Craiova Giurgiu Eforie Straight line distance to Bucharest 366 Bucharest 0 Craiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind
66 Greedy search Evaluation function h(n) (heuristic) = estimate of cost from n to goal E.g., h SLD (n) = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal 66
67 Greedy search example
68 Greedy search example [2] 366 Zerind Sibiu Timisoara
69 Greedy search example [3] 366 Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea
70 Greedy search example [4] 366 Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea Sibiu Bucharest
71 Properties of greedy search Complete? Time? Space? Optimal? 71
72 Properties of greedy search Complete: No can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... Complete in finite space with repeated-state checking. Time: O(b m ), but a good heuristic can give dramatic improvement Space: O(b m ) keeps all nodes in memory Optimal: No 72
73 A search Idea: avoid expanding paths that are already expensive Evaluation function f (n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost to goal from n f (n) = estimated total cost of path through n to goal A search uses an admissible heuristic i.e., h(n) h (n) where h (n) is the true cost from n. E.g., h SLD (n) never overestimates the actual road distance Theorem: A search is optimal 73
74 A search example
75 A search example [2] Zerind Sibiu Timisoara
76 A search example [3] Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea
77 A search example [4] Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea Craiova Pitesti Sibiu
78 A search example [5] Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea Craiova Pitesti Sibiu Rimnicu Vilcea Craiova Bucharest
79 A search example [6] Zerind Sibiu Timisoara Rimnicu Oradea Fagaras Vilcea Sibiu Bucharest Craiova Pitesti Sibiu Rimnicu Vilcea Craiova Bucharest
80 Optimality of A (standard proof) Suppose some suboptimal goal G 2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G 1. Start n G G 2 f (G 2 ) = g(g 2 ) since h(g 2 ) = 0 > g(g 1 ) since G 2 is suboptimal f (n) since h is admissible Since f (G 2 ) > f (n), A will never select G 2 for expansion 80
81 Optimality of A (more useful) Lemma: A expands nodes in order of increasing f value Gradually adds f -contours of nodes (cf. breadth-first adds layers) Contour i has all nodes with f = f i, where f i < f i+1 O Z N A 380 S 400 F I V T R L P H M U 420 B D E C G 81
82 Properties of A Complete? Time? Space? Optimal? 82
83 Properties of A Complete: Yes, unless there are infinitely many nodes with f f (G) Time: Exponential in [relative error in h length of soln.] Space: Keeps all nodes in memory Optimal: Yes cannot expand f i+1 until f i is finished 83
84 Admissible heuristics E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) Start State Goal State h 1 (S) =? h 2 (S) =? 84
85 Admissible heuristics E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) Start State Goal State h 1 (S) = 7 h 2 (S) = = 18 85
86 Dominance If h 2 (n) h 1 (n) for all n (both admissible) then h 2 dominates h 1 and is better for search Typical search costs: d = 14 d = 14 IDS = 3,473,941 nodes A (h 1 ) = 539 nodes A (h 2 ) = 113 nodes IDS = too many nodes A (h 1 ) = 39,135 nodes A (h 2 ) = 1,641 nodes 86
87 Relaxed problems Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h 1 (n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h 2 (n) gives the shortest solution Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem 87
88 Example: Travelling Salesperson Problem Find the shortest tour that visits each city exactly once Minimum spanning tree heuristic can be computed in O(n 2 ) and is a lower bound on the shortest (open) tour. 88
89 Example: n-queens Put n queens on an n n board with no two queens on the same row, column, or diagonal 89
90 Hill-climbing (or gradient ascent/descent) Like climbing Everest in thick fog with amnesia function Hill-Climbing( problem) returns a solution state inputs: problem, a problem local variables: current, a node next, a node current Make-Node(Initial-State[problem]) loop do next a highest-valued successor of current if Value[next] < Value[current] then return current current next end 90
91 Hill-climbing contd. Problem: depending on initial state, can get stuck on local maxima value global maximum local maximum states 91
92 Simulated annealing Idea: escape local maxima by allowing some bad moves but gradually decrease their size and frequency function Simulated-Annealing( problem, schedule) returns a solution state inputs: problem, a problem schedule, a mapping from time to temperature local variables: current, a node next, a node T, a temperature controlling the probability of downward steps current Make-Node(Initial-State[problem]) for t 1 to do T schedule[t] if T=0 then return current next a randomly selected successor of current E Value[next] Value[current] if E > 0 then current next else current next only with probability e E/T 92
93 Properties of simulated annealing At fixed temperature T, state occupation probability reaches Boltzman distribution: p(x) = αe E(x) kt T decreased slowly enough = always reach best state. Is this necessarily an interesting guarantee? Devised by Metropolis et al., 1953, for physical process modelling Widely used in VLSI layout, airline scheduling, etc. 93
94 Contents Problem Definition Uninformed search strategies BFS Uniform-Cost DFS Depth-Limited Iterative Deepening Informed Search Greedy A* Heuristics Hill-climbing aka gradient ascent/descent Simulated annealing Organizational 94
95 Links MIT online course on AI (available for free): Original version of these slides used at Berkeley by Russel in his AI course, based on the AI book of Norvig and Russel: 95
96 Info Summary Assignment code: REPO/assignment_6/src/*.lisp Assignment points: 7 points Assignment due: 28.11, Wednesday, 23:59 AM German time Next class: 29.11, 14:15 Next class topic: introduction to ROS. (Make sure your ROS and roslisp_repl are working.) 96
97 Q & A Thanks for your attention! 97
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