CS:4420 Artificial Intelligence
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1 S:4420 rtificial Intelligence Spring 2018 Uninformed Search esare Tinelli The University of Iowa opyright , esare Tinelli and Stuart Russell a a These notes were originally developed by Stuart Russell and are used with permission. They are copyrighted material and may not be used in other course settings outside of the University of Iowa in their current or modified form without the express written consent of the copyright holders. S:4420 Spring 2018 p.1/28
2 Readings hap. 3 of [Russell and Norvig, 2012] S:4420 Spring 2018 p.2/28
3 More on Graphs graph is a set of nodes and edges between them S F D E G graph is directed if its edges can be traversed only in a specified direction When an edge is directed from n i to n j it is uniquely identified by the pair (n i,n j ) n i is a parent (or predecessor) of n j n j is a child (or successor) of n i S:4420 Spring 2018 p.3/28
4 Directed Graphs S F D E G path, of length k 0, is a sequence (n 1,n 2 ),(n 2,n 3 ),...,(n k,n k+1 ) of k successive edges a Ex:, (S,D), (S,D),(D,E),(E,) For 1 i < j k +1, N i is a ancestor of N j ; N j is a descendant of N i graph is cyclic if it has a path starting and ending with the same node. Ex: (,D),(D,E),(E,) a Note that a path of length k > 0 contains k +1 nodes S:4420 Spring 2018 p.4/28
5 From Search Graphs to Search Trees The set of all possible paths of a graph can be represented as a tree tree is a directed acyclic graph all of whose nodes have at most one parent root of a tree is a node with no parents leaf is a node with no children The branching factor of a node is the number of its children Graphs can be turned into trees by duplicating nodes and breaking cyclic paths, if any S:4420 Spring 2018 p.5/28
6 From Graphs to Trees To unravel a graph into a tree choose a root node and trace every path from that node until you reach a leaf node or a node already in that path S depth 0 D depth 1 S F E D depth 2 D E G G E F depth 3 D... G depth 4 Note: must remember which nodes have been visited a node may get duplicated several times in the tree the tree has infinite paths if and only if the graph has infinite non-cyclic paths S:4420 Spring 2018 p.6/28
7 Tree Search lgorithms asic Idea: offline, simulated exploration of state space by generating successors of already-explored states function Tree-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 else choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add its successors to the tree done S:4420 Spring 2018 p.7/28
8 Tree Search Example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea S:4420 Spring 2018 p.8/28
9 Tree Search Example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea S:4420 Spring 2018 p.8/28
10 Tree Search Example rad Sibiu Timisoara Zerind rad Fagaras Oradea Rimnicu Vilcea rad Lugoj rad Oradea S:4420 Spring 2018 p.8/28
11 Implementation: states vs. nodes state is a (representation of) a physical configuration node is a data structure constituting part of a search tree (and includes such info as parent, children, depth, path cost g(x)) States do not have parents, children, depth, or path cost! parent, action State 5 4 Node depth = g = state S:4420 Spring 2018 p.9/28
12 Search Strategies strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: solution 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 ) S:4420 Spring 2018 p.10/28
13 Search Strategies Uninformed (or lind) Search Strategies Little or no information about the search space is available ll we know is how to generate new states and recognize a goal state Informed (or Heuristic) Search Strategies n estimate of the number of steps or the path cost from current state to goal state is available The estimate is not perfect (otherwise no search is needed!) but can help prune the search space considerably S:4420 Spring 2018 p.11/28
14 Some Uninformed Search Strategies readth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening (depth-first) search S:4420 Spring 2018 p.12/28
15 readth-first Search Strategy: Expand shallowest unexpanded node Implementation: the current set of unexpanded nodes, the fringe (or frontier), is processed as FIFO queue, i.e., new successors go at end S:4420 Spring 2018 p.13/28
16 readth-first Search Strategy: Expand shallowest unexpanded node Implementation: the current set of unexpanded nodes, the fringe (or frontier), is processed as FIFO queue, i.e., new successors go at end S:4420 Spring 2018 p.13/28
17 readth-first Search Strategy: Expand shallowest unexpanded node Implementation: the current set of unexpanded nodes, the fringe (or frontier), is processed as FIFO queue, i.e., new successors go at end S:4420 Spring 2018 p.13/28
18 readth-first Search Strategy: Expand shallowest unexpanded node Implementation: the current set of unexpanded nodes, the fringe (or frontier), is processed as FIFO queue, i.e., new successors go at end S:4420 Spring 2018 p.13/28
19 ost of readth-first Search Worst-case Time omplexity (no. of node expansions) ll nodes must be expanded to find a goal state. We must process these many nodes: O(1+b+b b d +b(b d 1) = O(b d+1 ) (exponential time) where b = maximum branching factor d = depth of shallowest goal state Note: The above assumes that the search space if finite. What if it is not? S:4420 Spring 2018 p.14/28
20 ost of readth-first Search Worst-case Space omplexity (no. of nodes in memory) ll nodes at depth d of the search tree are in the fringe when the procedure finds the goal state The number of nodes at depth d in a tree with branching factor b is O(b d+1 ) (exponential space) S:4420 Spring 2018 p.15/28
21 ost of readth-first Search Depth Nodes Time Memory millisecond 100 bytes seconds 11 kilobytes 4 11, seconds 1 megabyte minutes 111 megabytes hours 11 gigabytes days 1 terabyte years 111 terabytes years 11,111 terabytes b = 10, time/node=1ms, mem/node= 100bytes Exponential complexity problems become soon unmanageable Memory requirements are a bigger problem than time requirements S:4420 Spring 2018 p.16/28
22 Optimality of readth-first Search readth-first search is clearly complete. S:4420 Spring 2018 p.17/28
23 Optimality of readth-first Search readth-first search is clearly complete. Is it optimal? S:4420 Spring 2018 p.17/28
24 Optimality of readth-first Search readth-first search is clearly complete. Is it optimal? It depends S:4420 Spring 2018 p.17/28
25 Optimality of readth-first Search readth-first search is clearly complete. Is it optimal? It depends readth-first search always finds the shallowest goal state The path to that goal state, however, may have a higher cost than one to a deeper goal state S:4420 Spring 2018 p.17/28
26 Optimality of readth-first Search readth-first search is clearly complete. Is it optimal? It depends readth-first search always finds the shallowest goal state The path to that goal state, however, may have a higher cost than one to a deeper goal state D E ost of : 24 ost of E: 10+8 = 18 If we are looking for least-cost solutions, breadth-first is suboptimal unless all step costs are identical S:4420 Spring 2018 p.17/28
27 Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost S:4420 Spring 2018 p.18/28
28 Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost Equivalent to breadth-first if step costs all equal S:4420 Spring 2018 p.18/28
29 Uniform-ost Search: Example 71 Oradea Path cost = sum of step costs Neamt Zerind 75 rad Timisoara Dobreta 151 Sibiu 99 Fagaras 80 Rimnicu Vilcea Lugoj Mehadia 120 Pitesti ucharest raiova 90 Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie Exercise: Find cheapest route from Sibiu to ucharest S:4420 Spring 2018 p.19/28
30 Properties of Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost omplete? Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.20/28
31 Properties of Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost omplete? Yes (with step cost ǫ) Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.20/28
32 Properties of Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost omplete? Yes (with step cost ǫ) Time complexity? # of paths p with g(p) cost of optimal solution: O(b /ǫ ) where is the cost of the optimal solution Space complexity? Optimal? S:4420 Spring 2018 p.20/28
33 Properties of Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost omplete? Yes (with step cost ǫ) Time complexity? # of paths p with g(p) cost of optimal solution: O(b /ǫ ) where is the cost of the optimal solution Space complexity? Same as time complexity: O(b /ǫ ) Optimal? S:4420 Spring 2018 p.20/28
34 Properties of Uniform-ost Search ssumption: path cost function g such that g(p) g(p ) ǫ > 0 for all paths p and proper subpaths p of p Strategy: Expand least-cost unexpanded node Implementation: fringe = priority queue ordered by path cost omplete? Yes (with step cost ǫ) Time complexity? # of paths p with g(p) cost of optimal solution: O(b /ǫ ) where is the cost of the optimal solution Space complexity? Same as time complexity: O(b /ǫ ) Optimal? Yes, since nodes are expanded in increasing order of g S:4420 Spring 2018 p.20/28
35 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
36 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
37 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
38 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
39 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
40 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
41 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
42 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
43 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
44 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
45 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
46 Depth-First Search Strategy: Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front H I J K L M N O S:4420 Spring 2018 p.21/28
47 Properties of Depth-First Search omplete? Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.22/28
48 Properties of Depth-First Search omplete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.22/28
49 Properties of Depth-First Search omplete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time complexity? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space complexity? Optimal? S:4420 Spring 2018 p.22/28
50 Properties of Depth-First Search omplete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time complexity? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space complexity? O(bm), i.e., linear space! Optimal? S:4420 Spring 2018 p.22/28
51 Properties of Depth-First Search omplete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time complexity? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space complexity? O(bm), i.e., linear space! Optimal? No S:4420 Spring 2018 p.22/28
52 Depth-Limited Search = depth-first search with depth limit l, i.e., nodes at depth l have no successors function Depth-Limited-Search (problem, limit) return soln/fail/cutoff return Recursive-DLS(Make-Node(Initial-State(problem)), problem, limit) end function function Recursive-DLS (node, problem, limit) return soln/fail/cutoff cutoff-occurred := false; if (Goal-State(problem, State(node))) then return node; else if (Depth(node) == limit) then return cutoff; else for each successor in Expand(node, problem) do result := Recursive-DLS(successor, problem, limit) if (result == cutoff) then cutoff-occurred := true; else if (result!= fail) then return result; end for if (cutoff-occurred) then return cutoff; else return fail; end function S:4420 Spring 2018 p.23/28
53 Iterative Deepening Search function Iterative-Deepening-Search (problem) return soln for limit from 0 to MX-INT do result := Depth-Limited-Search(problem, limit) if (result!= cutoff) then return result end for end function S:4420 Spring 2018 p.24/28
54 Iterative Deepening Search Limit = 0 S:4420 Spring 2018 p.25/28
55 Iterative Deepening Search Limit = 1 S:4420 Spring 2018 p.25/28
56 Iterative Deepening Search Limit = 2 S:4420 Spring 2018 p.25/28
57 Iterative Deepening Search Limit = 3 H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O S:4420 Spring 2018 p.25/28
58 Properties of Iterative Deepening Search omplete? Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.26/28
59 Properties of Iterative Deepening Search omplete? Yes Time complexity? Space complexity? Optimal? S:4420 Spring 2018 p.26/28
60 Properties of Iterative Deepening Search omplete? Yes Time complexity? db 1 +(d 1)b b d = O(b d ) Space complexity? Optimal? S:4420 Spring 2018 p.26/28
61 Properties of Iterative Deepening Search omplete? Yes Time complexity? db 1 +(d 1)b b d = O(b d ) Space complexity? O(bd) Optimal? S:4420 Spring 2018 p.26/28
62 Properties of Iterative Deepening Search omplete? Yes Time complexity? db 1 +(d 1)b b d = O(b d ) Space complexity? O(bd) Optimal? Only if step costs are all identical S:4420 Spring 2018 p.26/28
63 Properties of Iterative Deepening Search omplete? Yes Time complexity? db 1 +(d 1)b b d = O(b d ) Space complexity? O(bd) Optimal? Only if step costs are all identical Numerical comparison between Iterative Deepening and readth First, with b = 10, d = 5, and solution at far right of search tree: N(ID) = ,000+20, ,000 = 123,450 N(F) = ,000+10, , ,990 = 1,111,100 Iterative deepening search is actually faster than breadth-first search! S:4420 Spring 2018 p.26/28
64 Properties of Iterative Deepening Search omplete? Yes Time complexity? db 1 +(d 1)b b d = O(b d ) Space complexity? O(bd) Optimal? Only if step costs are all identical Numerical comparison between Iterative Deepening and readth First, with b = 10, d = 5, and solution at far right of search tree: N(ID) = ,000+20, ,000 = 123,450 N(F) = ,000+10, , ,990 = 1,111,100 Iterative deepening search is actually faster than breadth-first search! It does better because other nodes at depth d are not expanded S:4420 Spring 2018 p.26/28
65 Summary of lgorithms riterion readth- Uniform- Depth- Depth- Iterative First ost First Limited Deepening omplete? Yes a Yes a,b No Yes, if l d Yes a Time b d+1 b /ǫ b m b l b d Space b d+1 b /ǫ bm bl bd Optimal? Yes c Yes No No Yes c b, branching factor d, depth of shallowest solution l, depth limit m, depth of search tree, cost of optimal solution a if b is finite b is step costs > ǫ for some ǫ > 0 c is all step costs are the same S:4420 Spring 2018 p.27/28
66 Repeated States Failure to detect repeated states can turn a linear problem into an exponential one! D (a) (b) (c) S:4420 Spring 2018 p.28/28
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