Problem-solving agents. Solving Problems by Searching. Outline. Example: Romania. Chapter 3

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1 Problem-solving agents olving Problems by earching hapter 3 function imple-problem-olving-gent( percept) returns an action static: seq, an action sequence, initially empty state, some description of the current world state goal, a goal, initially null problem, a problem formulation state Update-tate(state, percept) if seq is empty then goal Formulate-Goal(state) problem Formulate-Problem(state, goal) seq earch( problem) if seq is failure then return a null action action First(seq) seq est(seq) return action Note: this is offline problem solving; solution executed eyes closed. Online problem solving involves acting without complete knowledge. hapter 3 1 hapter 3 3 Outline Example: omania Problem-solving agents Problem types Problem formulation Example problems asic search algorithms On holiday in omania; currently in rad. Flight leaves tomorrow from ucharest Formulate goal: be in ucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, e.g., rad, ibiu, Fagaras, ucharest hapter 3 2 hapter 3 4

2 Example: omania Oradea 71 Neamt 75 Zerind 151 rad 140 ibiu 99 Fagaras Timisoara imnicu Vilcea 111 ugoj 97 Pitesti Mehadia 75 Dobreta ucharest raiova 90 Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie ingle-state, start in #5. olution?? Example: vacuum world hapter 3 5 hapter 3 7 Problem types Example: vacuum world Deterministic, fully observable = single-state problem gent knows exactly which state it will be in; solution is a sequence Non-observable = conformant problem gent may have no idea where it is; solution (if any) is a sequence Nondeterministic and/or partially observable = contingency problem percepts provide new information about current state solution is a contingent plan or a policy often interleave search, execution Unknown state space = exploration problem ( online ) ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? hapter 3 6 hapter 3 8

3 Example: vacuum world ingle-state problem formulation ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? [ight, uck, ef t, uck] ontingency Murphy s aw (non-deterministic): uck can dirty a clean carpet; start in #5 ocal sensing (partially-observable): dirt, location only, start in {#5,#7}. olution?? problem is defined by four items: initial state e.g., at rad successor function (x) = set of action state pairs e.g., (rad) = { rad Zerind,Zerind,...} goal test, can be explicit, e.g., x = at ucharest implicit, e.g., N odirt(x) path cost (additive) e.g., sum of distances, number of actions executed, etc. c(x,a,y) is the step cost, assumed to be 0 solution is a sequence of actions leading from the initial state to a goal state hapter 3 9 hapter 3 11 Example: vacuum world electing a state space ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? [ight, uck, ef t, uck] ontingency Murphy s aw (non-deterministic): uck can dirty a clean carpet; start in #5 ocal sensing (partially-observable): dirt, location only, start in {#5,#7}. olution?? [ight, while dirt do uck] [ight, if dirt then uck] eal world is absurdly complex state space must be abstracted for problem solving (bstract) state = set of real states (bstract) action = complex combination of real actions e.g., rad Zerind represents a complex set of possible routes, detours, rest stops, etc. For guaranteed realizability, any real state in rad must get to some real state in Zerind (bstract) solution = set of real paths that are solutions in the real world Each abstract action should be easier than the original problem! hapter 3 10 hapter 3 12

4 Example: vacuum world state space graph states?? actions?? goal test?? path cost?? hapter 3 13 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions?? goal test?? path cost?? hapter 3 14 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test?? path cost?? hapter 3 15 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test??: no dirt path cost?? hapter 3 16

5 Example: vacuum world state space graph Example: The 8-puzzle tart tate Goal tate states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test??: no dirt path cost??: 1 per action (0 for NoOp) states??: integer locations of tiles (ignore intermediate positions) actions?? goal test?? path cost?? hapter 3 17 hapter 3 19 Example: The 8-puzzle Example: The 8-puzzle tart tate Goal tate tart tate Goal tate states?? actions?? goal test?? path cost?? states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test?? path cost?? hapter 3 18 hapter 3 20

6 Example: The 8-puzzle Example: robotic assembly P tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given) path cost?? states??: real-valued coordinates of robot joint angles parts of the object to be assembled actions??: continuous motions of robot joints goal test??: complete assembly with no robot included! path cost??: time to execute hapter 3 21 hapter Example: The 8-puzzle 2 3 tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given) path cost??: 1 per move [Note: optimal solution of n-puzzle family is NP-hard] Tree search algorithms asic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) function Tree-earch( problem, strategy) returns a solution, or failure initialize the frontier using the initial state of problem loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier based on strategy if the node contains a goal state then return the corresponding solution else expand the chosen node and add the resulting nodes to the frontier end hapter 3 22 hapter 3 24

7 Tree search example Tree search example rad rad ibiu Timisoara Zerind ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea hapter 3 25 hapter 3 27 Tree search example states vs. nodes ibiu rad Timisoara Zerind state is a (representation of) a physical configuration node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x) tates do not have parents, children, depth, or path cost! parent, action rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea tate state Node depth = 6 g = 6 The Expand function creates new nodes, filling in the various fields and using the uccessorfn of the problem to create the corresponding states. hapter 3 26 hapter 3 28

8 earch strategies strategy is defined by picking the order of node expansion trategies 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 shallowest solution m maximum depth of the state space (may be ) Expand shallowest unexpanded node readth-first search frontier is a FIFO queue, i.e., new successors go at end hapter 3 29 hapter 3 31 Uninformed search strategies Uninformed strategies use only the information available in the problem definition readth-first search Uniform-cost search Expand shallowest unexpanded node readth-first search frontier is a FIFO queue, i.e., new successors go at end Depth-first search Depth-limited search Iterative deepening search hapter 3 30 hapter 3 32

9 readth-first search Properties of breadth-first search Expand shallowest unexpanded node frontier is a FIFO queue, i.e., new successors go at end omplete?? hapter 3 33 hapter 3 35 Expand shallowest unexpanded node readth-first search frontier is a FIFO queue, i.e., new successors go at end Properties of breadth-first search omplete?? Yes (if b is finite) Time?? hapter 3 34 hapter 3 36

10 Properties of breadth-first search omplete?? Yes (if b is finite) Time (# of visited nodes)?? 1+b+b 2 +b b d = O(b d ) Time (# of generated nodes)?? b+b 2 +b b d +(b d+1 b) = O(b d+1 ) pace?? Properties of breadth-first search omplete?? Yes (if b is finite) Time (# of generated nodes)?? b+b 2 +b b d +(b d+1 b) = O(b d+1 ) pace?? O(b d+1 ) (keeps every node in memory) Optimal?? Yes (if cost = 1 per step); not optimal in general pace is the big problem; can easily generate nodes at 100M/sec so 24hrs = 8640G. hapter 3 37 hapter 3 39 Properties of breadth-first search omplete?? Yes (if b is finite) Time (# of generated nodes)?? b+b 2 +b b d +(b d+1 b) = O(b d+1 ) pace?? O(b d+1 ) (keeps every node in memory) Optimal?? Expand least-cost unexpanded node Uniform-cost search frontier = queue ordered by path cost, lowest first Equivalent to breadth-first if step costs all equal omplete?? Yes, if step cost ǫ (lowest step cost) Time?? # of nodes with g cost of optimal solution, O(b /ǫ ) where is the cost of the optimal solution pace?? # of nodes with g cost of optimal solution, O(b /ǫ ) Optimal?? Yes nodes expanded in increasing order of g(n) hapter 3 38 hapter 3 40

11 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 41 hapter 3 43 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 42 hapter 3 44

12 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 45 hapter 3 47 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 46 hapter 3 48

13 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 49 hapter 3 51 Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front Expand deepest unexpanded node Depth-first search frontier = IFO queue, i.e., put successors at front H I J K M N O H I J K M N O hapter 3 50 hapter 3 52

14 Properties of depth-first search Properties of depth-first search omplete?? omplete?? Yes: in finite spaces No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first pace?? hapter 3 53 hapter 3 55 Properties of depth-first search omplete?? Yes: in finite spaces No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path Properties of depth-first search omplete?? Yes: in finite spaces No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path Time?? Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first pace?? O(bm), i.e., linear space! Optimal?? hapter 3 54 hapter 3 56

15 Properties of depth-first search omplete?? Yes: in finite spaces No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Iterative deepening search function Iterative-Deepening-earch( problem) returns a solution inputs: problem, a problem for depth 0 to do result Depth-imited-earch( problem, depth) if result cutoff then return result end pace?? O(bm), i.e., linear space! Optimal?? No hapter 3 57 hapter 3 59 Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors imit = 0 Iterative deepening search l = 0 ecursive implementation: function Depth-imited-earch( problem, limit) returns soln/fail/cutoff ecursive-d(make-node(initial-tate[problem]), problem, limit) function ecursive-d(node, problem, limit) returns soln/fail/cutoff cutoff-occurred? false if Goal-Test(problem, tate[node]) then return node else if Depth[node] = limit then return cutoff else for each successor in Expand(node, problem) do result ecursive-d(successor, problem, limit) if result = cutoff then cutoff-occurred? true else if result failure then return result if cutoff-occurred? then return cutoff else return failure hapter 3 58 hapter 3 60

16 Iterative deepening search l = 1 Iterative deepening search l = 3 imit = 1 imit = 3 H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O H I J K M N O hapter 3 61 hapter 3 63 Iterative deepening search l = 2 Properties of iterative deepening search imit = 2 omplete?? hapter 3 62 hapter 3 64

17 Properties of iterative deepening search omplete?? Yes Time?? Properties of iterative deepening search omplete?? Yes Time (# of generated nodes)?? db 1 +(d 1)b b d = O(b d ) pace?? O(bd) Optimal?? hapter 3 65 hapter 3 67 Properties of iterative deepening search omplete?? Yes Time (# of generated nodes)?? db 1 +(d 1)b b d = O(b d ) pace?? Properties of iterative deepening search omplete?? Yes Time (# of generated nodes)?? db 1 +(d 1)b b d = O(b d ) pace?? O(bd) Optimal?? Yes, if step cost = 1 an be modified to explore uniform-cost tree Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(ID) = ,000+20, ,000 = 123,450 N visited (F) = ,000+10, ,000 = 111,110 N generated (F) = ,000+10, , ,990 = 1,111,100 F can be modified to apply goal test when a node is generated hapter 3 66 hapter 3 68

18 ummary of algorithms riterion readth- Uniform- Depth- Depth- Iterative First ost First imited Deepening omplete? Yes Yes No Yes, if l d Yes Time (big-o) b d b /ǫ b m b l b d pace (big-o) b d b /ǫ bm bl bd Optimal? Yes Yes No No Yes Graph search function Graph-earch( problem) returns a solution, or failure frontier a list with node from the initial state of problem explored an empty set loop do if frontier is empty then return failure node emove-front(frontier) if node contains a goal state then return olution(node) add tate[node] to explored expand node add to frontier the resulting nodes that are not in explored or not in frontier or [better than the corresponding nodes in frontier in some algs] end frontier (aka fringe or open); explored (aka visited or closed) hapter 3 69 hapter 3 71 epeated states Failure to detect repeated states can turn a linear problem into an exponential one! Informed earch o far the search algorithms are uninformed independent to the problems Informed search incorporating knowledge related to the problem for guiding search D hapter 3 70 hapter 3 72

19 est-first search Idea: use an evaluation function for each node estimate of desirability Expand most desirable unexpanded node frontier is a queue sorted in decreasing order of desirability Greedy search Evaluation function h(n) (heuristic) = estimate of cost from n to the closest goal E.g., h D (n) = straight-line distance from n to ucharest Greedy search expands the node that appears to be closest to goal pecial cases: greedy search search hapter 3 73 hapter 3 75 omania with step costs in km Greedy search example Oradea 71 Neamt 87 Zerind Iasi rad ibiu 99 Fagaras Vaslui Timisoara imnicu Vilcea ugoj 97 Pitesti Hirsova Mehadia Urziceni ucharest 120 Dobreta 90 raiova Giurgiu Eforie traight line distance to ucharest rad 366 ucharest 0 raiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 ugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 imnicu Vilcea 193 ibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374 rad 366 hapter 3 74 hapter 3 76

20 Greedy search example Greedy search example rad rad ibiu Timisoara Zerind ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea ibiu ucharest hapter 3 77 hapter 3 79 Greedy search example Properties of greedy search rad omplete?? ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea hapter 3 78 hapter 3 80

21 Properties of greedy search omplete?? Yes omplete in finite space with repeated-state checking No can get stuck in loops, e.g., with Oradea as goal, Iasi Neamt Iasi Neamt Properties of greedy search omplete?? Yes omplete in finite space with repeated-state checking No can get stuck in loops, e.g., Iasi Neamt Iasi Neamt Time?? Time?? O(b m ), but a good heuristic can give dramatic improvement pace?? O(b m ), but a good heuristic can give dramatic improvement Optimal?? hapter 3 81 hapter 3 83 Properties of greedy search omplete?? Yes omplete in finite space with repeated-state checking No can get stuck in loops, e.g., Iasi Neamt Iasi Neamt Properties of greedy search omplete?? Yes omplete in finite space with repeated-state checking No can get stuck in loops, e.g., Iasi Neamt Iasi Neamt Time?? O(b m ), but a good heuristic can give dramatic improvement pace?? Time?? O(b m ), but a good heuristic can give dramatic improvement pace?? O(b m ), but a good heuristic can give dramatic improvement Optimal?? No hapter 3 82 hapter 3 84

22 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 search uses an admissible heuristic i.e., h(n) h (n) where h (n) is the true cost from n. (lso require h(n) 0, so h(g) = 0 for any goal G.) E.g., h D (n) never overestimates the actual road distance Theorem: search is optimal ibiu 393= search example rad Timisoara Zerind 447= = hapter 3 85 hapter 3 87 search example search example rad 366=0+366 rad ibiu Timisoara Zerind 447= = rad Fagaras Oradea imnicu Vilcea 646= = = = hapter 3 86 hapter 3 88

23 search example search example rad rad ibiu Timisoara Zerind 447= = ibiu Timisoara Zerind 447= = rad Fagaras 646= = Oradea 671= imnicu Vilcea rad 646= Fagaras Oradea 671= imnicu Vilcea raiova Pitesti ibiu 526= = = ibiu ucharest raiova Pitesti ibiu 591= = = = ucharest raiova imnicu Vilcea 418= = = hapter 3 89 hapter 3 91 rad 646= ibiu Fagaras ibiu ucharest Oradea 671= search example imnicu Vilcea rad Timisoara raiova Pitesti ibiu 591= = = = = Zerind 447= = Optimality of (standard proof) uppose some suboptimal goal G 2 has been generated and is in the queue. et n be an unexpanded node on a shortest path to an optimal goal G. n G Want to prove: f(g 2 ) > f(n) [ will never select G 2 for expansion] tart G 2 f(g 2 ) = g(g 2 ) since h(g 2 ) = 0 g(g 2 ) > g(g) since G 2 is suboptimal g(g) f(n) since h is admissible hapter 3 90 hapter 3 92

24 Properties of Properties of omplete?? omplete?? Yes, unless there are infinitely many nodes with f f(g) Time?? Exponential in [relative error in h d] pace?? hapter 3 93 hapter 3 95 Properties of omplete?? Yes, unless there are infinitely many nodes with f f(g) Time?? Properties of omplete?? Yes, unless there are infinitely many nodes with f f(g) Time?? Exponential in [relative error in h d] pace?? Exponential Optimal?? hapter 3 94 hapter 3 96

25 Properties of omplete?? Yes, unless there are infinitely many nodes with f f(g) Time?? Exponential in [relative error in h d] pace?? Exponential Optimal?? Yes cannot expand f i+1 until f i is finished, where f i+1 > f i * vs Uniform-cost earch f(n) = g(n)+h(n) Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? onsider h(n) is perfect, f(n) is? is the cost for the optimal solution: expands all nodes with f(n) < expands some nodes with f(n) = expands no nodes with f(n) > hapter 3 97 hapter 3 99 * vs Uniform-cost earch f(n) = g(n)+h(n) Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? * vs Uniform-cost earch f(n) = g(n)+h(n) Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? onsider h(n) is perfect, f(n) is the actual total path cost. hapter 3 98 hapter 3 100

26 f(n) = g(n)+h(n) * vs Uniform-cost earch Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? onsider h(n) is perfect, f(n) is the actual total path cost. If n doesn t lead to a goal state, f(n) = *? U? f(n) = g(n)+h(n) * vs Uniform-cost earch Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? onsider h(n) is perfect, f(n) is the actual total path cost. If n doesn t lead to a goal state, f(n) = * doesn t explore n. U doesn t know and keeps on exploring n (and its sucessors). If n doesn t lead to the optimal goal, but n does: f(n) > f(n ) * doesn t explore n and *only* explores n! U doesn t know and keeps on exploring n (and its sucessors). In terms of speed, the worst case for * is when h(n) is zero, but we don t use h(n) = 0. hapter hapter f(n) = g(n)+h(n) * vs Uniform-cost earch Isn t U just * with h(n) being zero (admissible)? oth are optimal, why is * usually faster? onsider h(n) is perfect, f(n) is the actual total path cost. If n doesn t lead to a goal state, f(n) = * doesn t explore n. U doesn t know and keeps on exploring n (and its sucessors). If n doesn t lead to the optimal goal, but n does: f(n) > f(n ) *? U? onsistency onsidern isasuccessorofn,aheuristicisconsistentif h(n) c(n,a,n )+h(n ), et s find the relationship between f(n) and f(n ): f(n ) = g(n )+h(n ) f(n ) = g(n)+c(n,a,n )+h(n ) f(n ) g(n)+h(n) [since h is consistent] f(n ) f(n) c(n,a,n ) n n h(n ) h(n) i.e. f(n) values for a sequence of nodes along *any* path are nondecreasing (similar to g(n) values in U). * using GPH-EH is optimal if h(n) is consistent (using a similar argument as U). G hapter hapter 3 104

27 Optimality of (consistent heuristics) emma: expands nodes in order of increasing f value Gradually adds f-contours of nodes (cf. breadth-first adds layers) ontour i has all nodes with f = f i, where f i < f i+1 T Z 380 O D M 400 F P 420 G N I U V H E E.g., for the 8-puzzle: dmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) h 1 () =?? h 2 () =?? tart tate Goal tate hapter hapter dmissible vs onsistent Heuristics onsistency is a slightly stronger/stricter requirement than admissibility. consistentheuristics admissibleheuristics dmissible heuristics are usually consistent. Not easy to concort admissible, but not consistent heuristics. E.g., for the 8-puzzle: dmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) tart tate Goal tate h 1 () =?? 6 h 2 () =?? = 14 hapter hapter 3 108

28 Dominance If h 2 (n) h 1 (n) for all n (both admissible) then h 2 dominates h 1 and is faster for search Typical search costs: d = 14 ID = 3,473,941 nodes (h 1 ) = 539 nodes (h 2 ) = 113 nodes d = 24 ID 54,000,000,000 nodes (h 1 ) = 39,135 nodes (h 2 ) = 1,641 nodes Given any admissible heuristics h a, h b, h(n) = max(h a (n),h b (n)) elaxed problems contd. Well-known example: travelling salesperson problem (TP) Find the shortest tour visiting all cities exactly once Minimum spanning tree can be computed in O(n 2 ) and is a lower bound on the shortest (open) tour is also admissible and dominates h a, h b hapter hapter elaxed problems dmissible 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 ummary Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored Variety of uninformed search strategies Iterative deepening search uses only linear space and not much more time than other uninformed algorithms Graph search can be exponentially more efficient than tree search hapter hapter 3 112

29 ummary Heuristic functions estimate costs of shortest paths Good heuristics can dramatically reduce search cost Greedy best-first search expands lowest h incomplete and not always optimal search expands lowest g +h complete and optimal also optimally efficient (up to tie-breaks, for forward search) dmissible heuristics can be derived from exact solution of relaxed problems hapter 3 113