Heuris'c search, A* CS171, Fall 2016 Introduc'on to Ar'ficial Intelligence Prof. Alexander Ihler. Reading: R&N

Transcription

1 Heuris'c search, A* CS171, Fall 2016 Introduc'on to Ar'ficial Intelligence Prof. Alexander Ihler Reading: R&N

2 Outline Review limita'ons of uninformed search methods Informed (or heuris/c) search Problem-specific heuris/cs to improve efficiency Best-first, A* (and if needed for memory limits, RBFS, SMA*) Techniques for genera'ng heuris'cs A* is op'mal with admissible (tree)/consistent (graph) heuris'cs A* is quick and easy to code, and oten works *very* well Heuris/cs A structured way to add smarts to your solu'on Provide *significant* speed-ups in prac'ce S'll have worst-case exponen'al 'me complexity In AI, NP-Complete means Formally interes'ng

3 Limita'ons of uninformed search Search space size makes search tedious Combinatorial explosion Ex: 8-Puzzle Average solu'on cost is ~ 22 steps Branching factor ~ 3 Exhaus've search to depth 22: 3.1x10 10 states 24-Puzzle: states (much worse!)

4 ! Recall: tree search Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Arad Lugoj Arad Oradea This strategy is what function TREE-SEARCH (problem, strategy) : returns a solution differentiates or failure! different initialize the search tree using the initial state of problem! search algorithms while (true):! if no candidates for expansion: return failure! choose a leaf node for expansion according to strategy! if the node contains a goal state: return the corresponding solution! else: expand the node and add the resulting nodes to the search tree!

5 Heuris'c func'on Idea: use a heuris'c func'on h(n) for each node g(n) = known path cost so far to node n h(n) = es#mate of (op'mal) cost to goal from node n f(n) = g(n)+h(n) = es#mate of total cost to goal through n f(n) provides an es'mate for the total cost Best first search implementa'on Order the nodes in fron'er by an evalua'on func'on Greedy Best-First: order by h(n) A* search: order by f(n) Search efficiency depends on heuris'c quality! The beeer your heuris'c, the faster your search!

6 Heuris'c func'on Heuris'c Def n: a commonsense rule or rules intended to increase the probability of solving some problem Same linguis'c root as Eureka = I have found it Using rules of thumb to find answers Heuris'c func'on h(n) Es'mate of (op'mal) remaining cost from n to goal Defined using only the state of node n h(n) = 0 if n is a goal node Example: straight line distance from n to Bucharest Not true state space distance, just es'mate! Actual distance can be higher Provides problem-specific knowledge to the search algorithm

7 Ex: 8-Puzzle 8-Puzzle Avg solu'on cost is about 22 steps Branching factor ~ 3 Exhaus've search to depth 22 = 3.1 x 10^10 states A good heuris'c f n can reduce the search process Two commonly used heuris'cs h 1 : the number of misplaced 'les h 1 (s) = 8 h 2 : sum of the distances of the 'les from their goal ( Manhaean distance ) h 2 (s) = = 18

8 Ex: Romania, straight-line distance Arad Oradea Neamt Zerind Iasi 140 Sibiu 99 Fagaras 92 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Giurgiu Eforie Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Eforie 161 Fagaras 176 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374

9 Rela'onship of search algorithms Nota'on g(n) = known cost so far to reach n h(n) = es'mated (op'mal) cost from n to goal f(n) = g(n)+h(n) = es'mated (op'mal) total cost through n Uniform cost search: sort fron'er by g(n) Greedy best-first search: sort fron'er by h(n) A* search: sort fron'er by f(n) Op'mal for admissible / consistent heuris'cs Generally the preferred heuris'c search framework Memory-efficient versions of A* are available: RBFS, SMA*

10 Greedy best-first search (some'mes just called best-first ) h(n) = es'mate of cost from n to goal Ex: h(n) = straight line distance from n to Bucharest Greedy best-first search expands the node that appears to be closest to goal Priority queue sort func'on = h(n)

11 Ex: GBFS for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Eforie 161 Fagaras 176 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374

12 Ex: GBFS for Romania GBFS: 450km Optimal path: 418 km Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Eforie 161 Fagaras 176 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374

13 Greedy best-first search With tree-search, will become stuck in this loop: Order of node expansion: S A D S A D S A D Path found: none Cost of path found: none h=5 D S h=6 h=7 A B C h=8 h=9 G h=0

14 Proper'es of greedy best-first search Complete? Tree version can get stuck in loops Graph version is complete in finite spaces Time? O(b m ) A good heuris'c can give drama'c improvement Space? O(b m ) Keeps all nodes in memory Op'mal? No Ex: Arad Sibiu Rimnicu Vilcea Pites' Bucharest shorter!

15 A * search Idea: avoid expanding paths that are already expensive Generally the preferred (simple) heuris'c search Op'mal if heuris'c is: admissible (tree search) / consistent (graph search) Evalua'on func'on f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = es'mated cost from n to goal f(n) = g(n)+h(n) = es'mated total cost of path through n to goal Priority queue sort func'on = f(n)

16 Admissible heuris'cs A heuris'c h(n) is admissible if for every node n, h(n) h*(n) h*(n) = the true cost to reach the goal state from n An admissible heuris'c never overes'mates the cost to reach the goal, i.e., it is op'mis'c (or, never pessimis'c) Ex: straight-line distance never overes'mates road distance Theorem: if h(n) is admissible, A* using Tree-Search is op'mal

17 Admissible heuris'cs Two commonly used heuris'cs h 1 : the number of misplaced 'les h 1 (s) = 8 h 2 : sum of the distances of the 'les from their goal ( Manhaean distance ) h 2 (s) = = 18

18 Consistent heuris'cs A heuris'c is consistent (or monotone) if for every node n, every successor n' of n generated by any ac'on a, h(n) c(n,a,n') + h(n') If h is consistent, we have f(n') = g(n') + h(n') = g(n) + c(n,a,n') + h(n') g(n) + h(n) = f(n) i.e., f(n) is non-decreasing along any path. (Triangle inequality) Consistent ) admissible (stronger condi'on) Theorem: If h(n) is consistent, A* using Graph-Search is op'mal

19 Op'mality condi'ons Tree search op'mal if admissible Graph search op'mal if consistent Why two different condi'ons? In graph search you oten find a long cheap path to a node ater a short expensive one, so you might have to update all of its descendants to use the new cheaper path cost so far A consistent heuris'c avoids this problem (it can t happen) Consistent is slightly stronger than admissible Almost all admissible heuris'cs are also consistent Could we do op'mal graph search with an admissible heuris'c? Yes, but you would have to do addi'onal work to update descendants when a cheaper path to a node is found A consistent heuris'c avoids this problem

20 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

21 Ex: A* for Romania (Simulated queue) Expanded: None Children: None Frontier: Arad/366 (0+366), Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

22 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

23 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Children: Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

24 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

25 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Sibiu/393 ( ), Children: Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

26 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

27 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Sibiu/393 ( ), RimnicuVilcea/413 ( ), Children: Craiova/526 ( ), Pitesti/417 ( ), Sibiu/553 ( ), Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Craiova/526 ( ), Pitesti/417 ( ), Sibiu/553 ( ), Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

28 Ex: A* for Romania Arad Zerind Oradea 140 Timisoara Dobreta Lugoj Mehadia Sibiu Fagaras Rimnicu Vilcea Cralova Pitesti Neamt Iasi Urziceni Bucharest Giurgiu Vaslui Hirsova 86 Eforie Note: search does not backtrack ; both routes are pursued. Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

29 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Sibiu/393 ( ), RimnicuVilcea/413 ( ), Fagaras/415 ( ), Children: Bucharest/450 (450+0), Sibiu/591 ( ), Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Craiova/526 ( ), Pitesti/417 ( ), Sibiu/553 ( ), Bucharest/450 (450+0), Sibiu/591 ( ), Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

30 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

31 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Sibiu/393 ( ), RimnicuVilcea/413 ( ), Fagaras/415 ( ), Pitesti/417 ( ), Children: Bucharest/418 (418+0), Craiova/615 ( ), RimnicuVilcea/607 ( ), Arad 118 Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Craiova/526 ( ), Pitesti/417 ( ), Sibiu/553 ( ), Bucharest/450 (450+0), Sibiu/591 ( ), Bucharest/418 (418+0), Craiova/ 615 ( ), RimnicuVilcea/607 ( ) 71 Zerind Oradea Timisoara Dobreta Lugoj Mehadia Sibiu Fagaras Rimnicu Vilcea Cralova Pitesti Neamt Iasi Urziceni Bucharest Giurgiu Vaslui Hirsova 86 Eforie Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

32 Ex: A* for Romania Oradea 71 Neamt 87 Zerind 151 Iasi 140 Arad Sibiu Fagaras 118 Vaslui Timisoara 80 Rimnicu Vilcea Lugoj Pitesti Mehadia 146 Hirsova 101 Urziceni Bucharest Dobreta 90 Cralova Eforie Giurgiu Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

33 Ex: A* for Romania (Simulated queue) Expanded: Arad/366 (0+366), Sibiu/393 ( ), RimnicuVilcea/413 ( ), Fagaras/415 ( ), Pitesti/417 ( ), Bucharest/418 (418+0) Children: None (goal test succeeds) Arad 118 Frontier: Arad/366 (0+366), Sibiu/393 ( ), Timisoara/447 ( ), Zerind/449 (+374), Arad/646 ( ), Fagaras/415 ( ), Oradea/671 ( ), RimnicuVilcea/413 ( ), Craiova/526 ( ), Pitesti/417 ( ), Sibiu/553 ( ), Bucharest/450 (450+0), Sibiu/591 ( ), Bucharest/418 (418+0), Craiova/ 615 ( ), RimnicuVilcea/607 ( ) 71 Zerind Oradea Timisoara Dobreta Lugoj Mehadia Sibiu Fagaras Rimnicu Vilcea Cralova Pitesti Neamt Iasi Urziceni Bucharest Giurgiu Vaslui Hirsova 86 Eforie Shorter, more expensive path remains on queue Cheaper path will be found & returned Straight-line dist to goal Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Fagaras 176 Lugoj 244 Mehadia 241 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Zerind 374

34 Contours of A* search For consistent heuris'c, A* expands in order of increasing f value Gradually adds f-contours of nodes Contour i has all nodes with f=f i, where f i < f i+1 Oradea Neamt Zerind Iasi Arad Sibiu Fagaras Timisoara Rimnicu Vilcea Vaslui Lugoj Mehadia Pitesti 85 Urziceni Hirsova Bucharest Dobreta Cralova Giurgiu Eforie

35 Proper'es of A* search Complete? Yes Unless infinitely many nodes with f < f(g) Cannot happen if step-cost ² > 0 Time/Space? O(b m ) Except if h(n) h*(n) O( log h*(n) ) Op'mal? Yes With: Tree-Search, admissible heuris'c; Graph-Search, consistent heuris'c Op'mally efficient? Yes No op'mal algorithm with same heuris'c is guaranteed to expand fewer nodes

36 Op'mality of A* Proof: Suppose some subop'mal goal G 2 has been generated & is on the fron'er. Let n be an unexpanded node on the path to an op'mal goal G Show: f(n) < f(g 2 ) (so, n is expanded before G 2 ) f(g 2 ) = g(g 2 ) since h(g 2 ) = 0 f(g) = g(g) since h(g) = 0 g(g 2 ) > g(g) since G 2 is suboptimal f(g 2 ) > f(g) from above, with h=0 h(n) h*(n) since h is admissible (under-estimate) g(n) + h(n) g(n) + h*(n) from above f(n) f(g) since g(n)+h(n)=f(n) & g(n)+h*(n)=f(g) f(n) < f(g2) from above

37 Memory-bounded heuris'c search Memory is a major limita'on of A* Usually run out of memory before run out of 'me How can we solve the memory problem? Idea: recursive best-first search (RBFS) Try something like depth-first search, but don t forget everything about the branches we have par'ally explored Remember the best f(n) value we have found so far in the branch we re dele'ng

38 RBFS: best alternative over frontier nodes, which are not children: i.e. do I want to back up? RBFS changes its mind very often in practice. This is because the f=g+h become more accurate (less optimistic) as we approach the goal. Hence, higher level nodes have smaller f-values and will be explored first. Problem: We should keep in memory whatever we can.

39 Simple Memory Bounded A* (SMA*) Memory limited, but uses available memory well: Like A*, but if memory full: delete the worst node (largest f-val) Like RBFS, remember the best descendent in deleted branch If there is a 'e (equal f-values) we delete the oldest nodes first. SMA* finds the op'mal reachable solu'on given memory constraint. Time can s'll be exponen'al. Best of search algorithms we ve seen A solution is not reachable if a single path from root to goal does not fit in memory Using memory avoids double work; heuris'c guides explora'on If memory is not a problem, basic A* is easy to code & performs well

40 SMA* Pseudocode Note: not in 2 nd edition of R&N function SMA*(problem) returns a solution sequence inputs: problem, a problem static: Queue, a queue of nodes ordered by f-cost Queue ß MAKE-QUEUE({MAKE-NODE(INITIAL-STATE[problem])}) loop do if Queue is empty then return failure n ß deepest least-f-cost node in Queue if GOAL-TEST(n) then return success s ß NEXT-SUCCESSOR(n) if s is not a goal and is at maximum depth then f(s) ß else f(s) ß MAX(f(n),g(s)+h(s)) if all of n s successors have been generated then update n s f-cost and those of its ancestors if necessary if SUCCESSORS(n) all in memory then remove n from Queue if memory is full then delete shallowest, highest-f-cost node in Queue remove it from its parent s successor list insert its parent on Queue if necessary insert s in Queue end

41 Simple memory-bounded A* (SMA*) (Example with 3-node memory) Search space g+h = f = goal 10+5=15 Progress of SMA*. Each node is labeled with its current f-cost. Values in parentheses show the value of the best forgotten descendant. B A 0+12= =13 G 12 best estimated solution so far for that node A 12 A maximal depth is 3, since memory limit is 3. This branch is now useless. A 13 best forgotten node A 13[15] 13 G C D 20+5= H I 16+2=18 15 B B G H 20+0= E F 30+5= = = J K 24+0= =29 A 15[15] G 24[ ] 15 A A 15[24] B 15 20[24] B 8 20[ ] A B G I C Algorithm can tell you when best solution found within memory constraint is optimal or not. 25 D 20

42 Heuris'c func'ons 8-Puzzle Avg solu'on cost is about 22 steps Branching factor ~ 3 Exhaus've search to depth 22 = 3.1 x 10^10 states A good heuris'c f n can reduce the search process True cost for this start & goal: 26 Two commonly used heuris'cs h 1 : the number of misplaced 'les h 1 (s) = 8 h 2 : sum of the distances of the 'les from their goal ( Manhaean distance ) h 2 (s) = = 18

43 Dominance Defini'on: If h 2 (n) h 1 (n) for all n then h 2 dominates h 1 h 2 is almost always beeer for search than h 1 h 2 is guaranteed to expand no more nodes than h 1 h 2 almost always expands fewer nodes than h 1 Not useful unless are h 1, h 2 are admissible / consistent Ex: 8-Puzzle / sliding 'les h 1 : the number of misplaced 'les h 2 : sum of the distances of the 'les from their goal

44 Ex: 8-Puzzle Average number of nodes expanded d IDS A*(h1) A*(h2) Average over 100 randomly generated 8-puzzle problems h1 = number of tiles in the wrong position h2 = sum of Manhattan distances

45 Effec've branching factor, b* Let A* generate N nodes to find a goal at depth d Effec've branching b* is the branching factor a uniform tree of depth d would have in order to contain N+1 nodes: For sufficiently hard problems, b* is oten fairly constant across different problem instances A good guide to the heuris'c s overall usefulness A good way to compare different heuris'cs

46 Designing heuris'cs OTen constructed via problem relaxa'ons A problem with fewer restric'ons on ac'ons Cost of an op'mal solu'on to a relaxed problem is an admissible heuris'c for the original problem Ex: 8-Puzzle Relax rules so a 'le can move anywhere: h 1 (n) Relax rules so 'le can move to any adjacent square: h 2 (n) A useful way to generate heuris'cs Ex: ABSOLVER (Priedi's 1993) discovered the first useful heuris'c for the Rubik s cube

47 More on heuris'cs Combining heuris'cs H(n) = max { h1(n), h2(n),, hk(n) } max chooses the least op'mis'c heuris'c at each node Paeern databases Solve a subproblem of the true problem ( = a lower bound on the cost of the true problem) Store the exact solu'on for each possible subproblem

48 Summary Uninformed search has uses but also severe limita'ons Heuris'cs are a structured way to make search smarter Informed (or heuris'c) search uses problem-specific heuris'cs to improve efficiency Best-first, A* (and if needed for memory, RBFS, SMA*) Techniques for genera'ng heuris'cs A* is op'mal with admissible (tree) / consistent (graph heuris'cs Can provide significant speed-ups in prac'ce Ex: 8-Puzzle, drama'c speed-up S'll worst-case exponen'al 'me complexity (NP-complete) Next: local search techniques (hill climbing, GAs, annealing ) Read R&N Ch 4 before next lecture

49 You should know evalua'on func'on f(n) and heuris'c func'on h(n) for each node n g(n) = known path cost so far to node n. h(n) = es'mate of (op'mal) cost to goal from node n. f(n) = g(n)+h(n) = es'mate of total cost to goal through node n. Heuris'c searches: Greedy-best-first, A* A* is op'mal with admissible (tree)/consistent (graph) heuris'cs Prove that A* is op'mal with admissible heuris'c for tree search Recognize when a heuris'c is admissible or consistent h 2 dominates h 1 iff h 2 (n) h 1 (n) for all n Effec've branching factor: b* Inven'ng heuris'cs: relaxed problems; max or convex combina'on