Chapter S:II (continued)

Transcription

1 Chapter S:II (continued) II. Baic Search Algorithm Sytematic Search Graph Theory Baic State Space Search Depth-Firt Search Backtracking Breadth-Firt Search Uniform-Cot Search AND-OR Graph Baic Depth-Firt Search of AND-OR Graph AND-OR Graph Search S:II-116 Baic Search Algorithm STEIN/LETTMANN

2 AND-OR Graph Baic Problem Reduction: A Powerful Tool in Problem Solving Example for problem decompoition in S:I Introduction Counterfeit Coin Problem Tic-Tac-Toe Game Example of problem decompoition in algorithm: divide-and-conquer Mergeort. Divide lit of object in two ubet, ort thee ublit eparately, and merge them; require O(n log n) comparion. Fat Median Computation the median of a lit of number. Special cae of the election problem determine k-th element in lit of number according to ome orting ; algorithm by by Blum, Floyd, Pratt, Rivet, and Tarjan require at mot 6n comparion. Fat Integer Multiplication. Karatuba algorithm (a2 k + b)(c2 k + d) = ac2 2k + (ac + bd (a b)(c d))2 k + bd; require O(n ) bit operation compared to O(n 2 ) required by a naive implementation. Matrix Multiplication. Straen algorithm require O(n ) arithmetic operation compared to O(n 3 ) required by a naive implementation. S:II-117 Baic Search Algorithm STEIN/LETTMANN

3 AND-OR Graph Baic Building Block of Problem Reduction Decompoition into ubproblem. A problem at hand i decompoed into a finite et of independently olvable ubproblem. Solving all ubproblem. All the ubproblem are olved independently, i.e., without interaction between olution procee for different ubproblem. Solution compoition. Solution to the ubproblem are ued to contruct a olution of the problem at hand in a (relatively) imple way. Decompoition Solving Compoition S:II-118 Baic Search Algorithm STEIN/LETTMANN

4 AND-OR Graph Baic Graph Repreentation of Problem Reduction Directed Hyperedge. A directed hyperedge (n, {n 1,..., n k }) with ingle tart node n and multiple end node n 1,..., n k repreent the decompoition of the problem repreented by n into ubproblem repreented by n 1,..., n k. Familie of AND link (AND edge). A family of directed edge (n, n 1),..., (n, n k ) repreent the decompoition into ubproblem. In graphical repreentation ibling AND link (n, n 1),..., (n, n k ) are connected by an arc. hyperedge hyperedge ibling link ibling link Q. I there a imple way to integrate the concept of hyperedge or familie of AND link into tate-pace graph? S:II-119 Baic Search Algorithm STEIN/LETTMANN

5 Remark: The above hyperedge are directed forward hyperedge, alo called F -arc (forward arc). [Gallo 1993] The concept of (cyclic/acyclic) path can be extended to directed hypergraph in a traight forward way. The definition of hyperpath, however, include the property of being acyclic and minimal. [Nielen/Anderen/Pretolani 2005] S:II-120 Baic Search Algorithm STEIN/LETTMANN

6 AND-OR Graph Baic Problem Reduction: Integration into State-Space Graph Idea: Ditinguih node intead of link. 1. AND node. Repreent a tate where only a pecific problem decompoition i poible. AND node have a ingle family of AND link a outgoing link. 2. OR node. Repreent a tate where only tate tranition are poible or a deciion to perform a pecific problem decompoition. OR node have only OR link a outgoing link. Definition 10 ((Canonical) Problem-Reduction Graph, AND-OR Graph) Let S be a tate-pace and l : S {AND, OR} be a labeling of tate in S that define dijoint et of AND-tate rep. OR-tate. A graph G = (S, E) with node et S and edge et E S S together with a labeling l and an OR-tate a tart node i called a (canonical) problem-reduction graph or an AND-OR graph. Uually, the labeling l will not be mentioned for AND-OR graph. S:II-121 Baic Search Algorithm STEIN/LETTMANN

7 AND-OR Graph Baic Canonical Repreentation of AND-OR Graph Graph tranformation. 1. Label all node a OR-node. 2. A long a there i an OR-node n with a family of outgoing AND link (n, n 1),..., (n, n k ), introduce a new intermediate AND-node n and replace the link by a new OR link (n, n ) and AND link (n, n 1),..., (n, n k ). Since a canonical repreentation can alway be generated, we will conider only AND-OR graph. S:II-122 Baic Search Algorithm STEIN/LETTMANN

8 Remark: Outgoing link of an OR-node are conidered a alternative tranformation of a problem. Outgoing link of an AND-node are conidered a a family of AND-link repreenting a problem decompoition, e.g. a ubet plitting. The canonical repreentation of AND-OR graph doe not require alternating node type. In particular, the root node may not be of OR type. A olution to ubproblem of a problem decompoition are combined to form the olution to the original problem, olution have to be repreented by more complex tructure than path in tate pace graph. A tate pace graph i an AND-OR graph without AND node. S:II-123 Baic Search Algorithm STEIN/LETTMANN

9 AND-OR Graph Baic AND-OR Graph Propertie Well-known concept from graph theory (path, tree,... ) can be ued a well in context of AND-OR graph. Example: Acyclic AND-OR Graph Let G = (S, E) with labeling l be an AND-OR-graph. The AND-OR graph i called acyclic or cycle-free if the underlying graph G i acyclic. If the pecial emantic of AND-node a tarting point of a problem decompoition i involved, redefinition would be neceary. Example: AND-OR Subgraph Let G = (S, E) with labeling l be an AND-OR-graph. A ubgraph G = (S, E ) of G with labeling l S i an AND-OR ubgraph of G with l, if G contain for each AND node in G either all or none of it outgoing edge in G. Uage: To avoid miundertanding, only tandard concept from graph theory are ued. Conequence reulting from the emantic of AND-node are alway mentioned explicitly. S:II-124 Baic Search Algorithm STEIN/LETTMANN

10 AND-OR Graph Baic Solution in AND-OR Graph Deirable characteritic: Compatibility. Path hould be permitted a olution if only OR node are contained. Contructability. There i a procedure to contruct a olution tepwie. Repreentability. Solution can be repreented a AND-OR tree. Totality. No (ub-)problem in a olution hould remain unolved. Finitene. Solution hould be finite tructure that contain for any ubproblem a finite number of tep to reach goal node. Reue. Solution hould provide only one (ub-)olution for problem that occur multiple time. Minimality. Solution hould not contain uperfluou part. S:II-125 Baic Search Algorithm STEIN/LETTMANN

11 AND-OR Graph Baic Solution Tree: A Generaliation of Solution Path AND-OR graph example: Solution tree: n 1 n 1 n 2 n 3 n 2 n 3 γ 1 n 4 γ 1 n 4 n 2 γ 2 γ 2 γ 1 Solution a et of path: {(, n 1, n 2, γ 1 ), (, n 1, n 3, n 4, n 2, γ 1 ), (, n 1, n 3, n 4, γ 2 )} Compact repreentation: AND-OR olution tree S:II-126 Baic Search Algorithm STEIN/LETTMANN

12 AND-OR Graph Baic Solution Tree in AND-OR Graph [ Solution Path Definition 11 (Solution Tree for an AND-OR Graph) in OR Graph] Let G be an AND-OR graph, and let n be a node in G. Alo, let ome additional olution contraint be given. An AND-OR tree H i called a olution tree for n in G iff ( ) the following condition hold: 1. H i finite tree. 2. H contain the node n a root node. 3. If H contain an inner node n, which i an intantiation of an OR node in G, then H contain exactly one link to an intantiation of a ucceor node of n in G and thi node intantiation. 4. If H contain an inner node n, which i an intantiation of an AND node in G, then H contain link to intantiation of all of it ucceor node of n in G and all thee node intantiation. 5. The leaf node (terminal node) in H are intantiation of goal node in G. 6. H atifie the olution contraint. S:II-127 Baic Search Algorithm STEIN/LETTMANN

13 Remark: For compatibility of olution in AND-OR graph with olution path in OR graph we allow multiple intance of node in a olution tree for repreenting loop. Thi conflict with the requet for reuing a olution for every intance of a problem. Bet-Firt earch algorithm for OR graph prune cyclic path. Therefore, compatibility can be retricted to cycle-free olution path: Any path from to a leaf node in a olution tree for for an AND-OR graph could be retricted to be cycle-free. Q. Doe thi retriction olve the conflict? S:II-128 Baic Search Algorithm STEIN/LETTMANN

14 AND-OR Graph Baic Solution Tree in AND-OR Graph (continued) Definition 12 (Bae of a Solution Tree for an AND-OR Graph) Let G be an AND-OR graph, and let n be a node in G. A olution-tree bae H for n in G i defined in the ame way a a olution tree for n in G except for condition 5 on leaf node and condition 6 on additional olution contraint. Leaf node mut not be dead end. Uage: Solution Tree We are intereted in finding a olution tree H for the tart node in G. Solution Tree Bae Algorithm maintain and extend a et of promiing olution-tree bae until a olution tree i found. Diadvantage: Multiple Occurrence of Node The ame problem (repreented by a node n G) can have to be olved multiple time (in different way), even in a ingle olution tree. S:II-129 Baic Search Algorithm STEIN/LETTMANN

15 AND-OR Graph Baic Requirement for an Algorithmization of AND-OR Graph Search Propertie of Search Space Graph 1. G i a canonical problem-reduction graph (directed AND-OR graph). 2. G i implicitly defined by (a) (b) a ingle tart node and a function ucceor(n) or next_ucceor(n) returning ucceor of a node. 3. For each node n it type l(n) (AND node / OR node) i known. 4. Computing ucceor alway return new clone of node. 5. G i locally finite. 6. For G a et Γ of goal node i given; in general Γ will not be ingleton. 7. G ha a function (.) returning true if a olution bae i a olution path. Tak: Determine in G a olution tree for. S:II-130 Baic Search Algorithm STEIN/LETTMANN

16 Remark: Node expanion work in the ame way for both type of node. Minimum requirement for a olution tree i that each terminal node of a olution-tree bae i a goal node. Function ucceor(n) rep. next_ucceor(n) return new clone for each ucceor node of an expanded node. An algorithm that doe not care whether two node repreent the ame tate will, therefore, conider an unfolding of the problem-reduction graph to a tree with root. S:II-131 Baic Search Algorithm STEIN/LETTMANN

17 AND-OR Graph Baic Generic Schema for AND-OR-Graph Tree Search Algorithm... from a olution-tree-bae-oriented perpective: 1. Initialize olution-tree-bae torage. 2. Loop. (a) Uing ome trategy elect a olution-tree bae to be extended. (b) (c) (d) Uing ome trategy elect an unexpanded node in thi bae. According to the node type extend the olution-tree bae by ucceor node in any poible way and tore the new candidate. Determine whether a olution tree i found. Uage: Search algorithm following thi chema maintain a et of olution-tree bae. Initially, only the tart node i available; node expanion i the baic tep. S:II-132 Baic Search Algorithm STEIN/LETTMANN

18 AND-OR Graph Baic Algorithm: Input: Output: Generic_AND-OR_Tree_Search. Start node repreenting the initial tate (problem). ucceor(n). Return the ucceor of node n. (b). Predicate that i True if olution-tree bae b i a olution tree. A olution tree b or the ymbol Fail. S:II-133 Baic Search Algorithm STEIN/LETTMANN

19 AND-OR Graph Baic [ Generic_OR_Search] Generic_AND-OR_Tree_Search(, ucceor, ) 1. b = olution_tree_bae(); // Initialize olution-tree bae b. IF (b) THEN RETURN(b); // Check if b i olution tree. 2. puh(b, OPEN); // Store b on OPEN waiting for extenion. 3. LOOP 4. IF (OPEN = ) THEN RETURN(Fail); 5. b = chooe(open); // Chooe a olution-tree bae b from OPEN. remove(b, OPEN); // Delete b from OPEN. n = chooe(b); // Chooe unexpanded tip node in b. 6. IF (i_and_node(n)) THEN b = add(b, n, ucceor(n)); // Expand n and extend b by AND edge. IF (b ) THEN RETURN(b ); // Check if b i olution tree. puh(b, OPEN); // Store b on OPEN waiting for extenion. ELSE FOREACH n IN ucceor(n) DO // Expand n. b = add(b, n, {n }); // Extend b by OR edge (n, n ). IF (b ) THEN RETURN(b ); // Check if b i olution tree. puh(b, OPEN); // Store b on OPEN waiting for extenion. ENDDO ENDIF 7. ENDLOOP S:II-134 Baic Search Algorithm STEIN/LETTMANN

20 Remark: Algorithm Generic_AND-OR_Search take a olution-tree-bae-oriented perpective. Generic_AND-OR_Search maintain and manipulate olution-tree bae (which are AND-OR tree with root ). In order to keep the peudo code hort, we do not cover the cae that a olution-tree bae choen in 5. ha no unexpanded tip node. Such a olution-tree bae can be dicarded. Remaining problem which till are to be olved can be found only among the tip node. So, a olution-tree bae may contain multiple open problem. Function add(b, n,.) return a repreentation of the extended olution-tree bae: node intance in ucceor(n) are added and the edge from n to thee node. For that purpoe, b i copied. So each olution-tree bae i repreented eparately, although they often hare initial ubtree. S:II-135 Baic Search Algorithm STEIN/LETTMANN

21 AND-OR Graph Baic Efficient Storage of Solution-Tree Bae OR-graph earch: Solution bae are path. Backpointer allow the recovery of a path tarting from it tip node. By haring of initial part, the maintained olution bae form a traveral tree. A olution bae i uniquely identified by the correponding tip node in the traveral tree. Idea: Reuing the backpointer concept of Baic_OR_Search allow haring of upper part of olution-tree bae a well. For the olution tree bae maintained by Generic_AND-OR_Tree_Search, the reulting tructure i a tree that conform to traveral tree in OR-graph earch. Q. Can we apply OR-graph earch to AND-OR graph to build uch tree? Q. Can we ue OR-graph earch to detect a olution-tree that i contained? Q. Can we ue OR-graph earch to detect, whether a olution tree i contained? S:II-136 Baic Search Algorithm STEIN/LETTMANN

22 AND-OR Graph Baic Example: Sharing initial ubtree in olution tree for AND-OR graph n 1 n 1 n 1 n 2 n 3 n 2 n 3 n 2 n 3 γ 1 n 4 γ 1 n 2 γ 1 n 4 n 2 γ 2 γ 1 γ 2 γ 1 Q. Which node intance are unified, which are kept eparate when haring initial ubtree? Q. What do the reulting tree look like for the maintained olution-tree bae in line 3 of algorithm Generic_AND-OR_Tree_Search? S:II-137 Baic Search Algorithm STEIN/LETTMANN

23 Depth-Firt Search of AND-OR Graph Difference to OR Graph Search Beide checking additional olution contraint, the different earch pace repreentation entail different termination tet: State-pace graph: analyze a ingle node (leaf node of a olution bae). AND-OR graph: analyze a et of node (leaf node of a particular olution-tree bae). S:II-138 Baic Search Algorithm STEIN/LETTMANN

24 Depth-Firt Search of AND-OR Graph Difference to OR Graph Search Beide checking additional olution contraint, the different earch pace repreentation entail different termination tet: State-pace graph: analyze a ingle node (leaf node of a olution bae). AND-OR graph: analyze a et of node (leaf node of a particular olution-tree bae). Recall the termination tet in AND-OR graph earch: We have to anwer the quetion: Allow the currently known ubgraph of the earch pace graph for the intantiation of a olution tree? To do: Propagate knowledge about olved or unolved problem upward from terminal node to the root node. Node whoe label have been propagated can be dicarded. Approach: Adapt [ DFS] (or [ BFS]) for the labeling. Label are propagated along backpointer. S:II-139 Baic Search Algorithm STEIN/LETTMANN

25 Depth-Firt Search of AND-OR Graph Solved Labeling The quetion whether or not an AND-OR graph G ha a olution tree can be anwered by applying the following labeling rule. Definition 13 (Solved-Labeling Procedure) Let G be an acyclic AND-OR graph. A node in G i labeled a olved, if one of the following condition i fulfilled. 1. A terminal node (leaf node) i labeled a olved, if it i a goal node (olved ret problem); otherwie it i labeled a unolvable. 2. A nonterminal OR node i labeled a olved a oon a one of it ucceor node i labeled a olved ; it i labeled a unolvable a oon a all of it ucceor node are labeled a unolvable. 3. A nonterminal AND node i labeled a olved a oon a all of it ucceor node are labeled a olved ; it i labeled a unolvable a oon a one of it ucceor node i labeled a unolvable. Note: No additional olution contraint are conidered here. S:II-140 Baic Search Algorithm STEIN/LETTMANN

26 Depth-Firt Search of AND-OR Graph Adaption of DFS Additional initialization. All node are unlabeled when generated. Thi include the tart node. If a node n i expanded, the number of it ucceor i tored a unlabeled_ucc(n). Replace termination condition. Intead of terminating with failure when OPEN i empty, terminate a oon a i labeled: IF i_labeled() RETURN(label()) (If OPEN i empty, tart node will be labeled unolvable.) Ue a depth bound. S:II-141 Baic Search Algorithm STEIN/LETTMANN

27 Depth-Firt Search of AND-OR Graph Adaption of DFS (continued) Perform labeling during node expanion (FOREACH loop): Intead of terminating with ucce when finding a goal node, thi cae i handled in the ame way a a dead end cae, i.e., the node i labeled and the label i propagated along the backpointer path uing the function propagate_label(n): FOREACH n IN ucceor(n) DO // Expand n. add_backpointer(n, n); IF i_goal_node(n ) THEN et_label(n, olved ); IF (n ) THEN et_label(n, unolvable ); IF i_labeled(n ) THEN propagate_label(n ); cleanup_cloed(); ELSE puh(n, OPEN); ENDIF ENDDO S:II-142 Baic Search Algorithm STEIN/LETTMANN

28 Depth-Firt Search of AND-OR Graph Adaption of DFS (continued) Propagation of label: Function propagate_label(n) will follow the backpointer. For the tart node the backpointer parent node i aumed to be null. propagate_label(n) l = label(n); p = backpointer_parent(n); WHILE p null AND i_unlabeled(p) DO // Proce parent node p. IF i_or_node(p) AND l = olved THEN et_label(p, olved ); IF i_or_node(p) AND l = unolvable THEN unlabeled_ucc(p) = unlabeled_ucc(p) 1; IF unlabeled_ucc(p) = 0 THEN et_label(p, unolvable ); ENDIF... // Cae i_and_node(p) analogouly. IF i_labeled(p) THEN l = label(p); p = backpointer_parent(p); ELSE p = null; ENDIF ENDDO S:II-143 Baic Search Algorithm STEIN/LETTMANN

29 Remark: We aume that (n) return true if n ha no ucceor. Then, no node without ucceor node will be expanded and thi cae doe not need to be covered. Since DFS i performed, only a path, i.e., a equence of node, i proceed in function propagate_label(n). However, the label of each node proceed by DFS will be et at mot once. Therefore, the overall effort in labeling i linear in the number of node (node intantiation) proceed by DFS. S:II-144 Baic Search Algorithm STEIN/LETTMANN

30 Depth-Firt Search of AND-OR Graph Example: Solved Labeling Uing DFS unolved / olved ret problem S:II-145 Baic Search Algorithm STEIN/LETTMANN

31 Depth-Firt Search of AND-OR Graph Example: Solved Labeling Uing DFS (continued) (1) S:II-146 Baic Search Algorithm STEIN/LETTMANN

32 Depth-Firt Search of AND-OR Graph Example: Solved Labeling Uing DFS (continued) (1) (1) (2) S:II-147 Baic Search Algorithm STEIN/LETTMANN

33 Depth-Firt Search of AND-OR Graph Example: Solved Labeling Uing DFS (continued) (1) (1) (2) (3) deleted node S:II-148 Baic Search Algorithm STEIN/LETTMANN

34 Depth-Firt Search of AND-OR Graph Example: Solved Labeling Uing DFS (continued) (1) (1) (2) (3) (3) (4) + current path deleted node deleted node S:II-149 Baic Search Algorithm STEIN/LETTMANN

35 Remark: Information olved rep. unolvable for a node i propagated to it parent node following the backpointer. Once the final label i determined, it can propagated upward and earch of ibling node can be left undone (ee tranition from (2) to (3) in the above example). Depth-firt earch realize a recurive olution tree earch, imilar to the inductive part of the olution tree definition. If a olution tree ha to fulfill additional contraint (due to optimality requirement or particularitie of the domain), a DFS-baed olved labeling can be applied only if the impoed contraint fulfillment can be checked recurively a well. S:II-150 Baic Search Algorithm STEIN/LETTMANN

36 Depth-Firt Search of AND-OR Graph Solution Tree Labeling Information collected during DFS earch: Solved-labeling procedure: The information propagated i boolean: either unolvable or olvable. Solution-labeling procedure: The information propagated i a proof tree or null. Scope of the analyi by DFS earch: Solved-labeling procedure: The label of the tart node indicate the exitence of olution tree. OR-node are not fully analyzed once olvable i known. Solution-labeling procedure: If more than one olution tree i required, the information propagated i a lit of all poible proof tree. Thi lit may be empty. OR-node have to be fully analyzed. Note: No additional olution contraint are conidered here. S:II-151 Baic Search Algorithm STEIN/LETTMANN

37 Depth-Firt Search of AND-OR Graph Example: Solution Labeling Uing DFS (1) (L (L)) + - S:II-152 Baic Search Algorithm STEIN/LETTMANN

38 Depth-Firt Search of AND-OR Graph Example: Solution Labeling Uing DFS (continued) (1) (1) (2) (L (L)) (L (L)) (L (L)) (R (L)) S:II-153 Baic Search Algorithm STEIN/LETTMANN

39 Depth-Firt Search of AND-OR Graph Example: Solution Labeling Uing DFS (continued) (1) (1) (2) (L (L)) (L (L)) (L (L)) (R (L)) (1) (3) (R (L (L)) (R (L))) (L (L)) (L (L)) (R (L)) + - S:II-154 Baic Search Algorithm STEIN/LETTMANN

40 Depth-Firt Search of AND-OR Graph Example: Solution Labeling Uing DFS (continued) (1) (1) (2) (L (L)) (L (L)) (L (L)) (R (L)) (1) (3) (R (L (L)) (R (L))) (1) (4) (R (L (L)) (R (L))) (L (L)) (L (L)) (R (L)) (L (L)) S:II-155 Baic Search Algorithm STEIN/LETTMANN

41 Remark: If in DFS node are dicarded oon after their label have been propagated, a code for recontructing the olution graph need to be propagated a well. In the hown illutration the code decribe the choen operator (L or R), wherea the neting level of the parenthee indicate the depth of the tree. Intead of propagating a ingle olution tree, DFS could alo propagate a lit of all poible olution tree. In inner OR node union of thee lit have to be built, in inner AND node a Carteian product i neceary. If a olution tree ha to fulfill additional contraint (due to optimality requirement or particularitie of the domain), a DFS-baed olution labeling can be applied only if the impoed contraint fulfillment can be checked recurively a well. If a recurive contraint analyi i infeaible, we have to determine all olution tree in olution labeling and check them one by one. S:II-156 Baic Search Algorithm STEIN/LETTMANN

42 Depth-Firt Search of AND-OR Graph Redundant Problem Solving DFS cannot exploit the compact encoding of problem-reduction repreentation. Reaon: AND-OR graph contain only a ingle intance of identical ubproblem, DFS maintain only a path from to the current node. Identical ubproblem may be encountered on different path and olved everal time by DFS. If AND-OR graph are earched with DFS, redundancy cannot be avoided. S:II-157 Baic Search Algorithm STEIN/LETTMANN

43 Depth-Firt Search of AND-OR Graph Redundant Problem Solving DFS cannot exploit the compact encoding of problem-reduction repreentation. Reaon: AND-OR graph contain only a ingle intance of identical ubproblem, DFS maintain only a path from to the current node. Identical ubproblem may be encountered on different path and olved everal time by DFS. If AND-OR graph are earched with DFS, redundancy cannot be avoided. AND-OR graph (left) and it exploration (= unfolding) with DFS (right): t 0 t 1 A B A B A S:II-158 Baic Search Algorithm STEIN/LETTMANN

44 Remark: With a complete occurrence check, i.e., an occurrence check that conider all explored node o far, recurring ubproblem can be identified. A complete occurrence check entail a memory conumption imilar to BFS, which render uch a check very unattractive if not impoible. In mot cae the mall memory footprint of DFS along with a uually manageable effort for reolving (a mall number of) recurring problem make DFS uperior to BFS when earching AND-OR graph. Recall that by omitting an occurrence check we cannot identify loop in the earch pace graph. The path from to the current node, i.e., the partial olution path, i alo called traveral path or backpointer path [Pearl 84]. Recall that DFS tore the partial olution path on the CLOSED lit (with exeption of the lat node which i in OPEN). A partial occurrence check, which i limited to the node on the traveral path, i both computationally manageable and prohibit infinite loop caued by recurrent problem olving: Node that are encountered twice on the traveral path are dicarded. S:II-159 Baic Search Algorithm STEIN/LETTMANN

45 Depth-Firt Search of AND-OR Graph Dealing with Cycle (1) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): A A B + B... + A S:II-160 Baic Search Algorithm STEIN/LETTMANN

46 Depth-Firt Search of AND-OR Graph Dealing with Cycle (1) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): A A B + B... + (partial) ocurrence check A + S:II-161 Baic Search Algorithm STEIN/LETTMANN

47 Depth-Firt Search of AND-OR Graph Dealing with Cycle (2) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): A B + + S:II-162 Baic Search Algorithm STEIN/LETTMANN

48 Depth-Firt Search of AND-OR Graph Dealing with Cycle (2) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): t 0 t 1 A B A B + + B A (partial) ocurrence check A + S:II-163 Baic Search Algorithm STEIN/LETTMANN

49 Depth-Firt Search of AND-OR Graph Dealing with Cycle (2) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): t 0 t 1 A B A B + + B A (partial) ocurrence check A + B S:II-164 Baic Search Algorithm STEIN/LETTMANN

50 Depth-Firt Search of AND-OR Graph Dealing with Cycle (2) AND-OR graph (left) and it exploration (= unfolding) with DFS (right): t 0 t 1 A B A B + + B A (partial) ocurrence check A B + + S:II-165 Baic Search Algorithm STEIN/LETTMANN

51 Remark: Q. Why doe the dicarding of recurring node (partial occurrence check) not compromie the completene of olved-labeling? Recapitulation. When earching an AND-OR graph with DFS, a partial occurrence check cannot addre the iue of redundant problem olving in general. Given the previou AND-OR graph (2), breadth-firt earch would have found a olution in depth 2 already. S:II-166 Baic Search Algorithm STEIN/LETTMANN

52 AND-OR Graph Search Enfolding of Solution Tree: Compact Repreentation a AND-OR Graph Advantage: Avoiding multiple olving of the ame problem. AND-OR graph example: Two poible olution graph: Idea: Multiple intance of node from the AND-OR graph G are merged. Reulting AND-OR graph i a ubgraph of the AND-OR graph G. S:II-167 Baic Search Algorithm STEIN/LETTMANN

53 AND-OR Graph Search Enfolding of Solution Tree Enfolding of olution tree can reult in cyclic AND-OR graph (no unique tructure): n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Solution graph for?? S:II-168 Baic Search Algorithm STEIN/LETTMANN

54 AND-OR Graph Search Enfolding of Solution Tree Enfolding of olution tree can reult in cyclic AND-OR graph (no unique tructure): n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Acyclic Solution olution graph for for?? Conequence: AND-OR olution graph are required to be acyclic. S:II-169 Baic Search Algorithm STEIN/LETTMANN

55 AND-OR Graph Search Enfolding of Solution Tree Enfolding of olution tree can reult in cyclic AND-OR graph (no unique tructure): n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Acyclic Solution olution graph for for?? Conequence: AND-OR olution graph are required to be acyclic. Problem: Folding AND-OR olution tree can reult in cyclic AND-OR graph. S:II-170 Baic Search Algorithm STEIN/LETTMANN

56 AND-OR Graph Search Enfolding of Solution Tree (continued) Syntheized olution graph are not necearily minimum: n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Solution graph for?? S:II-171 Baic Search Algorithm STEIN/LETTMANN

57 AND-OR Graph Search Enfolding of Solution Tree (continued) Syntheized olution graph are not necearily minimum: n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Minimum Solution olution graph graph for for?? S:II-172 Baic Search Algorithm STEIN/LETTMANN

58 AND-OR Graph Search Enfolding of Solution Tree (continued) Syntheized olution graph are not necearily minimum: n 1 n 2 AND-OR graph Solution graph for n 1 Solution graph for n 2 Minimum Solution olution graph graph for for?? Problem: A definition of olution graph will not be contructive. S:II-173 Baic Search Algorithm STEIN/LETTMANN

59 AND-OR Graph Search Solution Graph in AND-OR Graph [ Solution Path Definition 14 (Solution Graph in AND-OR Graph) in OR Graph] Let G be an AND-OR graph, and let n be a node in G. Alo, let ome additional olution contraint be given. A ubgraph H of G i called a olution graph for n in G iff ( ) the following condition hold: 1. H i finite and acyclic. 2. H contain the node n. 3. If H contain an inner OR node n, then H contain exactly one link to a ucceor node of n in G and thi ucceor node. 4. If H contain an inner AND node n, then H contain all link to it ucceor node of n in G and all thee ucceor node. 5. The leaf node in H are goal node in G. 6. H atifie the additional olution contraint. 7. H i minimal: it contain no additional node or edge. The acyclicity condition on H can be omitted if earch i retricted to acyclic AND-OR graph G. S:II-174 Baic Search Algorithm STEIN/LETTMANN

60 Remark: If H i a olution graph for a node n in G and if n i ome node in H, then the ubgraph of H rooted at n, H, i a olution graph for a node n in G. Thi ubgraph i called the olution graph in H induced by n. Each olution graph define a decompoition hierarchy of the problem aociated with n. The intantiation of a olution graph i baed on local deciion. E.g., a olution graph for ome AND node i given by combining olution graph of it ucceor. A olution graph i compact in the ene that it doe not contain multiple intance of identical ret problem. A legitimate olution in a problem-reduction graph mut be finite. Thi property i implicitly fulfilled ince the problem-reduction graph itelf i created by a earch algorithm within a finite amount of time. Becaue of condition (1) in the above definition, earching for a olution graph in an AND-OR graph correpond to earching for a cycle-free olution path in an OR graph. S:II-175 Baic Search Algorithm STEIN/LETTMANN

61 Remark: One of the main advantage of problem decompoition approache i that olution to ubproblem can be reued if uch ubproblem occur multiple time. Therefore, the contructability requirement alo conflict with the reue apect. A contructional procedure for bottom-up olution graph ynthei could look like follow: 1. If n Γ, i.e., n i a node that repreent a olved ret problem, then H n := {n}, i a olution graph for n in G. 2. If n i an OR node with ucceor n in G and H n = V, E i a olution graph for n in G, then H n := V {n}, E {(n, n )} i a olution graph for n in G. 3. If n i an AND node with ucceor n 1,..., n k in G and H n = V i i, E i i a olution graph for n i in G, i = 1,..., k, then H n := V 1... V k {n}, E 1... E k {(n, n 1),..., (n, n k )} i a olution graph for n in G. Q. Why i the above proce problematic? Following the above inductive decription, a recurive procedure can be implemented that contruct uch graph. By contruction thee graph will be finite and contain tart node. Iteratively eliminating edge and node from that graph will lead to a olution graph (apart from additional olution contraint that might not be met). An example of additional olution contraint i a maximum edge cot retriction for olution graph. S:II-176 Baic Search Algorithm STEIN/LETTMANN

62 AND-OR Graph Search Solution Graph in AND-OR Graph (continued) Definition 15 (Solution Bae in an AND-OR Graph) Let G be an AND-OR graph, and let n be a node in G. A olution bae H for n in G i defined in the ame way a a olution graph for n in G except for condition 5 on leaf node and condition 6 on additional olution contraint. Leaf node mut not be dead end. Uage. Solution Graph We are intereted in finding a olution graph H for the tart node in G. Solution Bae Algorithm maintain and extend a et of promiing olution bae until a olution graph i found. Note. Solution graph and olution bae are ubgraph of graph G. S:II-177 Baic Search Algorithm STEIN/LETTMANN

63 Remark: Simply put, a ubgraph of a earch pace graph i called olution bae if it can be extended toward a olution (graph). Obviouly, all olution graph contained in G are alo olution bae. If H i a olution bae for a node n in G and if n i ome node in H, then the ubgraph of H rooted at n, H, i a olution bae for a node n in G. Thi ubgraph i ometime called the olution bae in H induced by n. The fact whether a problem-reduction graph G contain a olution bae H for the tart node can be checked by recurively applying the propagation rule of the [olved-labeling procedure]. S:II-178 Baic Search Algorithm STEIN/LETTMANN

64 AND-OR Graph Search Generic Schema for AND-OR-Graph Search Algorithm... from a olution-bae-oriented perpective: 1. Initialize olution-bae torage. 2. Loop. (a) Uing ome trategy elect a olution bae to be extended. (b) (c) (d) Uing ome trategy elect an unexpanded node in thi bae. According to the node type extend the olution bae by ucceor node in any poible way and tore the new candidate. Determine whether a olution graph i found. Uage: Search algorithm following thi chema maintain a et of olution bae. Initially, only the tart node i available; node expanion i the baic tep. S:II-179 Baic Search Algorithm STEIN/LETTMANN

65 AND-OR Graph Search Algorithm: Input: Output: Generic_AND-OR_Search. Start node repreenting the initial tate (problem). ucceor(n). Return the ucceor of node n. (b). Predicate that i True if olution bae b i a olution tree. A olution graph b or the ymbol Fail. S:II-180 Baic Search Algorithm STEIN/LETTMANN

66 AND-OR Graph Search [Generic_AND_OR_Tree_Search] Generic_AND-OR_Search(, ucceor, ) 1. b = olution_bae(); // Initialize olution bae b. IF (b) THEN RETURN(b); // Check if b i olution graph. 2. puh(b, OPEN); // Store b on OPEN waiting for extenion. 3. LOOP 4. IF (OPEN = ) THEN RETURN(Fail); 5. b = chooe(open); // Chooe a olution bae b from OPEN. remove(b, OPEN); // Delete b from OPEN. n = chooe(b); // Chooe unexpanded tip node in b. [ Generic_OR_Search] 6. IF (i_and_node(n)) THEN b = add(b, n, ucceor(n)); // Expand n and extend b by AND edge. IF (b ) THEN RETURN(b ); // Check if b i olution graph. puh(b, OPEN); // Store b on OPEN waiting for extenion. ELSE FOREACH n IN ucceor(n) DO // Expand n. b = add(b, n, {n }); // Extend b by OR edge (n, n ). IF (b ) THEN RETURN(b ); // Check if b i olution graph. puh(b, OPEN); // Store b on OPEN waiting for extenion. ENDDO ENDIF 7. ENDLOOP S:II-181 Baic Search Algorithm STEIN/LETTMANN

67 Remark: Algorithm Generic_AND-OR_Search take a olution-bae-oriented perpective. Generic_AND-OR_Search maintain and manipulate olution bae (which are AND-OR graph with root ). In order to keep the peudo code hort, we do not cover the cae that a olution bae choen in 5. ha no unexpanded tip node. Such a olution bae can be dicarded. Remaining problem which till are to be olved can be found only among the tip node. So, a olution bae may contain multiple open problem. Function add(b, n,.) return a repreentation of the extended olution bae: node intance in ucceor(n) will be unified with intance already contained in b, the edge to the direct ucceor have to be added in any cae. For that purpoe, b i copied. So each olution bae i repreented eparately, although they often hare ubgraph. To be precie, function add(b, n,.) hould include a check whether the reulting olution bae are till acyclic. Cyclic olution bae have to be dicarded. (In order to avoid a tet for cycle, it i often aumed that the earch pace graph G i acyclic.) Solution bae handled in Generic_AND-OR_Search contain at mot one intance per node of the underlying earch pace graph. Neverthele we till aume that function ucceor(n) return new intantiation (clone) for each ucceor node of an expanded node. S:II-182 Baic Search Algorithm STEIN/LETTMANN

68 AND-OR Graph Search Efficient Storage of Solution Graph AND-OR graph example: Two poible olution graph: Solution bae can hare multiple and completely different part. Node expanion can happen at any terminal node in a olution bae. There i no partial order on olution bae. Q. I there a benefit of uing backpointer? S:II-183 Baic Search Algorithm STEIN/LETTMANN

69 AND-OR Graph Search Solution Graph and Backpointer: Sharing Initial Part AND-OR graph example: Two poible olution graph: S:II-184 Baic Search Algorithm STEIN/LETTMANN

70 AND-OR Graph Search Solution Graph and Backpointer: Sharing Initial Part AND-OR graph example: Two poible olution graph: hared olution bae Backpointer can be ued to tore olution bae efficiently if a olution bae i included completely. Problem. Which olution bae are hared depend on the order of node expanion. S:II-185 Baic Search Algorithm STEIN/LETTMANN

71 AND-OR Graph Search Solution Graph and Backpointer: Sharing Solution Graph AND-OR graph example: Two poible olution graph: S:II-186 Baic Search Algorithm STEIN/LETTMANN

72 AND-OR Graph Search Solution Graph and Backpointer: Sharing Solution Graph AND-OR graph example: Two poible olution graph: hared olution graph Solution graph for ubproblem occur in different olution graph. Problem. For haring of olution graph more than one backpointer per node i needed. S:II-187 Baic Search Algorithm STEIN/LETTMANN

73 AND-OR Graph Search Efficient Storage of Solution Bae Idea: Since each olution bae i a ubgraph of G, thee ubgraph can eaily be combined and for a finite ubgraph G e of G, the explored part of G. G e i tored explicitly, i.e. edge from node to direct ucceor are tored. Advantage: Only a ingle intance i tored for each node of G in G e. Extenion by node expanion become available to all olution bae that are currently contained in G e. DFS can be applied to G e, e.g. to determine olution graph that are contained. S:II-188 Baic Search Algorithm STEIN/LETTMANN

74 AND-OR Graph Search Efficient Storage of Solution Bae Idea: Since each olution bae i a ubgraph of G, thee ubgraph can eaily be combined and for a finite ubgraph G e of G, the explored part of G. G e i tored explicitly, i.e. edge from node to direct ucceor are tored. Advantage: Only a ingle intance i tored for each node of G in G e. Extenion by node expanion become available to all olution bae that are currently contained in G e. DFS can be applied to G e, e.g. to determine olution graph that are contained. Idea: Backpointer connect each node with all of it expanded parent in G e. combined and for a finite ubgraph G e of G, the explored part of G. G e i tored explicitly, i.e. edge from node to direct ucceor are tored. Advantage: Sharing of olution ubgraph i poible. Information can be propagated in any poible way in direction to the tart node. S:II-189 Baic Search Algorithm STEIN/LETTMANN

75 AND-OR Graph Search Foundation of AND-OR Graph Search General Structure. Algorithm maintain the explicitly explored part G e of the AND-OR graph G. Initially, only the tart node i available; node expanion i the baic tep. Unexpanded node are tored on OPEN, expanded node on CLOSED. For each expanded node pointer to each of it ucceor node are tored. With each generated node backpointer to each of it parent node are tored. Solution bae maintained are the olution bae for in G e with tip node in OPEN. Advantage. Algorithm mut not handle multiple intance of node. Graph algorithm can be ued a ubroutine to proce G e. Diadvantage: Solution bae / olution graph have to be computed from G e. Solution bae, e.g. cyclic olution bae, cannot be dicarded eparately. Therefore, AND-OR graph earch i now retricted to acyclic AND-OR graph. S:II-190 Baic Search Algorithm STEIN/LETTMANN

76 AND-OR Graph Search Algorithm: Input: Output: Baic_AND-OR_Search. Start node repreenting the initial problem. ucceor(n). Return the ucceor of node n. i_goal_node(n). Predicate that i True if n i a goal node. A olution graph or the ymbol Fail. S:II-191 Baic Search Algorithm STEIN/LETTMANN

77 AND-OR Graph Search [ Baic_OR_Search ] Baic_AND-OR_Search(, ucceor, ) 1. inert(, OPEN); add_node(, G e ); // G e i the explored part of G. 2. LOOP 3. IF (OPEN = ) THEN RETURN(Fail); 4. H = chooe_olution_bae(, G e ); // Chooe olution bae for in G e. n = chooe(open, H); // Chooe OPEN tip node in H. remove(n, OPEN); puh(n, CLOSED); 5. FOREACH n IN ucceor(n) DO ENDDO 6. ENDLOOP IF (n OPEN OR n CLOSED) // Intance of n een before? THEN // Ue old intance of n intead. n = retrieve(n, OPEN CLOSED); ELSE // n encode an intance of a new tate. inert(n, OPEN); add_node(n, G e ); ENDIF add_backpointer(n, n); IF ( i_goal_node(n ) OR i_labeled(n ) ) // I n olvable? THEN propagate_label(n ); IF i_olved() THEN RETURN(compute_olution_graph(G e )); ENDIF S:II-192 Baic Search Algorithm STEIN/LETTMANN

78 Remark: Algorithm Baic_AND-OR_Search take a graph-oriented perpective: A olution bae H that need expanion can be computed from G e if there are OPEN node available. A ingle OPEN node uually doe not repreent a olution bae (if nonterminal AND node i contained in the backpointer path). Here, OPEN can be een a the earch frontier, the border line between the explored and the unexplored part of the underlying earch pace graph. Remaining problem which till are to be olved can be found only among the tip node of a olution bae H. However, a olution bae may contain multiple open problem. Expanding a node affect all olution bae that contain thi node a a tip node. Function propagate_label(n) will propagate information about olved or unolvable ubproblem. Compared to [olved labeling in DFS], not only a path, but a directed acyclic graph i defined by the backpointer tarting in a node n. Even if the propagation of information olvable guarantee that i olved, the found olution graph doe not have to be an extenion of H. Therefore, function chooe_olution_graph(g e )) earche for a olution graph in G e, e.g. uing DFS. Again, no additional olution contraint are conidered here. S:II-193 Baic Search Algorithm STEIN/LETTMANN

79 Remark (continued): Teting whether are graph i acyclic can be done in linear time (earch for a topological orting). However, G e i the union of all olution bae under conideration. Extending one olution bae by node expanion will affect multiple olution bae available in G e. For ome of them, thi extenion may lead to a cycle, for ome other not. But there i no way to dicard ingle olution bae from G e. Cyclic olution bae will have to be ruled out again and again. So, there i no eay way to ue multiple backpoiter and imultaneouly to avoid cycle. Thi i again a jutification for auming that the earch pace graph G i acyclic. S:II-194 Baic Search Algorithm STEIN/LETTMANN