Additive AND/OR Graphs

A d d i t i v e AND/OR graphs are de f ined as AND/ /ORgraphs w i t hou t c i r c u i t s , which can be cons idered as fo lded AND/OR t r e e s ; i . e . the cost of a common subproblem is added to the cost as many t imes as the subproblem occurs , but i t is computed on ly once. A d d i t i v e AND/OR graphs are n a t u r a l l y obta ined by r e i n t e r p r e t i n g the dy namic programming method in the l i g h t of the prob lem-reduct ion approach. An example of t h i s r educ t i on i s g i v e n . A top-down and a bottom-up method are p ro posed f o r searching a d d i t i v e AND/OR graphs. These methods a re , r e s p e c t i v e l y , extensions of the "arrow" method proposed by N i l sson f o r searching AND/OR t r ees and D i j k s t r a ' s a l g o r i t h m f o r f i n d i n g the sho r tes t p a t h . A proof is g iven t h a t the two methods f i n d an op t ima l s o l u t i o n whenever a s o l u t i o n e x i s t s . 1) i n t r o d u c t i o n I n the l i t e r a t u r e o n a r t i f i c i a l i n t e l l i gence, AND/OR t rees have proved to be a good formal ism f o r rep resen t i ng the problem-reduct i o n approach to problem so l v i ng . Usua l l y , the search is f o r any s o l u t i o n t r e e , but in a paper by N i l sson the problem is presented of f i n d i n g the best s o l u t i o n t r e e , where arcs have a g iven c o s t , and the cost of a t r e e is s imply the sum of the cos ts of the a r c s . N i l s son g ives the re an a l g o r i t h m which as sumes a v a i l a b l e , f o r each node, an est imate of the cos t o f the "opt imal s o l u t i o n t r e e roo ted a t t h a t node. An a l g o r i t h m f o r searching AND/OR graphs has been proposed by Chang and S l a g l e 3 . Here a s o l u t i o n graph is de f ined in the usual way, and, as f o r t r e e s , the cos t of a s o l u t i o n graph is the sum of the cos ts of i t s a r c s . In t h i s paper we in t roduce a new type of AND/OR graphs c a l l e d a d d i t i v e , which can be considered as fo lded AND/OR t r e e s , i . e . t r ees where d i f f e r e n t nodes have been recognized to be roo ts of equal subtrees and have been i d e n t i f i e d , thus genera t ing AND/OR graphs w i thou t c i r c u i t s . However, the cost of a s o l u t i o n subgraph is de f ined as equal to the cost o f the unfo lded equ iva len t t r e e . In o ther words, the cost of a common subproblem is added to the cost as many t imes as the subproblem occurs , but i t i s computed on l y once. A d d i t i v e AND/OR graphs are n a t u r a l l y obta ined by r e i n t e r p r e t i n g the dynamic programming method o f o p t i m i z a t i o n in the l i g h t o f the p rob lem-reduc t ion approach 4 . For g i v i n g an idea of the technique of r e d u c t i o n , we w i l l consider here the w e l l -known problem of ha lanc ing b ina ry search t r e e s , solved by Knuth v i i t h a (modif ied) dynamic programming a l g o r i t h m " . Here a number o f ordered i tems ( l e x i c o g r a p h i c a l l y ordered words, f o r ins tance) are g i v e n , together w i t h p r o b a b i l i t i e s of a new i tem occurence to be any of the g iven i tems or to be a new i tem located in any in te rmed ia te p o s i t i o n . For example, the data mean t h a t there are 3 p r o b a b i l i t i e s out of 31 t h a t a new word w i l l be "do" and 3 n r o h a h i l i t i e s t h a t a new word w i l l be any word between "do" and " i f " . A b ina ry search t r e e f o r these data is a t r e e o f the type shown in F i g . 1 . For ins tance i f the new word "a r ray " i s generated, i t s proper l o c a t i o n is found by means of three t e s t s , namely comparisons w i t h the given words " i f " and "beg in" r e s p e c t i v e l y and a terminat i o n t e s t . Given a search t r e e , the average number of t e s t s M necessary f o r reaching a node, is then g iven by summing up the products of the number mj. of t e s t s requ i red f o r any node i and i t s p r o b a b i l i t y p1 For i ns tance , the average number of t e s t s f o r the t r e e in F i g . 1 i s The problem is to f i n d a search t r e e w i t h min imal cos t M. For i ns tance , the t r e e of Fig.1 i s op t ima l f o r the data ( 1 ) . An equ i va len t f o rmu la t i on of the same problem considers f requencies ins tead of p r o b a b i l i t i e s . I n t h i s case the cost i s c a l l e d weighted path l e n g t h . Two p r o p e r t i e s a l l ow the use of a dynamic programming technique i n t h i s case. F i r s t , i f T = (a A 8) is an op t ima l t r e e rooted in A and having a and 6 as sub t rees , then both o and B are o p t i m a l . Fur thermore, the data f o r a and S are d i s j o i n t subs t r i ngs of the data f o r T. Second, i f fA , fa , fB ' L a 'B are the f requencies of A,o and 8 and the weighted In the dynamic programming algori thm, a l l the subproblems of the given problem are considered, whose data are substrings of the given data. Such problems are divided in lev e l s , according to the length of the data, and solved in increasing order* A problem at a leve l can be solved by picking up a root in a l l possible ways* (say JO and thus decomposing the problem in k pairs of subproblems at lower leve ls . The cost of every decomposition is computed using (2)and a best decomposition is chosen. The optimal search tree problem described above is a good example of the gener al case where, at each stage, the computat i on of each a l ternat ive requires the sum of the costs of one or more ( in f ac t , two) sub problems. In th i s case, the structure of the problem is conveniently ref lected in to an addi t ive AND/OR graph, whose AND nodes correspond to subproblem cost sums, and OR nodes to a l t e r native select ions. For instance, the AND/OR graph for the optimal search tree problem, with data (1) , is shown in F ig . 2. There,AND nodes are marked wi th c i r c les and OR nodes (corresponding to subproblems) wi th squares. The optimal solut ion graph, corresponding to the search tree in F ig . 1, is blackened. Reduction of dynamic programming to addit1 ve AND/OR graphs can imply several advantages. F i r s t , the AND/OR graph expresses the structure of the problem in the form of a p a r t i a l ordering of subproblems. Dynamic pro gramming solves a l l the subproblems bottom-up in some s ta t i c order which respects the p a r t i a l orderings. However,in Section 4 of th is paper a bottom-up algorithm is descr i bed, which solves every time the cheapest avai lable subproblem, thus considering, in general, only a subset of subproblems in a (*) Actua l ly , Knuth gives a modified dynamic programming algorithm which excludes a p r i o r i most of the decompositions, using a par t i cu la r monotonieity property. dynamic order. Second, in many cases estimates of the subproblem costs are avai lab le , which can be used for d i rec t ing a top-down search. In f a c t , in Section 3 we give an extension of Nl lsson's t ree algorithm to the addi t ive graph case, which, if the estimate is a lower bound of the minimal cost , is guaranteed to f i nd the o p t i mal solut ion graph. The algorithm is s l i g h t l y s imp l i f ied i f the estimate sa t i s f i es a "consistency" constra int . F ina l l y , various known heur is t ic techniques can be applied to a d d i t i ve AND/OR graphs,which can f ind a good solut i on where the exact dynamic programming algor i thm is too Expensive. We emphasize that th i s paper extends to general dynamic programming a reduction technique which is well-known in the so cal led "sequential" case. There, each a l ternat ive can be computed by summing a constant to the cost of one simpler subproblem, thus reducing an addi t ive AND/OR graph to an OR graph. In th i s case, a solut ion tree reduces to a path and thus shortest path algorithms apply, l i ke D i j ks t ra ' s algorithm in the uninformed case and the algorithm by Hart, Ni lsson, and Raphael i f an estimate is avai lab le . F ina l l y , note that in the shortest path case no d i s t i nc t i on between top-down and bottom-up is needed, while i t is suggestive in the general case. 2) Addit ive AND/OR Graphs In t h i s paper, we shal l consider AND/OR graphs without cycles. An example is given in F ig . 3. Node A is cal led the s ta r t node and represents the problem to be solved. Node A is cal led an OR node,(*) because a solut ion is constructed by select ing ei ther i t s successor B or i t s successor C. Node B is cal led an AND node, because a l l of the suboroblems represent ed by i t s successors D and E must be solved in order to solve problem B. An AND node is i n d i cate by a l ine across the arcs connecting it to the successor nodes. Nodes G and H are c a l led terminal nodes and correspond to problems wi th known so lu t ion . We assume that a l l the nonterminal nodes are e i ther OR nodes or AND nodes, that i s , there are no nodes corresponding to problems which can be solved by solving some of the i r successors. We give here a de f i n i t i on of OR nodes and AND nodes, which is opposite to the one given by Ni lsson . We do so, because we want each node to be ei ther an OR node or an AND node, instead, wi th Nl lsson's d e f i n i t i o n , we could have a node which is at the same time an OR node because of one parent and an AND node because of another parent. W e s h a l l b e c o n c e r n e d w i t h A N D / O R g r a p h s i m p l i c i t l y s p e c i f i e d by a s t a r t node s and a s u c c e s s o r o p e r a t o r T . A p p l i c a t i o n o f r t o any node n q e n e r a t e s a f i n i t e number o f s u c c e s s o r s o f n and a l a b e l s n e c i f y i n o w h e t h e r t h e s u c c e s s o r s a r e AND nodes or OR n o d e s , A s o l u t i o n g r a p h of an AND/OR g r a n h G w i t h s t a r t node s i s any suhoranh o f P c o n t a i n i n g s and h a v i n o t h e f o l l o w i n g p r o p e r t i e s ; (1) Suppose node n of P is an OR node and i s i n c l u d e d i n t h e s o l u t i o n o r a p h . Then one and o n l y one o f t h e s u c c e s s o r s o f n i s a l s o i n c l u d e d i n t h e s o l u t i o n g r a p h . (2) Suppose node n of R is an AND node and i s i n c l u d e d i n t h e s o l u t i o n g r a p h . Then a l l o f t h e s u c c e s s o r s o f n a r e a l s o i n c l u d e d i n t h e s o l u t i o n o r a n h . (3) The s o l u t i o n g r a n h i s f i n i t e , meanlno t h a t i t ends i n a s e t o f t e r m i n a l n o d e s , F ig . 4 shows two s o l u t i o n o ranhs o f t h e A N D / O R q r a p h i n F i g . 3 . I n g e n e r a l , a c o s t i s a s s o c i a t e d w i t h e v e r y a r c o f a n AND/OR G r p h . L e t t h e f u n c t i o n c ( n , n . ) g i v e t h e c o s t t o h e a s s o c i a t e d w i t h t h e a re c o n n e c t i n g node n i w i t h one o f i t s s u c c e s s o r s n j . F o r a d d i t i v e AND/OR o ranhs t h e c o s t o f a s o l u t i o n g r a n h i s r e c u r s i v e l y d e f i n e d a s f o l l o w s : (1) The c o s t c * 0 is a s s o c i a t e d w i t h e v e r v t e r m i n a l node i n t h e s o l u t i o n o r a n h . (2) L e t c 1 , — , c k b e t h e c o s t s a s s o c i a t e d w i t h t h e k s u c c e s s o r s o f node n i n t h e s o l u t i o n o r a n h . Then w e a s s o c i a t e w i t h n t h e c o s t c o i v e n b y ( * ) k C = 2 _ ( c , + c ( n , n ))