An Improved Di-Directional Heuristic Search Algorithm

A m o d i f i c a t i o n o f P o h l ' s b i d i r e c t i o n a l h e u r i s t i c search a l g o r i t h m i s d e s r i b e d t o g e t h e r w i t h a s i m p l i f i e d i m p l e m e n t a t i o n . Theorems are p roved about c o n d i t i o n s y i e l d i n g s h o r t e s t p a t h s . The r e s u l t s a re g i v e n o f a wors t case a n a l y s i s o f d i f f e r e n t a l g o r i t h m s s u g g e s t i n g a r anko rde r o f t h e i r q u a l i t y . Resu l t s t h a t Pohl had o b t a i n e d w i t h a u n i d i r e c t i o n a l h e u r i s t i c search a l g o r i t h m on the 1 5 p u z z l e are compared w i t h the r e s u l t s o b t a i n e d by the new s i m p l i f i e d a l g o r i t h m . keywords: a r t i f i c i a l i n t e l l i g e n c e , b i d i r e c t i o n a l n e u r i s t i c s e a r c h , f r o n t t o f r o n t g u i d i n g , p a t h f i n d i n g . 1 . I n t r o d u c t i o n A s i s w e l l known, see e . g . / 1 / , a h e u r i s t i c f u n c t i o n can be used to gu ide u n i d i r e c t i o n a l s e a r c h . The p r o p e r t y o f end ing w i t h a s h o r t e s t p a t h b e tween s t a r t s t a t e and g o a l s t a t e , as can be found w i t h b read th f i r s t search w i t h o u t h e u r i s t i c f u n c t i o n , can even b e p rese rved i f t h e h e u r i s t i c f u n c t i o n is a l ower bound on t h e r e a l minimum e f f o r t to be made. In /2/ and / 3 / an a l g o r i t h m is d e s c r i b e d which g e n e r a l i z e s t o t h e b i d i r e c t i o n a l case . T h i s a l g o r i t h m i n f a c t pe r fo rms two i n d e pendent u n i d i r e c t i o n a l s e a r c h e s , a f o r w a r d search gu ided to the g o a l node, and a backward search gu ided t o the s t a r t node. The d isadvan tage o f t h i s i s , t h a t i n a search space where more t h a n one pa th e x i s t s from t h e s t a r t node to the g o a l node , t h e two searches o f t e n proceed a l o n g two d i f f e r e n t p a t h s , s o t h a t t h e two se ts o f c l o s e d nodes ( f o r the t e r m i n o l o g y o f ' c l o s e d ' , ' n o d e s ' and o t h e r t e c h n i c a l j a r g o n see / 1 / ) grow i n t o n e a r l y complete u n i d i r e c t i o n a l t r e e s b e f o r e i n t e r s e c t i n g each o t h e r , i n s t e a d o f mee t ing i n t h e ' m i d d l e ' o f t h e space. In the nex t s e c t i o n we d i scuss ano ther a l g o r i t h m t h a t remedies t h i s d e f i c i e n c y and we g i v e a s i m p l i f i c a t i o n o f t h a t a l g o r i t h m which has been i m p lemented in a F o r t r a n program. In s e c t i o n 3 we compare our r e s u l t s w i t h t h e r e s u l t s o f P o h l ' s u n i d i r e c t i o n a l h e u r i s t i c search o n t h e 15 -puzz le . In s e c t i o n 4 we men t ion some p o s s i b l e improve ments o f t he implemented a l g o r i t h m . 309 Before g o i n g i n t o a p r e c i s e d e s c r i p t i o n o f t h e BHFFA, we g i v e an i n t u i t i v e s k e t c h . Suppose we have t h e s i t u a t i o n as in f i g . l , where S and T are t h e s e t s of c l osed nodes and 3 and T are t h e two f r o n t s of open nodes , and we dec ide to e x pand a node f rom S. In P o h l ' s BHPA a l g o r i t h m a node was chosen in S wh ich had a minimum va lue of t he f u n c t i o n g +h s where g was the c u r r e n t s s s mimimum to t h e s t a r t node & and h was an e s t i mator o f t h e d i s t a n c e f rom the node to t h e goa l node t. The h we use is d i f f e r e n t , h is a m i — s s nimum of H+g over every node in the o p p o s i t e f r o n t T , where g i s l i k e g but w i t h r e s p e c t to t h e goa l node t and H is an e s t i m a t o r of t h e s h o r t e s t d i s t a n c e between a r b i t r a r y p a i r s o f nodes. Immed ia te ly the d isadvan tage o f t h i s a l g o r i t h m w i t h respec t to BHPA must be c l e a r , s i n c e t h e c a l c u l a t i o n o f t he f u n c t i o n h i n our a l g o r i t h m i s much more c o m p l i c a t e d t han the c a l c u l a t i o n o f t h e d i s t a n c e to t in BHPA. On t h e o t h e r hand t h e r e are g a i n s , bu t we d e f e r d i s c u s s i n g them t o t h e next s e c t i o n . I n