# Depth Increment in IDA Ling Zhao Dept of

- Slides: 15

Depth Increment in IDA* Ling Zhao Dept. of Computing Science University of Alberta July 4, 2003

Outline l Introduction l Examples l Modifications on IDA* l Analytical Model l Experimental results l Work to do l Conclusions

Introduction l In some single-agent search problems with very low branching factors (less than 2), the generic IDA* typically visits many more nodes than A*. l Branching factor: b solution length: s A* -- bs IDA* -- b 1+b 2 +… +bs = (bs+1 -b)/(b-1) when bs is large enough, IDA* visits about 1/(b-1) times more nodes than A*.

Examples Overhead = 1/(1 -b) PCP: b = 1. 120 overhead = 8. 33 Pathfinding: b is very close to 1 (? ) overhead > 30

How to solve the problem? l Depth is increased by more than one in each iteration. cons: less redundant work pros: may need to search nodes with f value greater than solution length. Modification on generic IDA*(ensure optimality): when a solution is found, update the search threshold and keep searching.

Analytical model Assumptions: l search tree grows perfectly in an exponential series (1<b<2, s is very large) l need to find all optimal solutions l depth increment is a constant

A* - bs Analytical model IDA* - (bs+1 -b)/(b-1) -> bs+1/(b-1) IDA*’- (b(i+1)c- bc)/ (bc -1) -> b(i+1)c/ (bc -1) (ic = k+s) = bs * bk+c / (bc -1) overhead = bk+c / (bc -1) -1 Iteration 1: bc c solution length s Iteration 2 : b 2 c Iteration i-1: b(i-1)c k Iteration i: bic

overhead = bk+c / (bc -1) – 1 worst case (k=c): Min(overhead)=3 when bc = 2 best case (k=0): Min(overhead)->0 when bc -> +infinity average case (k=c/2): Min(overhead)=1. 598 when bc = 3 Iteration 1: bc c solution length s Iteration 2 : b 2 c Iteration i-1: b(i-1)c k Iteration i: bic

overhead = bk+c / (bc -1) – 1 another average case: only shadowed area is visited (suppose it is 50% of the over-searched area), and k=c/2 Min(overhead) = 1. 1 when bc =2. 28 Iteration 1: bc c solution length s Iteration 2 : b 2 c Iteration i-1: b(i-1)c k Iteration i: bic

Experimental results l 200 difficult PCP instances k = 45%i x = 20% of over-searched area From the model: Min(overhead) = 0. 81 when bc = 7. 70 Empirically: Min(overhead)=0. 67 when c=20

Empirical Results

Further thoughts Decrease overhead of IDA*’ 1. node overhead - better heuristics (look ahead) 2. right increment - very specific to problems 3. variable increment? ? ?

Work to do l Test the impact from better heuristics (lookahead) l Apply the idea to other problems - pathfinding? - two-player games with low bf - sliding puzzles (15, 24 -puzzles)? l Better statistical methods to compare analytical and empirical results

Summary Search problems with very low branching factors receive relatively less attention. difficult problem => high/medium bf l PCP is a good domain to test algorithms for problems with low bf l IDA* with large depth increment performs much better in some problems with low branching factor (1<b<2) and typically very long solution lengths. l IDA*’ is almost cost-free l

A Simple Example of PCP Instance 1 2 3 100 1 00 Rules: 100 pair 1: 1 1001 pair 3: 1001100 pair 1: 1001 pair 1: 100100 10011 1. select pairs 1001001 pair 3: 1001100 2. make the concatenated top string and bottom string identical. pair 2: 100100100 pair 2: 1001100100100

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