Combining Front-to-End Perimeter Search and Pattern Databases

Combining Front-to-End Perimeter Search and Pattern Databases CMPUT 652 Eddie Rafols

Motivation From Russell, 1992

Allocating Memory • More memory for Open/Closed Lists • Caching • Perimeter Search • Pattern Databases

Perimeter Search • Generate a perimeter around the Goal • Any path to Goal must pass through a perimeter node Bi • We know the optimal path from Bi to Goal • Stopping condition for IDA* is now: • If A is a perimeter node and g(Start,A)+g*(A,Goal)  Bound

Perimeter Search • Traditional Search O(bd) • Perimeter Search O(br+bd-r) • Large potential savings!

Perimeter Search • Can be used to improve our heuristic • Kaindl & Kainz, 1997 • Add Method • Max Method

Pattern Databases • Provide us with a consistent estimate of the distance from any given state to the goal* *This point will become relevant in a few slides

Approach • Generate a pattern database to provide a heuristic • Use Kaindl & Kainz’s techniques to improve on heuristic values (Add,Max) • Determine how perimeter search and PDBs can most effectively be combined via empirical testing

A Digression… • Among other things, the Max method requires: • h(Start, A), where A is a search node • h(Start, Bi), where Bi is a perimeter node

A Digression… • Among other things, the Max method requires: • h(Start, A), where A is a search node • h(Start, Bi), where Bi is a perimeter node • We are not explicitly given this information in a PDB.

A Digression... • Recall: Alternate PDB lookups • If we are dealing with a state space where distances are symmetric and ‘tile’-independant, we can use this technique

A Digression... • ex. Pancake Problem

A Digression... • However, this technique may provide inconsistent heuristics ϕ1 ϕ2 ϕ1  ϕ2

A Digression... • Kaindl’s proof of the max method relies on a consistent heuristic • ...but we can still use our pattern database • We just have to use it correctly

A Digression... • Distances are symmetric, therefore • h*(Start, A) = h*(A, Start) • h*(Start, Bi) = h*(Bi, Start) • We can map the Start state to the Goal state • In this case, when we do alternate lookups on A and Bi, we are using the same mapping! • Our heuristic is now consistent!

A Digression... ϕ ϕ ϕ

A Third Heuristic? • Since we have the mechanisms in place, why not use alternate lookup to get h’(Goal, A)? • Turns out that this is the exact same lookup as h(A,Goal)

Combining the Heuristics • The Add method lets us adjust our normal PDB estimate: • h’1(A,goal) = h(A,goal) +  • The Max method gives us another heuristic: • h’2(A,goal)=mini(h(Bi,start) +g*(Bi,goal))-h(A, start) • Our final heuristic: • H(A,goal)=max(h’1(A,goal),h’2(A,goal))

Hypotheses • All memory used on perimeter, expect poor performance • All memory used on PDB, expect good performance, but not the best possible • Small perimeters combined with large PDBs should outperform large perimeters with small PDBs

Method • Do a binary search in the memory space. • Test ‘pure’ perimeter search and ‘pure’ PDB search • Give half the memory to the winner, compare whether a PDB or a perimeter would be a more effective use of the remaining memory • Repeat until perimeter search becomes a more effective use of memory

Results

Discussion • Discouraging results • Using “extra” space for a PDB seems to provide better results across the board

Discussion • Adding a perimeter does not appear to have a significant effect

Discussion • Empirically, the Add method is always returning  = 0 • A directed strategy for perimeter creation is likely needed for this method to have any effect

Discussion • Further experiments show that given a fixed PDB, as perimeters increase in size, there is a negligible performance increase

An Idea • Is the heuristic effectively causing paths to perimeter nodes to be pruned? • This means that performance is only being improved along a narrow search path • Can we generate the perimeter ‘intelligently’ to make it more useful?

Questions?

Combining Front-to-End Perimeter Search and Pattern Databases