External A* Stefan Edelkamp, Shahid Jabbar (ich) University of Dortmund, Germany and Stefan Schrödl (DaimlerChrysler, CA)

Shahid Jabbar (Dortmund)External A*2 A* Algorithm  A.k.a Goal-directed dijkstra  A heuristic estimate is used to guide the search. E.g. Straight line distance from the current node to the goal in case of a graph with a geometric layout.  Reweighting: w’(u,v) = w(u,v) – h(u) + h(v)

Shahid Jabbar (Dortmund)External A*3 Problems ( It’s a big big world for a small small memory )  A* needs to store all the states during exploration.  A* generates large amount of duplicates that can be removed using an internal hash table – only if it can fit in the main memory.  A* do not exhibit any locality of expansion. For large state spaces, standard virtual memory management can result in excessive page faults.

Shahid Jabbar (Dortmund)External A*4 A bit of History  Munagala and Ranade: Generated states flushed to the disk for every BFS level. No hash table. Duplicates are removed by sorting the nodes according to the indices and doing an scan and compaction phase. Before expanding a layer t, the nodes in the layer t-1 and t-2 are subtracted from t. O(|V| + sort(|V| + |E|)) I/Os.

Shahid Jabbar (Dortmund)External A*5 A bit of History (contd…)  Further improved by Mehlhorn & Meyer to O(√(|V| ∙ scan(|V| + |E|)) + sort(|V| + |E|)) I/Os.  Korf presented External BFS for implicit graphs with the name Delayed duplicate detection for frontier search. Keep a level in the main memory until it exceeds a certain bound. If it does, sort it and flush to the disk. When a level is finished, merge the presorted buffers to get a sorted file and remove the duplicates

Shahid Jabbar (Dortmund)External A*6 Restriction on the domain  Implicit state space – generated on the fly –> no adjacency list  Unweighted  Undirected  Consistent Heuristic

Shahid Jabbar (Dortmund)External A*7 Take a closer look  Implicit, unweighted, undirected graphs  Consistent heurisitc estimates. => ∆h ={-1,0,1} g h Ah ha! It’s a Bucket of states

Shahid Jabbar (Dortmund)External A*8 Bucket  A Bucket is a set of states, residing on the disk, having the same (g, h) value, Where, g = number of transitions needed to transform the initial state to the states of the bucket, and h = Estimated distance of the bucket’s state to the goal  No state is inserted again in a bucket that is expanded  If Active (being read or written), represented internally by a small buffer. File on disk Buffer in internal memory Insert stateFlush when full

Shahid Jabbar (Dortmund)External A*9  Buckets represent temporal locality – cache efficient order of expansion.  If we store the states in the same bucket together we can exploit the spatial locality.  Munagala and Ranade’s BFS and Korf’s delayed duplicate detection for implicit graphs. External A*

Shahid Jabbar (Dortmund)External A*10 External A* - pseudocode Procedure External A* Bucket(0, h(I))  {I} f min  h(I) while (f min ≠ ∞) g  min{i | Bucket(i, f min − i) ≠  } while (g min ≤ f min ) h  f min − g Bucket(g, h)  remove duplicates from Bucket(g, h) Bucket(g, h)  Bucket(g, h) \ (Bucket(g − 1, h) U Bucket(g − 2, h)) // Subtraction A(f min ),A(f min + 1),A(f min + 2)  N(Bucket(g, h)) // Generate Neighbours Bucket(g + 1, h + 1)  A(f min + 2) Bucket(g + 1, h)  A(f min + 1) U Bucket(g + 1, h) Bucket(g + 1, h − 1)  A(f min ) U Bucket(g + 1, h − 1) g  g + 1 f min  min{i + j > f min | Bucket(i, j) ≠  } U {∞}

Shahid Jabbar (Dortmund)External A*11 Complexity Analysis  Internal A* => Each edge is looked at most once.  Duplicates Removal: Sorting the green bucket having one state for every edge from the 3 black buckets. Scanning and compaction.  O(sort(|E|))  Subtraction: Removing states of orange buckets (duplicates free) from the green one.  O(scan(|V|) + scan(|E|))

Shahid Jabbar (Dortmund)External A*12 I/O Performance of External A* Theorem: The complexity of External A* in an implicit unweighted and undirected graph with a consistent estimate is bounded by O(sort(|E|) + scan(|V|)) I/Os.

Shahid Jabbar (Dortmund)External A*13 15-Puzzle

Shahid Jabbar (Dortmund)External A*14 Experimental Results – Test Instances S. No.InstanceInit. Estim ate Opt. Sol. Length (12) (16) (14) (60) (88)

Shahid Jabbar (Dortmund)External A*15 Test Run – Generated states

Shahid Jabbar (Dortmund)External A*16 Test Run - Duplicates

Shahid Jabbar (Dortmund)External A*17 Exp. Results – Generated nodes S.No.IDA*External A*% gain Space (GB) 4546,344493, ,984,0515,180, ,369,596,778297,583, ,337,690,3312,269,240, ,009,130,7482,956, 384,

Shahid Jabbar (Dortmund)External A*18 Cache-Efficient Behaviour: CPU Internal Buffers File on disk

Shahid Jabbar (Dortmund)External A*19 Conclusion  A* for secondary storage with an I/O complexity of O(sort(|E|) + scan(|V|)).  Given that Delayed Duplication Detection has to be performed, the bound is I/O optimal.  File-based priority queue.  Hash table replaced by Delayed Duplicate Detection.  Successfully implemented to solve Korf’s Largest instance in secondary memory.  In case of non-uniformly weighted graphs with small integer weights in {1, …, C}. O(sort(|E|) + C ∙ scan(|V|))