More advanced aspects of search Extensions of A*

Iterated deepening A* Simplified Memory-bounded A*

Iterative-deepening A*

4 Memory problems with A*  A* is similar to breadth-first: Breadth-first d = 1 d = 2 d = 3 d = 4 Expand by depth-layers  Here: 2 extensions of A* that improve memory usage. f1 f2 f3 f4 A* Expands by f-contours

5Depth-first in each f- contour  Perform depth-first search LIMITED to some f-bound.  If goal found: ok.  Else: increase de f- bound and restart. Iterative deepening A* f4 How to establish the f-bounds? - initially: f(S) - initially: f(S) generate all successors record the minimal f(succ) > f(S) Continue with minimal f(succ) instead of f(S) f3 f2 f1

6 Example: Sf=100 Af=120Bf=130Cf=120 Df=140Gf=125Ef=140Ff=125 f-limited, f-bound = 100 f-new = 120

7 Example: Sf=100 Af=120Bf=130Cf=120 Df=140Gf=125Ef=140Ff=125 f-limited, f-bound = 120 f-new = 125

8 Example: Sf=100 Af=120Bf=130Cf=120 Df=140Gf=125Ef=140Ff=125 f-limited, f-bound = 125 SUCCESS

9 f -limited search: 1. QUEUE <-- path only containing the root; f-bound ; f-bound ; f-new <--  f-new <--  2. WHILE QUEUE is not empty AND goal is not reached DO remove the first path from the QUEUE; DO remove the first path from the QUEUE; create new paths (to all children); create new paths (to all children); reject the new paths with loops; reject the new paths with loops; add the new paths with f(path)  f-bound add the new paths with f(path)  f-bound to front of QUEUE; to front of QUEUE; f-new <-- minimum of current f-new and f-new <-- minimum of current f-new and of the minimum of new f-values which are of the minimum of new f-values which are larger than f-bound larger than f-bound 3. IF goal reached THEN success; ELSE report f-new ;

10 Iterative deepening A*: 1. f-bound <-- f(S) 2. WHILE goal is not reached DO perform f-limited search; DO perform f-limited search; f-bound <-- f-new f-bound <-- f-new

11 Properties of IDA*  Complete and optimal:  under the same conditions as for A*  Memory:  Let  be the minimal cost of an arc:  == O( b* (cost(B) /  ) )  Speed:  depends very strongly on the number of f-contours there are !!  In the worst case: f(p)  f(q) for every 2 paths:  ….+ N = O(N 2 )

12 Why is this optimal, even without monotonicity ?? S AB CDFE  In absence of Monotonicity:  we can have search spaces like:  If f can decrease,  how can we be sure that the first goal reached is the optimal one ???

13 Properties: practical  If there are only a reduced number of different contours:  IDA* is one of the very best optimal search techniques !  Example: the 8-puzzle  But: also for MANY other practical problems  Else, the gain of the extended f-contour is not sufficient to compensate recalculating the previous  In such cases:  increase f-bound by a fixed number  at each iteration:  effects: less re-computations, BUT: optimality is lost: obtained solution can deviate up to 

Simplified Memory-bounded A*

15 Simplified Memory-bounded A*  Fairly complex algorithm.  If memory is full and we need to generate an extra node (C):  Remove the highest f- value leaf from QUEUE (A).  Remember the f-value of the best ‘forgotten’ child in each parent node (15 in S). S AB C memory of 3 nodes only  Optimizes A* to work within reduced memory.  Key idea: (15) 18 B

16 Generate children 1 by 1  When expanding a node (S), only add its children 1 at a time to QUEUE.  we use left-to-right  Avoids memory overflow and allows monitoring of whether we need to delete another node S AB 13 First add A, later B AB

17 Too long path: give up  If extending a node would produce a path longer than memory: give up on this path (C).  Set the f-value of the node (C) to   (to remember that we can’t find a path here) S B C memory of 3 nodes only D 18  B C

18 Adjust f-values S AB  If all children M of a node N have been explored and for all M:  f(S...M)  f(S...N)  then reset:  f(S…N) = min { f(S…M) | M child of N}  A path through N needs to go through 1 of its children ! better estimate for f(S) 15

19 SMA*: an example: S AB CG1DG2 EG3G4F 0+12=12 8+5= = = =1820+0= = = = = = S12S12 A 15 S12A 15 B 13 S 13 B 13 A 15 A 15 D 18 (15) 1312

20 S 13 B D  13 (15) S AB CG1DG2 EG3G4F0+12=128+5= = = =1820+0= = = = = = S 13 (15) B 13 D  Example: continued D  (()(() G S 15 B 24 (()(()(15)G2 24 (15) G2 24 (()(() A 15 S 15 A 15 B 24 B 24 (24) C 25  S 15 (24) A C  15 C  ()() G

21 SMA*: properties:  Complete: If available memory allows to store the shortest path.  Optimal: If available memory allows to store the best path.  Otherwise: returns the best path that fits in memory.  Memory: Uses whatever memory available.  Speed: If enough memory to store entire tree: same as A*