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OPTIMAL Search

OPTIMAL Search. Uniform Cost Branch and Bound Introducing Underestimates Path Deletion A*. When cost of TRAVERSING the path should be minimized (even at expense of more complicated SEARCHING ) :. 4. 4. 3. 5. 5. 4. 3. 2. 4. 4. 3. S. 4. A. D. 2. 5. 5. A. B. C. B. D. A.

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OPTIMAL Search

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  1. OPTIMAL Search Uniform Cost Branch and Bound Introducing Underestimates Path Deletion A* When cost of TRAVERSING the path should be minimized (even at expense of more complicatedSEARCHING) :

  2. 4 4 3 5 5 4 3 2 4 4 3 S 4 A D 2 5 5 A B C B D A E 4 5 S 5 2 4 4 G C E E B B F 2 4 5 4 4 5 4 4 D E F 3 D F B F C E A C G 3 4 3 4 G C G F 3 G Re-introduce the costs of arcs in the NET

  3. S 3 4 4 3 A D 5 4 5 2 6 9 B 7 A 8 D E 5 4 5 4 5 4 11 10 11 12 13 C B 13 B F E E 3 B 13 F F C A C D E G C F G G G Uniform cost algorithm= uniformed best-first • At each step, select the node with the lowest accumul-ated cost.

  4. (BY ACCUMULATED COST) Uniform cost algorithm: 1. QUEUE <-- path only containing the root; 2. WHILEQUEUE is not empty AND goal is not reached DO remove the first path from the QUEUE; create new paths (to all children); reject the new paths with loops; add the new paths and sort the entire QUEUE; 3. IF goal reached THEN success; ELSE failure;

  5. S 1 1 5 5 A C 1 2 5 B 10 D 5 15 E 100 5 F 20 G 5 Problem: NOT always optimal! • Uniform cost returns the path with cost 102, while there is a path with cost 25.

  6. S 2 3 A B 0.5 3.5 3 2 D 3 5 C 5 G 6 X ignore E G X ignore First goal reached The Branch-and-Bound principle • Use any (complete) search method to find a path. • Remove all partial paths that have an accumulated cost larger or equal than the found path. • Continue search for the next path. • Iterate.

  7. S 1 1 5 5 A C 1 2 5 B 10 D 5 15 E 100 5 F 20 G 5 102 25 A weak integration of branch and bound in uniform cost: • Change the termination condition: • only terminate when a path to a goal node HAS BECOME THE BEST PATH.

  8. Optimal Uniform cost version: 1. QUEUE <-- path only containing the root; 2. WHILE QUEUE is not empty ANDfirst path does not reach goal remove the first path from the QUEUE; create new paths (to all children); reject the new paths with loops; add the new paths and sort the entire QUEUE; 3. IF goal reached THEN success; ELSE failure; (by accumulated cost)

  9. S 3 4 A D S 4 A A D 7 8 B D S A D D 7 8 9 6 A B D E S A D 9 7 8 A E E B D 10 11 B F Example:

  10. S A D 9 8 A D B B E 11 10 B F 11 12 C E S A D 9 A D D B E 11 10 B F 11 12 10 C E E S A D A A D B E 11 10 B F 11 13 12 10 C E E B S A D A D B E 11 10 B F 11 13 12 C E E E B 15 14 B F

  11. S A D A D B E 11 B F F 11 13 12 C E E B 15 14 13 B F G S A D A D B E B B F 11 13 12 C E E B 15 14 15 15 13 B F A C G S A D A D B E B F 13 C E E E B 14 15 15 15 13 16 B F 14 D F A C G Don’t stop yet! STOP!

  12. Properties of extended uniform cost : • Optimal path: • If there exists a number  > 0, such that every arc has cost  , and if the branching factor is finite, • Thenextended uniform cost finds the optimal path (if one exists). • Memory and speed: • In the worst case, at least as bad as for breadth-first: • needs additional sorting step after each path- expansion ! • How to improve ?

  13. f(path) = cost(path) + h(endpoint_path) + Extension with heuristic estimates: • Replace the ‘accumulated cost’ in the ‘extended uniform cost’ algorithm by a function: • where: cost(path) = the accumulated cost of the partial path h(T) =a heuristic estimate of the cost remaining from T to a goal node f(path) =an estimate of the cost of a path extending the current path to reach a goal.

  14. G 3 C F 4 4 B 5 E 4 2 10.4 6.7 A B C 4 5 D A 11 S 4 G 8.9 3 F 6.9 S 3 D E Example: Reconsider the straight-line distance: • h(T) = the straight-line distance from T to G

  15. S 3 + 10.4 = 13.4 4 + 8.9 = 12.9 A D S 13.4 A D D 9 + 10.4 = 19.4 6 + 6.9 = 12.9 A E S 13.4 A D 19.4 A E E 11 + 6.7 = 17.7 B 10 + 3.0 = 13.0 F S 13.4 A D 19.4 A E 17.7 B F F 13 + 0.0 = 13.0 G STOP!

  16. (by f = cost + h) Estimate-extended Uniform cost algorithm: 1. QUEUE <-- path only containing the root; 2. WHILE QUEUE is not empty ANDfirst path does not reach goal remove the first path from the QUEUE; create new paths (to all children); reject the new paths with loops; add the new paths and sort the entire QUEUE; 3. IF goal reached THEN success; ELSE failure;

  17. IF for all T: h(T) is an UNDERestimate of the remaining cost to a goal node • THEN estimate-extended uniform cost is optimal. S includes underestimated remaining cost 13.4 A D 19.4 A E 17.7 B F with real remaining cost, this path must at least be 13.4 13 G G Optimality • With the same condition on the arcs-costs and on the branching factor: • Intuition:

  18. 2 3 1 1 3 2 1 1 Real remaining costs A B C 1 1 Over- estimate 1 5 2 4 1 1 1 S D E G 3 2 1 1 3 1 4 2 1 1 F H I 1+3 2+2 3+1 A A B B C C 1+5 4 S D G 1+4 Not the optimal path! F More on underestimates: • Example: • If h is NOT an underestimate:

  19. 2 3 0 1 1 0 1 1 Real remaining costs A B C 1 1 Under- estimate 1 2 2 1 1 1 1 S D E G 3 2 1 1 2 1 3 1 1 1 F H I 1+1 2+0 3+0 A A B B C C 1+2 4 3 ! E E G S D D G 1+3 F More on underestimates: • Example: • If h is an underestimate: 2+1 Bad underestimates always get cor- rected by increasing accumulated cost

  20. Speed and memory • In the worst case: no improvement over ‘branch and bounded extended uniform cost’ • Just take h = 0 everywhere. • For good heuristic functions: search may expand much less nodes ! • See our running example. • BUT: the cost of computing such functions may be high • Trade-off

  21. S 3 4 A A D 7 8 B D Accumulated distance • Principle: • the minimum distance from S to Gvia I = (min. dist. from S to I) + (min. dist. from I to G) An orthogonal extension:path deletion • Discard redundant paths: • in the ‘branch and bound extended uniform cost’ : X discard !

  22. S A D Q P 7 8 9 6 A B D E More precisely: • IF the QUEUE contains: • a path P terminating in I, with cost cost_P • a path Q containing I, with cost cost_Q • cost_Pcost_Q • THEN • delete P X

  23. S 3 4 A D S 4 A D 7 8 B D X S A D 7 9 6 A B D E X X S A D 7 A E B D X X 10 11 B F X

  24. S A D A D B E X X 10 B F 11 12 C E X X S A D A D B E X X B F 11 C E X 13 X G Note how this optimization reduces the number of expansions VERY much, compared to ‘branch and bound extended uniform cost’. 5 expansions less !

  25. A* search • IS: • Branch and bound extended, • Heuristic Underestimate extended, • Redundant path deletion extended, • Uniform Cost Search. • Note that redundant path deletion is based on the accumulated costs only, so that there is no problem combining it with heuristic underestimates.

  26. A* algorithm: 1. QUEUE <-- path only containing the root; 2. WHILE QUEUE is not empty ANDfirst path does not reach goal remove the first path from the QUEUE; create new paths (to all children); reject the new paths with loops; add the new paths and sort the entire QUEUE; IFQUEUE contains path P terminating in I, with cost cost_P, and path Q containing I, with cost cost_Q AND cost_Pcost_Q THEN delete P 3. IF goal reached THEN success; ELSE failure; (by f = cost + h)

  27. S 3 + 10.4 = 13.4 4 + 8.9 = 12.9 A D S 13.4 A D 9 + 10.4 = 19.4 6 + 6.9 = 12.9 A E X Only difference is here. S 13.4 A D A E X B 10 + 3.0 = 13.0 F 11 + 6.7 = 17.7 S 13.4 A D A E STOP! X 17.7 B F 13 + 0.0 = 13.0 G

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