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Heuristic Search Methods : 

Heuristic Search Methods Methods that use a heuristic function to provide specific knowledge about the problem: Heuristic Functions Hill climbing Beam search Hill climbing (2) Greedy search

Heuristic Functions : 

2 Heuristic Functions To further improve the quality of the previous methods, we need to include problem-specific knowledge on the problem. How to do this in such a way that the algorithms remain generally applicable ???

Examples (1): : 

3 Examples (1): Imagine the problem of finding a route on a road map and that the NET below is the road map:

Examples (2): 8-puzzle : 

4 Examples (2): 8-puzzle f1(T) = the number correctly placed tiles on the board: Most often, ‘distance to goal’ heuristics are more useful !

Examples (3):Manhattan distance : 

5 Examples (3):Manhattan distance f3(T) = the sum of ( the horizontal + vertical distance that each tile is away from its final destination): gives a better estimate of distance from the goal node = 1 + 4 + 2 + 3 = 10

Examples (4):Chess: : 

6 Examples (4):Chess: F(T) = (Value count of black pieces) - (Value count of white pieces) f

Hill climbing : 

Hill climbing A basic heuristic search method: depth-first + heuristic

Hill climbing_1 : 

8 Hill climbing_1 Perform depth-first, BUT: instead of left-to-right selection, FIRST select the child with the best heuristic value Example: using the straight-line distance: 8.9 10.4 10.4 6.9 6.7 3.0

Hill climbing_1 algorithm: : 

9 Hill climbing_1 algorithm: 1. QUEUE <-- path only containing the root; 2. WHILE QUEUE 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; sort new paths (HEURISTIC) ; add the new paths to front of QUEUE; 3. IF goal reached THEN success; ELSE failure;

Beam search : 

Beam search Narrowing the width of the breadth-first search

Beam search (1): : 

11 Beam search (1): Assume a pre-fixed WIDTH (example : 2 ) Perform breadth-first, BUT: Only keep the WIDTH best new nodes depending on heuristic at each new level.

Beam search (2): : 

12 Beam search (2): Optimi-zation: ignore leafs that are not goal nodes (see C)

Beam search algorithm: : 

13 Beam search algorithm: See exercises! Properties: Completeness: Hill climbing: YES (backtracking), Beam search: NO Speed/Memory: Hill climbing: same as Depth-first (in worst case) Beam search: QUEUE always has length WIDTH, so memory usage is constant = WIDTH, time is of the order of WIDTH*m*b or WIDTH*d*b if no solution is found

Hill climbing_2 : 

14 Hill climbing_2 == Beam search with a width of 1. Inspiring Example: climbing a hill in the fog. Heuristic function: check the change in altitude in 4 directions: the strongest increase is the direction in which to move next. Is identical to Hill climbing_1, except for dropping the backtracking. Produces a number of classical problems:

Problems with Hill climbing_2: : 

15 Problems with Hill climbing_2:

Comments: : 

16 Comments: Foothills are local minima: hill climbing_2 can’t detect the difference. Plateaus don’t allow you to progress in any direction. Foothills and plateaus require random jumps to be combined with the hill climbing algorithm. Ridges neither: the directions you have fixed in advance all move downwards for this surface. Ridges require new rules, more directly targeted to the goal, to be introduced (new directions to move) .

Local search : 

17 Local search Hill climbing_2 is an example of local search. In local search, we only keep track of 1 path and use it to compute a new path in the next step. QUEUE is always of the form: ( p) Another example: MINIMAL COST search: If p is the current path: the next path extends p by adding the node with the smallest cost from the endpoint of p

Greedy search : 

Greedy search Always expand the heuristically best nodes first.

Greedy search, orHeuristic best-first search: : 

19 Greedy search, orHeuristic best-first search: At each step, select the node with the best (in this case: lowest) heuristic value.

Greedy search algorithm: : 

20 Greedy search algorithm: 1. QUEUE <-- path only containing the root; 2. WHILE QUEUE 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;

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