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Advanced Search Strategies in AI: A* Algorithm, Beam Search, and Genetic Algorithms

This lecture explores advanced search strategies in artificial intelligence, focusing on the A* algorithm, beam search, and genetic algorithms. Learn how the A* algorithm uses heuristics for optimal pathfinding in problems like the 8-puzzle, comparing various heuristics for efficiency. Discover beam search's practicality with a fixed number of promising nodes and the mechanism of genetic algorithms involving mutation and crossover operators. The session includes practical lab assignments designed to implement and analyze these algorithms in solving complex problems.

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Advanced Search Strategies in AI: A* Algorithm, Beam Search, and Genetic Algorithms

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  1. CS621 : Artificial Intelligence Pushpak BhattacharyyaCSE Dept., IIT Bombay Lecture 4: Search

  2. Graph Search S 1 10 5 3 A B C 10 9 4 5 6 3 I H D E 2 6 3 2 7 J F G(oal)

  3. 8-puzzle problem 1 2 3 4 3 6 1 4 2 8 5 6 5 8 7 7 S0 G Tile movement represented as the movement of the blank space. Operators: L : Blank moves left R : Blank moves right U : Blank moves up D : Blank moves down C(L) = C(R) = C(U) = C(D) = 1

  4. 8-puzzle: heuristics Example: 8 puzzle s n g h*(n) = actual no. of moves to transform n to g h1(n) = no. of tiles displaced from their destined position. h2(n) = sum of Manhattan distances of tiles from their destined position. h1(n) ≤ h*(n) and h1(n) ≤ h*(n) h* h2 h1 Comparison

  5. Power of Heuristic

  6. Theorem A version A2* of A* that has a “better” heuristic than another version A1* of A* performs at least “as well as” A1* Meaning of “better” h2(n) > h1(n) for all n Meaning of “as well as” A1* expands at least all the nodes of A2* h*(n) h2*(n) h1*(n) For all nodes n, except the goal node

  7. Proof by induction on the search tree of A2*. A* on termination carves out a tree out of G Induction on the depth k of the search tree of A2*. A1* before termination expands all the nodes of depth k in the search tree of A2*. k=0. True since start node S is expanded by both Suppose A1* terminates without expanding a node n at depth (k+1) of A2* search tree. Since A1* has seen all the parents of n seen by A2* g1(n) <= g2(n) (1)

  8. Since A1* has terminated without expanding n, f1(n) >= f*(S) (2) Any node whose f value is strictly less than f*(S) has to be expanded. Since A2* has expanded n f2(n) <= f*(S) (3) S k+1 G From (1), (2), and (3) h1(n) >= h2(n) which is a contradiction. Therefore, A1* has to expand all nodes that A2* has expanded. Exercise If better means h2(n) > h1(n) for some n and h2(n) = h1(n) for others, then Can you prove the result ?

  9. Monotone Restriction • Very reasonable assumption • That is the estimated cost of reaching the goal node for a particular node is no more than the cost of reaching a child and the estimated cost of reaching the goal from the child • This obviates the need for redirecting parent pointer for a less costly path in the closed list • This means that for any node taken up for expansion, the optimal path is already found n1 C(n1,n2) n2 h(n1) h(n2) g

  10. Other search strategies

  11. Beam Search • Run A*, but with a “Beam width K” • That is, for each iteration keep K most promising nodes and throw away other nodes • Incomplete and non-optimal, but fast

  12. Genetic Algorithm Based • Use the operators of mutation and cross over • Mutation: flip a bit randomly to produce a new state • Cross over: produce by crossing portion of two states a new state

  13. Blank 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 GA for 8-puzzle 4 3 6 1 2 8 5 7 Represented as 0100 0011 0110 0010 0001 1000 0111 0000 0101 1 6 2 3 4 7 8 5 9

  14. Mutation and Crossover Mutation: flip any bit randomly A 0100 0011 0110 0010 0001 1000 0111 0000 0101 B 0100 0111 0110 0010 0001 1000 0111 0000 0101 Crossover: cross a set of bit 1000 0111 0000 0101 0001 1000 0111 0000 0101 First 16 bits of A crossed With last 16 bits of B 0100 0111 0110 0010 0001 0100 0011 0110 0010

  15. Care to be exercised • The bit strings are called chromosomes • Mutation and crossover produce bit strings which are state descriptions • But care has to be exercised to see that the produced state is • Legal • Reachable

  16. Lab assignment • Implement A* algorithm for the following problems: • 8 puzzle • Specifications: • Try different heuristics and compare with baseline case, i.e., the breadth first search. • Violate the condition h ≤ h*. See if the optimal path is still found. Observe the speedup. • Now implement for the same problem • Beam Search • Genetic Algorithm

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