1. Heuristic Search In addition to depth-first search, breadth-first search, bound depth-first search, and iterative deepening, we can also use informed or heuristic search. We need an estimate, h(n), of the distance from state n to a goal state. We will then use priority-first search with the priority being f(n) = g(n) + h(n) where f(n) is the known distance from the start state to state n.

2. Example – the Eight Puzzle

3. The Eight Puzzle The rule is that you can only move one tile at a time and you must move into the space. Alternatively, you can think of a move as moving the space up, down, left, or right one square (if possible). The search space is the graph of possible states of the puzzle, which can can specified as a permutation of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 (where 9 is the space).

4. Heuristics for the Eight Puzzle h 1 (n) = the total number of tiles out of place. h 2 (n) = the sum of the distances that each tile must be moved to the proper place (Manhattan distance). h 3 (n) = p 1 (p 2 -1 (p 1 (n))) where p 1 is the starting permutation and p 2 is the goal permutation. h 4 (n) = 0. (Blind search)

5. Results h1(n): 110 nodes visited, 183 nodes created. h2(n): 55 nodes visited, 88 nodes created. h3(n): 1583 nodes visited, 2574 nodes created. h4(n): 1811 nodes visited, 2968 nodes created.

6. Code pq.add(start->number,start->distance); while(!pq.empty()) { int nodeNum, priority; pq.remove(nodeNum,priority); numVisited++; if(nodeNum == goal->number) return priority; Node *node = nodes[nodeNum]; //node->print(priority); int g = priority - node->distance + 1; node->priority = priority; node->mark = 2;

7. Code (cont'd) for(int i=0;i numNeighbors();i++) { Node *neighbor = node->neighbor(i); int f = g + neighbor->distance; if(neighbor->mark == 0 || (neighbor->mark == 2 && neighbor->priority > f)) { neighbor->mark = 1; pq.add(neighbor->number,f); neighbor->previous = node; } else if(neighbor->mark == 1 && pq.getPriority(neighbor- >number) > f) { pq.update(neighbor->number,f); neighbor->previous = node; } return INT_MAX;

8. Definitions g*(n) = the shortest path from the start state to state n. h*(n) = the shortest path from state n to a goal state. f*(n) = g*(n) + h*(n) = the shortest path from the start state to a goal state that passes thru state n. A search algorithm is admissible if it is guaranteed to find a minimal solution whenever one exists.

9. A*-Algorithms Using priority-first search with the priority f(n) = g(n) + h(n) is called Algorithm A. If, in addition, we have that h(n) <= h*(n), the procedure is called an A*-Algorithm. A*-algorithms are all admissible. BFS is Algorithm A with h(n) = 0, and hence admissible.

10. Monotonicity A heuristic function h is monotone if For all states n i and n j, where n j is a descendant of n i, h(n i ) – h(n j ) <= cost(n i,n j ). The heuristic evaluation of any goal state is zero, i.e., h(Goal) = 0. This means that the difference between the heuristic measure for a state and its successor is bound by the actual cost of going from the state to its successor.

11. Why do we care? Monotonicity ensures that the first time a state is reached, we've found the shortest path to that state. Subsequent visits to the state are longer paths, and so, do not have to be reconsidered. The upshot is that although we have to check if a state has been visited, once it is, we never have to consider it again.

12. Informedness For two A* heuristics, h 1 and h 2, if h 1 (n) <= h 2 (n) for all states n in the search space, then heuristic h 2 is said to be more informed than h 1.

13. Two Player Games We can use state space search for two (or more) player games as well. We assume that we have two players, MAX and MIN, who alternate turns. We assume one player must win (no ties).The state space search graph is then stratified into layers, alternate layers belonging to each player. At each level, the nodes are labeled by the player whose turn it is.

14. Minimax Procedue Leaves are labeled according to who wins: A win for MAX is labeled 1 and a win for MIN is labeled 0. An internal MAX node is labeled by the maximum value of its children. An internal MIN node is labeled by the minimum value of its children. We work from the bottom up.

15. Fixed Ply Depth Minimax This assumes that we can create the complete graph. If we can't do that, we can generate the graph to a fixed depth, and then arbitrarily cutoff search. We must estimate the goodness of the leaves using a heuristic measure rather than knowing if they are wins or losses.

18. Alpha-Beta Pruning With alpha-beta pruning, we do not have to search the entire space. We search the state-space in depth- first fashion to a limited depth. Leaves are assigned values given a heuristic function. The values are propagated back up the tree. However, once MAX has a node with a given value (called alpha), we know that MAX will never choose a move which has a smaller value.

19. Alpha-Beta Pruning (cont'd) If another child of the given node will yield a smaller (or equal) value, we can cut off the search and not expand the entire tree under that child. Since the child is a MIN node, we know that MIN will always choose the move with the smallest value. So, if a child of the MIN node has value less than alpha, the MIN node is guaranteed to have a value less than alpha, and MAX will not choose that node.