Aug 28, 2006
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- Recap goal-based agents: they
        - formulate goal
                (Goal Formulation)
        - formulate state, actions/operators
                (Problem Formulation)
        - try to achieve goal
                (Search)

- Search
        - ubiquitous in AI
        - need a good understanding of it before
          moving onto other topics

- Example search problems
        - 8-queens problem
        - 8-puzzle problem
        - traveling salesperson problem (TSP)

- Formalizing a search problem
        GIVEN:
        - what is the start state?
        - what are the available operators?
        - what is the goal test?
                - i.e., how do you know you have
                  succeeded?
		- "are we there yet?"
        FIND:
        a path/state that satisfies the goal test
        OPTIONAL
        and that is optimal

- Revisiting the problems
        - goal is a state
                - 8-queens problem
        - goal is a path
                - 8-puzzle problem
        - goal is an optimal path
                - TSP

- Two basic uninformed search methods
        - breadth first search (BFS)
        - uniform cost search (UCS)

- Pseudocode for generic BFS algorithm

      fringe := make-queue(initial-node)
      loop
		if empty?(fringe) then return FAIL
		else
		    X := remove-front(fringe)
		    if isa-goal?(state(X)) then return X
		    else
			fringe := insert(fringe,expand(X))
      end loop

- Breadth first search
        - expand nodes of one level completely before
          moving to the next
        - time complexity
		- nodes expanded: O(b^d)
		- nodes generated: O(b^(d+1))
        - space complexity
		- nodes expanded: O(b^d)
		- nodes generated: O(b^(d+1))
        - complete?: yes
                - i.e., guaranteed to find a solution
        - finds optimal solution?: yes
                - because nodes at lower depths
                  are expanded before ones at higher
                  depths
        - Note: memory requirement is killing!

- In the above
        - b: branching factor
        - d: depth at which solution is found

- BFS assumptions
        - equates cost to depth
        - what if different edges in the search tree have
          different costs?

- Solution: uniform cost search (UCS)
        - expand nodes based on cost
                - has a tendency to flip-flop its way
        - path costs must be non-decreasing down the
          tree, for UCS to make sense
	- easy to get by making the min. cost to be
	  a small positive number
        - has same worst-case time and space
          complexity as BFS

- Data structure support for search algorithms
        - BFS: use queue
        - UCS: use priority queue

- Next idea
        - DFS: uses a stack!

- More search algorithms: Depth First Search (DFS)
	- uses a stack in place of a queue
	- primary advantage is its space complexity
	- problems: does not provide optimal solution
	- more problems: can go into an infinite loop

- DFS features
	- time complexity: O(b^m)
	- space complexity: O(bm)
		- where m is the max depth of search tree

- DBS: Depth Bounded Search
	- uses a limit "l" to prevent infinite loops
	- complete only if l is chosen large enough so that
	  l >= d (but then need not be optimal)
	- if l < d, not complete
	- time complexity: O(b^l)
	- space complexity: O(bl)

- Thus far
	- BFS and UCS are good for "guarantees"
	- DFS and DBS good for space complexity

- DFID: Depth First Iterative Deepening
	- best of both worlds

- DFID Details
	- performs a series of DBSs with increasing l
	- guaranteed to find solution and,
	- since at each time it is doing only a DFS,
	  has the nice space complexity we want
	- time complexity: O(b^d)
	- space complexity: O(bd)
	- complete and optimal

- Still..
	- Q: won't DFID be wasting too much time expanding
	  nodes over and over again?
	- A: not really, asymptotically the bulk of the work
	  is really in the "last iteration"

- All search algorithms studied thus far can be categorized as
	- "uninformed search"

- Informed search algorithms
	- e.g., "psychic search"
		- more next class!