1. Informed Search
  2. Evaluation Function
  3. Best-first search
    1. Greedy search
    2. A*
    3. An example of two searches
  4. Problems with heuristic search

Informed Search

One advantage of Breadth-first and Depth-first Search is their generality. A disadvantage is that they don't use any problem-specific information to guide them.

We now move away from these uninformed search methods and begin to investigate informed (or heuristic) search techniques.

The general ideas behind informed search techniques are the following:

Algorithms for informed search are still general. The problem-specific information is encoded in an evaluation function

Evaluation Function

Informed search techniques require an evaluation function. The evaluation is applied to states (nodes) in the search tree.

Best-first search

In best-first search, the general idea is to always expand the most desirable node on the frontier (fringe). The frontier of the search space includes all nodes that are currently available for expansion.

Types of best-first search include:

Greedy search

For a node n in the search tree,

Let h(n) = an estimate of the cost of the cheapest path from n to a goal state. This is the (heuristic) evaluation function. (Note that h(n) is 0 when n is a goal state.)

Greedy search selects the next node to expand based upon the function h.

Greedy search can work quite well, but it does have problems:

That is, if h(n) greatly underestimates the distance of certain nodes to the goal, the algorithm can begin to follow paths that do not lead to the goal.

Evaluating Greedy search on the four criteria:

A* search

A* search corrects for the problem of "greed". In evaluating a given node/state, it takes into account both the distance traveled so far and the estimate of distance to the goal.

For a node n in the search tree,

let f(n) = g(n) + h(n),

where g(n) = the cost of getting to n from the initial state.

and h(n) is as before.

We can think of f(n) as an estimate of the cost of the cheapest solution from the initial state to a goal through n.

The wonderful thing about this search is that if h is admissible, then the search will be optimal and complete! i.e., the best solution will be found. (For the math majors among you, yes there is a proof of this!)

Def. An admissible heuristic is one that never overestimates the cost to reach the goal.

Some examples of admissible heuristics:

An example of the two searches

Let’s compare the behavior of Greedy search and A* search on the 5-puzzle.

We’ll define g(n) and h(n) as follows:

Let g(n) be the number of steps taken from the initial state to n.

Let h(n) be the number of tiles out of place.

Problems with heuristic (informed) search

In spite of its optimality and completeness, A* still has problems:

For most problems, the number of nodes on the frontier of the search space is still exponential in the length of the solution. That is, the search tree can still grow to be as "bushy" as in Breadth-first Search. So there can be problems with respect to the amount of memory needed to run A* search.

General drawbacks of heuristic (informed) search include the following:

When at a loss for a good heuristic function, consider a relaxed version of the problem. An exact solution to a relaxed problem might be a good heuristic for the real problem. For example, consider a sliding tile puzzle. One relaxed version of the puzzle is one in which the tiles can simply be picked up and put into place. Therefore, one possible heuristic is to count the number of tiles out of position, since simply placing them would solve the relaxed problem.