A Star Search

By Sohaib Chaudhery,UE Okara Campus! 1

“A* Search in Artificial Intelligence”

By:Sohaib Saleem

To:Resp. Inam-ul-Haq


Uniform Cost Search Uniform Cost is a blind search algorithm that is optimal according to any specified path

length function. Do not have additional info about states beyond problem def. Total search space is looked for solution No info is used to determine preference of one child over other. Example: 1. Breadth First Search(BFS), Depth First Search(DFS), Depth Limited

Search (DLS).

A

B

C

E

D HF

GState Space without any extra information associated with each state


Informed/Heuristic Search

Some info about problem space(heuristic) is used to compute preference among the children for exploration and expansion.

Examples: 1. Best First Search, 2. Problem Decomposition, A*, Mean end Analysis

The assumption behind blind search is that we have no way of telling whether a particular search direction is likely to lead us to the goal or not

The key idea behind informed or heuristic search algorithms is to exploit a task specific measure of goodness to try to either reach the goal more quickly or find a more desirable goal state.

Heuristic: From the Greek for “find”, “discover”.


Informed/Heuristic Search(conti…) Heuristic function: It maps each state to a numerical value which depicts goodness of a node.

H(n)=value Where , H() is a heuristic function and ‘n’ is the current state. Ex: in travelling salesperson problem heuristic value associated with each

node(city) might reflect estimated distance of the current node from the goal node.

The heuristic we use here is called HSLD Straight line Distance heuristic.


A* Search A* search is a combination of lowest-cost-first and best-first searches that

considers both path cost and heuristic information in its selection of which path to expand.

For each path on the frontier, A* uses an estimate of the total path cost from a start node to a goal node constrained to start along that path.

It uses cost(p), the cost of the path found, as well as the heuristic function h(p), the estimated path cost from the end of p to the goal.

For any path p on the frontier, define f(p)=cost(p)+h(p). This is an estimate of the total path cost to follow path p then go to a goal node.

If n is the node at the end of path p, this can be depicted as follows:

actual estimate start ------------> n --------------------> goal cost(p) h(p) -------------------------------------------->

f(p)


A * Search(conti…) It is best-known form of Best First search. It avoids expanding paths that are

already expensive, but expands most promising paths first. Idea: minimize the total estimated solution cost f(n) = g(n) + h(n),

where g(n) the cost (so far) to reach the node h(n) estimated cost to get from the node to the goal f(n) estimated total cost of path through n to goal. It is implemented using

priority queue by increasing f(n). Minimize the total path cost to reach the goal.

A* is complete, optimal, and optimally efficient among all optimal search algorithms.


A * Search(conti…)At S we observe that the best node is A with a value of 4 so we move to 4.


A * Search(conti…)


A * Search(conti…)Now we move to D from S.


A* Search Example: Romania tour


Search Terminology Problem Space − It is the environment in which the search takes place. (A set of

states and set of operators to change those states) Problem Instance − It is Initial state + Goal state. Problem Space Graph − It represents problem state. States are shown by nodes and

operators are shown by edges. Depth of a problem − Length of a shortest path or shortest sequence of operators

from Initial State to goal state. Space Complexity − The maximum number of nodes that are stored in memory. Time Complexity − The maximum number of nodes that are created. Admissibility − A property of an algorithm to always find an optimal solution. Branching Factor − The average number of child nodes in the problem space graph. Depth − Length of the shortest path from initial state to goal state.


Admissible heuristics

A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n e.g., Straight-Line Distance an admissible heuristic never overestimates the cost

to reach the goal, i.e., it is optimistic

THEOREM: If h(n) is admissible, A* using Tree-Search is optimal


Optimality of A*

A* expands nodes in order of increasing f value


Evaluation: A* search□ Complete

■ Yes□ Optimal

■ Yes□ Space

■ Keeps all nodes in memory□ Time

■ Exponential


Thank-You

A Star Search

Education

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