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Submitted By Md. Wasim Akram

ID- UG 02-22-09-016

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A* Search Algorithm

What is A*?

A* is one of the many search algorithms thattakes an input, evaluates a number of

possible paths and returns a solution

The goal of this presentation is to providebackground information, details and an

example to demonstrate A* search

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What is A* Search Good For?

If there is solution A* will find it, which makesthis search algorithm complete

Similar to greedy best-first search but ismore accurate because A* takes intoaccount the nodes that have already beentraversed

(Best-first search is a search algorithmwhich explores a graph by expanding themost promising node chosen according to aspecified rule)

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Path Scoring: How to Calculate the

Path Cost

A* figures the least-cost path to the node whichmakes it a best first search algorithm.

Uses the formula F(x)=g(x)+h(x) G(x) is the total distance from the initial position to

the current position. H(x) is the heuristic function that is used to

approximate distance from the current location tothe goal state. This function is distinct because it is a mere estimation

rather than an exact value.

The more accurate the heuristic the better the faster thegoal state is reach and with much more accuracy.

F(x) = g(x)+h(x) this is the current approximationof the shortest path to the goal.

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The Open List (Priority Queue) and

the Closed List

When starting at the initial node this searchkeeps track of the nodes to be traversed. This is known as the open set and it keeps track of

nodes in order of the lowest f(x) to the highest f(x). In order of priority each node is removed from the

queue and then the values for f(x) and h(x)neighboring nodes around the removed node areupdated.

This search continues until it reaches the nodewith the lowest f(x) value, which is called the goal

node. H(x) at the goal is zero which is an admissibleheuristic because the path to the goal is clearlynot an overestimate.

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The Open and Closed List (contd)

If the heuristic function hneveroverestimates the minimum cost ofreaching the goal, also known as being

admissible, then A* is also known to beadmissible.

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Example Using A* Search

Here we are using Disneyland Paris toprovide an example to traverse nodes froman initial state to a goal state.

The main entrance is the initial node

The Magic Kingdom is the goal state

There are two paths that can be taken and aremarked by nodes

Each node will have the f(x), g(x), and h(x).

Then it will show at each node and indicatewhich is the next node that it will traversebased on least path cost.

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Disneyland Paris

Say you are at the entrance ofDisneyland Paris and you are trying toget to the Magic Kingdom.

There are two distinct paths that overlapin the center.

Using A* I will demonstrate how the

algorithm traverses nodes and reachesthe final destination.

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As you can see the initial node and the goal state are labeled accordingly.

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The first path is highlighted in with green nodes.

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The second path is illustrated with purple nodes and overlaps with the firstpath

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In this stage all of the values are shown for f(x), g(x), & h(x).Note that in this example the heuristic is not monotonic and therefore we arenot using a closed set to keep track of the nodes traversed. Since the closed

set is not keeping track of the visited nodes the values of the neighboring

nodes of the current node remain the same.

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The first node traversed is the one with the lowest f(x) value

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The only option is the purple node where F(x)=80

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There are two options in this case:1.The purple one with the higher F(x)2.The green one with the lower F(x)As mentioned before A* will choose the one with the lowest F(x)

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Finally we see that A* found the node with the smallest f(x) value. When thegoal node is popped off the priority queue then the search stops.

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Pseudocode

unction A*(start,goal)closedset := the empty set // The set of nodes already evaluated.

openset := {start} // The set of tentative nodes to be evaluated,initially containing the start node

came_from := the empty map // The map of navigated nodes.

g_score[start] := 0 // Cost from start along best known path.

h_score[start] := heuristic_cost_estimate(start, goal)f_score[start] := h_score[start] // Estimated total cost from start to

goal through y.

while openset is not emptyx := the node in openset having the lowest f_score[] value

if x = goalreturn reconstruct_path(came_from, came_from[goal])

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remove x from openset

add x to closedset

foreach y in neighbor_nodes(x)

if y in closedset

continue

tentative_g_score := g_score[x] + dist_between(x,y)

if y not in openset

add y to openset

tentative_is_better := true

else if tentative_g_score < g_score[y]

tentative_is_better := true

else

tentative_is_better := false

if tentative_is_better = true

came_from[y] := x

g_score[y] := tentative_g_score

h_score[y] := heuristic_cost_estimate(y, goal)

f_score[y] := g_score[y] + h_score[y]

return failure

function reconstruct_path(came_from, current_node)

if came_from[current_node] is set

p = reconstruct_path(came_from, came_from[current_node])

return (p + current_node)

else

return current_node

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Complexity

The time complexity of A* depends on the heuristic. In the

worst case, the number of nodes expanded is exponential in

the length of the solution (the shortest path), but it ispolynomial when the search space is a tree, there is a single

goal state, and the heuristic function h meets the following

condition:

| h(x) h * (x) | = O(logh * (x))

where h * is the optimal heuristic, the exact cost to get from x

to the goal. In other words, the error of h will not grow fasterthan the logarithm of the perfect heuristic h * that returns

the true distance from x to the goal

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Conclusion

Recap on A*:

Uses the formula f(x)=h(x)+g(x)

Traverses the node with the lowest f(x) value

Very useful in searching for a goal state

References:

Book: Artificial Intelligence: A ModernApproach (Second Edition) pp 97-101

Wikipedia.com