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The Search Problem

Starting from a node n find the shortest path to a goal node g

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Shortest Path

Consider the following weighted undirected graph:

We want:

A path 5 🡪 v1 🡪 v2 🡪 … 🡪 20

Such that g(20) = cost(5🡪v1) + cost(v1🡪 v2) + … + cost ( 🡪20) is minimum

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Djikstra Algorithm

Greedy algorithm: from the candidate nodes select the one that has a path with minimum cost from the starting node

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Djikstra Algorithm

Given a Graph G = (V,E) and T a subset of V, the fringe of T, is defined as:

Fringe(T) = { (w,x) in E : w ∈ T and x ∈V − T}

Djikstra’s algorithm pick the edge v in Fringe(T) that has minimum distance to the starting node

g(v) is minimum

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Example

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Properties

Dijkstra’s is a greedy algorithm

Why Dijkstra’s Algorithm works?

V  T

The path from u to every node in T is the minimum path

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Example

What does Djikstra’s algorithm will do? (minimizing g(n))

Problem: Visit too many nodes, some clearly out of the question

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Complexity

Actual complexity is O(|E|log₂|E|)

Is this good?

Actually it is bad for very large graphs!

Branching factor: b

….

# nodes = b^{(# levels)}

Another Example: think of the search space in chess

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Better Solution: Make a ‘hunch”!

Use heuristics to guide the search

Heuristic: estimation or “hunch” of how to search for a solution

We define a heuristic function:

h(n) = “estimate of the cost of the cheapest path from the starting node to the goal node”

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Lets Try A Heuristic

Heuristic: minimize h(n) = “Euclidean distance to destination”

Problem: not optimal (through Rimmici Viicea and Pitesti is shorter)

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The A* Search

Difficulty: we want to still be able to generate the path with minimum cost

A* is an algorithm that:

Uses heuristic to guide search
While ensuring that it will compute a path with minimum cost

A* computes the function f(n) = g(n) + h(n)

“actual cost”

“estimated cost”

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

g(n) = “cost from the starting node to reach n”
h(n) = “estimate of the cost of the cheapest path from n to the goal node”

g(n)

h(n)

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Example

A*: minimize f(n) = g(n) + h(n)

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Properties of A*

A* generates an optimal solution if h(n) is an admissible heuristic and the search space is a tree:

h(n) is admissible if it never overestimates the cost to reach the destination node

A* generates an optimal solution if h(n) is a consistent heuristic and the search space is a graph:

h(n) is consistent if for every node n and for every successor node n’ of n:

h(n) ≤ c(n,n’) + h(n’)

n’

h(n)

c(n,n’)

h(n’)

If h(n) is consistent then h(n) is admissible
Frequently when h(n) is admissible, it is also consistent

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Admissible Heuristics

A heuristic is admissible if it is too optimistic, estimating the cost to be smaller than it actually is.

Example:

In the road map domain,

h(n) = “Euclidean distance to destination”

is admissible as normally cities are not connected by roads that make straight lines

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How to Create Admissible Heuristics

Relax the conditions of the problem

This will result in admissible heuristics!

Lets look at an 8-puzzle game:

http://www.permadi.com/java/puzzle8/

Possible heuristics?

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Example: Admissible Heuristics in 8-Puzzle Game

Heuristic: a tile A can be moved to any tile B

H1(n) = “number of misplaced tiles in board n”

Heuristic: a tile A can be moved to a tile B if B is adjacent to A

H2(n) = “sum of distances of misplaced tiles to goal positions in board n”

Some experimental results reported in Russell & Norvig (2002):

A* with h2 performs up to 10 times better than A* with h1
A* with h2 performs up to 36,000 times better than a classical uninformed search algorithm (iterative deepening)

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Using A* in Planning

h(n) = “# of goals remaining to be satisfied” g(n) = “# of steps so far”

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A* in Games

Path finding (duh!)

We will have several presentations on the topic
We will see that sometimes even A* speed improvements are not sufficient
Additional improvements are required

A* can be used for planning moves computer-controlled player (e.g., chess)

F.E.A.R. uses A* to plan its search