Informed Search I (Beginning of AIMA Chapter 4.1)

CIS 391

Fall 2008

CIS 391 - Intro to AI 2

OutlinePART I Informed = use problem-specific knowledge Best-first search and its variants A* - Optimal Search using Knowledge

Proof of Optimality of A* A* for maneuvering AI agents in games Heuristic functions? How to invent them

PART II Local search and optimization

• Hill climbing, local beam search, genetic algorithms,… Local search in continuous spaces Online search agents

Today!!

CIS 391 - Intro to AI 3

Breadth First Search for Games, Robots, …

Pink: Starting Point Blue: Goal Teal: Scanned squares

• Darker: Closer to starting point…

Graphics from http://theory.stanford.edu/~amitp/GameProgramming/

(A great site for practical AI & game Programming

CIS 391 - Intro to AI 4

An optimal informed search algorithm

We add a heuristic estimate of distance to the goal

Yellow: examined nodes with high estimated distance Blue: examined nodes with low estimated distance

CIS 391 - Intro to AI 5

Breadth first in a world with obstacles

CIS 391 - Intro to AI 6

Informed search in a world with obstacles

CIS 391 - Intro to AI 7

Is Uniform Cost Search the best we can do? Consider finding a route from Bucharest to Arad..

Arad

118

CIS 391 - Intro to AI 8

A Better Idea… Node expansion based on some estimate of distance to

the goal, extending current path

General approach of informed search:• Best-first search: node is selected for expansion based on an

evaluation function f(n)

Implementation:• Sort fringe queue in decreasing order of desirability.

• Special cases: greedy search, A* search

CIS 391 - Intro to AI 9

Arad

118

Romania with city-to-city distances & straight-line distances in km

CIS 391 - Intro to AI 10

A heuristic function

Let evaluation function f(n) = h(n) (heuristic)• h(n) = estimated cost of the cheapest path from node n to

goal node.• If n is goal then h(n)=0

Here: hSLD(n) = straight-line distance from n to Bucharest

[dictionary]“A rule of thumb, simplification, or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood.”

Heuristic knowledge is useful, but not necessarily correct.Heuristic algorithms use heuristic knowledge to solve a problem.A heuristic function here takes a state and returns an estimate of the distance to the goal.

Heureka! ---Archimedes

CIS 391 - Intro to AI 11

Greedy best-first search (BFS)

Expands the node that appears to be closest to goal

Expands nodes based on f(n) = h(n)• (Again, h(n)): Some heuristic estimate of cost from n to goal)

Ignores cost so far to get to that node (g(n)) Here, h(n) = hSLD(n) = straight-line distance from n

to Bucharest

CIS 391 - Intro to AI 12

Greedy best-first search example

Initial State = Arad Goal State = Bucharest

Open queue:

Arad

CIS 391 - Intro to AI 13

Greedy best-first search exampleOpen queue:

Sibiu

Timisoara

Zerind

CIS 391 - Intro to AI 14

Greedy best-first search exampleOpen queue:

Fararas

Rimnicu Vilcea

Timisoara

Arad

Zerind

Oradea

CIS 391 - Intro to AI 15

Greedy best-first search example

Goal reached !!

Open queue:

Bucharest

Rimnicu Vilcea

Timisoara

Sibiu

Arad

Zerind

Oradea

CIS 391 - Intro to AI 16

Properties of greedy best-first search

Optimal? • No!

—Found: Arad Sibiu Fagaras Bucharest (450km) —Shorter: Arad Sibiu Rimnicu Vilcea Pitesti Bucharest (418km)

Arad

118

CIS 391 - Intro to AI 17

Properties of greedy best-first search Complete?

• No – can get stuck in loops,

• e.g., Iasi Neamt Iasi Neamt …

GoalInitial

CIS 391 - Intro to AI 18

Properties of greedy best-first search

Complete? • No – can get stuck in loops, • e.g., Iasi Neamt Iasi Neamt

Time? • O(bm) – worst case (like Depth First Search)• But a good heuristic can give dramatic improvement

Space? • O(bm) – keeps all nodes in memory

Optimal? No

CIS 391 - Intro to AI 19

A* search

Best-known form of best-first search. Key Idea: avoid expanding paths that are already

expensive. Evaluation function f(n)=g(n) + h(n)

• 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

Implementation: Expand the node n with minimum f(n)

CIS 391 - Intro to AI 20

Admissible heuristics

h(n) must be an admissible heuristic • A heuristic is admissible if it never overestimates

the cost to reach the goal; i.e. it is optimistic• Formally:

n, where n is a node:

1. h(n) h*(n) where h*(n) is the true cost from n

2. h(n) 0 so h(G)=0 for any goal G.

• Example:

hSLD(n) never overestimates the actual road

distance

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

CIS 391 - Intro to AI 21

A* search exampleOpen queue:

Arad

We start with our initial state Arad. We make a node and add it to the open queue. Since it’s the only thing on the open queue, we expand the node.

Implementation: The open queue is just a priority queue (or heap) that sorts the nodes inside of it according to their g()+h() score.

CIS 391 - Intro to AI 22

A* search exampleOpen queue:

Sibiu

Timisoara

Zerind

We add the three nodes we found to the open queue.

We sort them according to the g()+h() calculation.

CIS 391 - Intro to AI 23

A* search exampleOpen queue:

Rimricu Vicea

Fagaras

Timisoara

Zerind

Arad

Oradea

When we expand Sibiu, we run into Arad again. Note that we’ve already expanded this node once; but we still add it to the open queue again.

CIS 391 - Intro to AI 24

A* search exampleOpen queue:

Rimricu Vicea

Fagaras

Timisoara

Zerind

Arad

Oradea

We see that Rimricu Vicea is at the top of the open queue; so, it’s the next node we will expand.

CIS 391 - Intro to AI 25

A* search exampleOpen queue:

Fagaras

Pitesti

Timisoara

Zerind

Craiova

Sibiu

Arad

Oradea

We expand Rimricu Vicea.

CIS 391 - Intro to AI 26

A* search exampleOpen queue:

Fagaras

Pitesti

Timisoara

Zerind

Craiova

Sibiu

Arad

Oradea

Fagaras will be the next node we should expand – it’s at the top of the sorted open queue.

CIS 391 - Intro to AI 27

A* search exampleOpen queue:

Pitesti

Timisoara

Zerind

Bucharest

Craiova

Sibiu

Sibiu

Arad

Oradea

When we expand Fagaras, we find Bucharest, but we’re not done. The algorithm doesn’t end until we “expand” the goal node – it has to be at the top of the open queue.

CIS 391 - Intro to AI 28

A* search exampleOpen queue:

Pitesti

Timisoara

Zerind

Bucharest

Craiova

Sibiu

Sibiu

Arad

Oradea

It looks like Pitesti is the next node we should expand.

CIS 391 - Intro to AI 29

A* search exampleOpen queue:

Bucharest

Timisoara

Zerind

Bucharest

Craiova

Rimricu Vicea

Sibiu

Sibiu

Craiova

Arad

Oradea

Note that we just found a better value for Bucharest! We also found a worse value for Craiova.

CIS 391 - Intro to AI 30

A* search exampleOpen queue:

Bucharest

Timisoara

Zerind

Bucharest

Craiova

Rimricu Vicea

Sibiu

Sibiu

Craiova

Arad

Oradea

Now it looks like Bucharest is at the top of the open queue…

CIS 391 - Intro to AI 31

A* search exampleOpen queue:

Bucharest

Timisoara

Zerind

Bucharest

Craiova

Rimricu Vicea

Sibiu

Sibiu

Craiova

Arad

Oradea

Now we “expand” the node for Bucharest.

We’re done! (And we know the path that we’ve found is optimal.)