Best-First Search: The A* Algorithm - NTNU · Incremental -vs- Local Search Incremental Local The problem-solver works with partial -vs- complete solutions. Keith L. Downing Best-First

Best-First Search: The A* Algorithm

Keith L. Downing

August 22, 2017

Keith L. Downing Best-First Search: The A* Algorithm

Search is..

A (preferably methodical) process for finding something.

Searching for:

pre-existing entities (information, objects, etc.)strategies for creating/designing entities.

Examples:

Web search (for just about anything)AI search - creates something, doesn’t just find it.Web search for AI search algorithms.

Uninformed -vs - Informed: Do points in the search space giveinformation that helps the searcher to determine the next step?

Partial -vs- Complete solutions (i.e., attempts): Could the current stateof search always be considered a complete solution, though notnecessarily good or optimal? Or is it often a partial state that must beincrementally enhanced to become a solution?

E.g. 2 approaches to origami.Closed-loop -vs- Open-loop control


Incremental -vs- Local Search

Incremental Local

The problem-solver works with partial -vs- complete solutions.


8 Puzzle: An AI Classic

1

2

34

56

78

1

234

56

78

Move Up

1

2

34

56 78

MoveDown

1

2

34

5678

Move Left

1

7

38

2456

Goal State

Current State

For this puzzle, a complete solution is a sequence of moves that achieves thegoal configuration. Typically, this is solved using incremental methods thatadd one move at a time to a partial sequence.


The Horizon of a Search Graph

Root Node Contains Starting Search State Open, Unexpanded

Closed, ExpandedExpansion = Generating Child Nodes

Expansion


3 Common Incremental Search Algorithms

Depth-First Breadth-First

Best-First

General General

Problem-Specific

"Intelligent"


Stacks, Queues and Agendas

The key decision that differentiates depth-first, breadth-first andbest-first search is: What node on the OPEN list should beexpanded next?

Depth-First: OPEN = a stack (LIFO - Last in, first out)Breadth-First: OPEN = a queue (FIFO - First in, first out)Best-First: OPEN = a sorted list known as the agenda,where sorting is based on the node’s evaluation, whichreflects the algorithm’s knowledge about that search stateand its potential to be extended into an optimal solution.


Shakey the Robot: Stanford Research Institute, 1960’s

Needed a good route-planning algorithm.Hart, Nilsson, Raphael (1968), A formal basis for the heuristicdetermination of minimum cost paths, in IEEE Transactions SystemScience and CyberneticsStill used by the vehicle-navigation industry.https://www.youtube.com/watch?v=mQ7M-zhiu7U


The Evaluation Function in Best-First Search

x

GoalState

f(x) = g(x) + h(x)

h(x)

g(x)

Concrete Estimate


The Heuristic Function: h(X)

1

2

34

56

78

1

7

38

2456

Goal StateCurrent State

Distance ??

1 35

8

Euclidiean Distance

1 35

8

Manhattan Distance

The heuristic function estimates the distance (typically in terms of the numberof moves) from the current state to the goal.


The A* Algorithm

DEFINE best-first-search()1 CLOSED← /0; OPEN← /02 Generate the initial node, n0, for the start state.3 g(n0)← 0; Calculate h(n0)4 f(n0)← g(n0) + h(n0);5 Push n0 onto OPEN.6 Do Agenda Loop


The Agenda Loop (Open List = Agenda)

If OPEN = /0 return FAIL

X← pop(OPEN)

push(X,CLOSED)

If X is a solution, return (X, SUCCEED)

SUCC← generate-all-successors(X)

For each S ∈ SUCC do:

If node S* has previously been created, and if state(S*) =state(S), then S← S*.push(S,kids(X))If not(S ∈ OPEN) and not(S ∈ CLOSED) do:

attach-and-eval(S,X)insert(S,OPEN) ;; OPEN is sorted by ascending f value.

else if g(X) + arc-cost(X,S) < g(S) then (found cheaper pathto S):

attach-and-eval(S,X)If S ∈ CLOSED then propagate-path-improvements(S)


Node Expansion in A*

A(10)

B(9) C(9)

D(8) E(10)

Agenda(Open Nodes)

C

D

E

Closed Nodes

A

B

2

1

1

1

F(7) G(7) H(8)

A(10)

B(9) C(9)

D(8) E(10)

2

1

1

1

2 1 1

Expand D

Agenda(Open Nodes)

Closed Nodes

A

B

D

C

E

H

G

F


Updating g values in A*

P1g = 4 h = 20

C1g = 5 h = 19

P2g = 2 h = 22

C2g = 6 h = 18

C1g = 3 h = 19

C2g = 4 h = 18

Before After


Admissable Heuristics for A*An admissable heuristic NEVER overestimates distance to the goal.

W

X

Y

Z

Agenda

W

X

Time

1

11

1

2

2

1

V

f(w) = 5 + 0

f(v) <= 3 + 2

h(x) <= 2

h(v) <= 2

V

X W

Z

W

Y

f(x) <= 2 + 2f(y) <= 3 + 1f(z) = 4

SolutionFound


Losing AdmissabilityIf h over-estimates for state x, node X dies on the agenda despite being onoptimal path.

W

X

Y

Z

f(x) = 2 + 4

Agenda

W

X

Time

1

11

1

2

2

1

V

f(w) = 5 + 0

f(v) = 3 + 2

h(x) = 4

h(v) = 2

V

X

W

X

W

X

SolutionFound


Rush Hour

4

0

3

21

5

0, 2, 2, 20, 0, 4, 30, 3, 4, 20, 4, 1, 21, 2, 0, 21, 4, 2, 2

ScenarioSpecification

Exit

What is g?Find an admissible h.


Missionaries and Cannibals: A Classic AI Puzzle

M

M

MC C

C

3 x (M,C) => Boat holds 2(4 or 5) x (M,C) => Boat holds 3(6 or more) x (M,C) => Boat holds 4



The Switchboard Puzzle

Start

End

Visit every peg from start to end. Neighbor distance is D, buteach turn requires an extra amount, W, for wrapping.



Example Demos

Missionaries and CannibalsRush HourNavigation


Documents

Best-First Search: The A* Algorithm - NTNU · Incremental -vs- Local Search Incremental Local The problem-solver works with partial -vs- complete solutions. Keith L. Downing Best-First