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

August 20, 2014

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 give information that helps the searcher to determine the next step? Partial -vs- Complete solutions (i.e., attempts): Could the current state of search always be considered a complete solution, though not necessarily good or optimal? Or is it often a partial state that must be incrementally enhanced to become a solution?

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

Best-First Search: The A* Algorithm

Incremental -vs- Local Search

Incremental

Local

The problem-solver works with partial -vs- complete solutions. Keith L. Downing

Best-First Search: The A* Algorithm

8 Puzzle: An AI Classic Goal State

1 2 3 8

Current State

4

7 6 5

1 5 3 4 6 2 8 7 Move Left Move Up Move Down

1 5 4 6 3 2 8 7

1 5 3 4

1 5 3

6

2 8 7

4 6 7 2 8

For this puzzle, a complete solution is a sequence of moves that achieves the goal configuration. Typically, this is solved using incremental methods that add one move at a time to a partial sequence. Keith L. Downing

Best-First Search: The A* Algorithm

The Horizon of a Search Graph Root Node Contains Starting Search State Expansion = Generating Child Nodes

Open, Unexpanded Closed, Expanded

Expansion

Keith L. Downing

Best-First Search: The A* Algorithm

3 Common Incremental Search Algorithms General

General

Breadth-First

Depth-First

Problem-Specific "Intelligent"

Best-First

Keith L. Downing

Best-First Search: The A* Algorithm

Stacks, Queues and Agendas

The key decision that differentiates depth-first, breadth-first and best-first search is: these algorithms: What node on the OPEN list should be expanded 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, which reflects the algorithm’s knowledge about that search state and its potential to be extended into an optimal solution.

Keith L. Downing

Best-First Search: The A* Algorithm

Shakey the Robot: Stanford Research Institute, 1960’s

Needed a good route-planning algorithm. Hart, Nilsson, Raphael (1968), A formal basis for the heuristic determination of minimum cost paths, in IEEE Transactions System Science and Cybernetics Still used by the vehicle-navigation industry. Keith L. Downing

Best-First Search: The A* Algorithm

The Evaluation Function in Best-First Search

g(x)

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

Concrete

Estimate

h(x)

Goal State

Keith L. Downing

Best-First Search: The A* Algorithm

The Heuristic Function: h(X)

Current State

Goal State

1 5 3

Distance ??

1 2 3

4 6

8

2 8 7

7 6 5

Euclidiean Distance

4

Manhattan Distance

1 5 3

1 5 3

8

8

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

Keith L. Downing

Best-First Search: The A* Algorithm

The A* Algorithm

DEFINE best-first-search() 1

CLOSED ← 0; / OPEN ← 0/

2

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

Keith L. Downing

Best-First Search: The A* Algorithm

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 path to S): attach-and-eval(S,X) If S ∈ CLOSED then propagate-path-improvements(S) Keith L. Downing

Best-First Search: The A* Algorithm

Node Expansion in A* A (10)

A (10)

1

1

1 C(9)

B(9)

1

2 E (10)

D(8)

1

C(9)

B(9)

1

Expand D

2

D(8)

2

F(7)

E(10)

1

1 G(7)

H(8)

D C

B

E

A

Agenda Closed Nodes (Open Nodes)

Keith L. Downing

G C H

D

F

B

E

A

Agenda Closed Nodes (OpenSearch: Nodes)The A* Algorithm Best-First

Updating g values in A*

P1

g=4

h = 20

P2

g=2

h = 22

Before

After

C1

g=5

C1

h = 19

g=3

h = 19

C2

g=6

C2

h = 18

g=4

Keith L. Downing

h = 18

Best-First Search: The A* Algorithm

Admissable Heuristics for A* An admissable heuristic NEVER overestimates distance to the goal. 1

2

f(x) ≤ 2 + 2 1

X

Y

1

f(z) = 4 + 0 1 Z

V

2

f(v) ≤ 3 + 2 f(w) = 5 + 0

f(y) ≤ 3 + 1

1

W

Agenda

V X

Y

W

W

Z

X W

Time

Keith L. Downing

Best-First Search: The A* Algorithm

Rush Hour

Scenario Specification

4 3 Exit

0 5

0, 2, 2, 2 0, 0, 4, 3 0, 3, 4, 2 0, 4, 1, 2

1

2

1, 2, 0, 2 1, 4, 2, 2

What is g? Find an admissible h.

Keith L. Downing

Best-First Search: The A* Algorithm

Missionaries and Cannibals: A Classic AI Puzzle

M C M C C M 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

What is g? Find an admissible h. Keith L. Downing

Best-First Search: The A* Algorithm

The Switchboard Puzzle

Start

End

Visit every peg from start to end. Neighbor distance is D, but each turn requires an extra amount, W, for wrapping. What is g? Find an admissible h.

Keith L. Downing

Best-First Search: The A* Algorithm

Example Demos

Missionaries and Cannibals Rush Hour Navigation

Keith L. Downing

Best-First Search: The A* Algorithm