Download A* Search Algorithm in Artificial Intelligence and more Exams Algorithms and Programming in PDF only on Docsity!
Search
Lecture Attendance: http://bit.ly/3GEMoKZState space graph.Given a task, model of world appropriate to task.Search Tree.Representation of all plans
(also paths through state space graph.) Uninformed Search.Depth First Search.
Good space bound, āleftmostā exploration seems bad. Breadth First Search.
Bad space bound, finds plan with fewest number of actions. Uniform Cost Search.
Bad space bounds, finds optimal plan. Demo: L3D
Informed Search
Search Heuristics
A heuristic is:^ I
A function that
estimates
how close a state
is to a goal I^
Designed for a particular search problem I^
Examples:
Euclidean distance for pathing.
Manhattan distance.
Example: Heuristic Function
Example: Heuristic Function
Heuristic: the number of the largest pancake that is still out of place
Greedy Search
Greedy Search
Expand the node that seems closest to the goal?
What can go wrong?
Greedy: 140 + 99 + 211 = 450Better: 140+80+97+101 = 418
Greedy Search
Strategy: expand a node that (you think) is closest to a goal state.Heuristic: estimate of distance to nearest goal for each state
Wildest dream:Perfect heuristic. A common case:
Best-first takes you straight to the (wrong) goal Worst-case:
Like a badly-guided DFS
[Democontours greedy empty (L3D3)][Demo: contours greedy pacman small maze (L3D4)]
Video of Demo Contours Greedy (Empty)
Video of Demo Contours Greedy (pacman/small)
A* Search
A* Search
UCS
Greedy
A*
Optimality of A* Tree Search
Optimality of A* Tree Search: Blocking
Claim: Goal
A^
is expanded before
B
with
g
(B
)^ >
g(
A)
Proof:
Definition of f-cost:
g
(n
h(
n)
Admissibility of
h(
n)
ā h (n
h = 0 at a goal
Imagine B is on the fringe.
Some ancestor
n^
of A is on the fringe, too (maybe A!)
Claim:
n^
will be expanded before B
f(n) is less or equal to f(A)
since
f^ (
n) =
g(
n) +
h(
n)
f^ (
A)
f(A) is less than f(B)
since
g(
A) +
<^
g(
B) +
h(
B)
All ancestors of A expand before B.A expands before B.
A* search is optimal.
Properties of A*
UCS
A*
UCS vs A* Contours
Uniform-cost expands equally in allādirectionsā.A* expands mainly toward the goal, butdoes hedge its bets to ensure optimality
[Demo: contours UCS / greedy / A* empty (L3D1)][Demo: contours A* pacmansmall maze (L3D5)]
Video of Demo Contours (Empty) ā UCS
Video of Demo Contours (Empty) ā Greedy
Video of Demo Contours (Empty) ā A*
Video of Demo Contours: Pacman A*
Comparison
Uniform Cost Greedy
A*
A* Applications
A* Applications
Video gamesPathing / routing problemsResource planning problemsRobot motion planningLanguage analysisMachine translationSpeech recognition[Demo: UCS / A * pacman tinymaze (L3D6,L3D7)][Demo: guess algorithm EmptyShallow/Deep (L3D8)]
Video of Demo Pacman (Tiny Maze) ā UCS / A*
8 Puzzle III
How about using the actual cost as aheuristic?Would it be admissible?Would we save on nodes expanded?Whatās wrong with it?
With A*: a trade-off between quality of estimate and work per node^ I
As heuristics get closer to the true cost, expand fewer nodes, butmore work per node to compute the heuristic itself.
Semi-Lattice of Heuristics
Trivial Heuristics, Dominance
exact^ zero ha
hb
hc
max(
ha
,^ h
)b
Dominance:
ha
hc
if ān
:^ h
(na
)^ ā„
hc
(n
Heuristics form a semi-lattice:Max of admissible heuristics is admissible.Trivial heuristicsBottom of lattice is the zero heuristic.
(what does this give us?)Top of lattice is the exact heuristic
Graph Search
Tree Search: Extra Work!
Search TreeState Graph
Failure to detect repeated statescan cause exponentially morework.
Graph Search
In BFS, for example, we shouldnāt bother expanding the circled nodes(why?)
Graph Search
Idea: never expand a state twiceHow to implement:Tree search + set of expanded states (āclosed setā)Expand the search tree node-by-node, but...Before expanding a node,
check if state was never been expanded beforeIf yes skip it, else add to closed set and expand. Important:
store the closed set as a set, not a list
Can graph search wreck completeness? Why/why not?How about optimality?
A* Graph Search Gone Wrong?
Is^
h(
Ā·)^ admissible? Yes. Will exploring w.r..t
h(
B) +
g(
n)
be optimal?
Expand
S
A^ and
B^
in fringe!
Expands
B
, since
h(
B) +
g(
B) =
<^
h(
A) +
g(
A)
C^
in fringe with key, 3
+^
h(
C) =
G^
in fringe with key, 5. Could have been there in 4.
Consistency of Heuristics
A h =^
h^ =
h^ =
C G
Main idea: est. heuristic costs
actual costs
Admissibility:
h
(x)
cost to goal.
Consistency:
h
(x)
h(
y)
cost
(x
,^ y
heuristic āarcā cost
actual arc cost
Consistent
admissible? Yes? No?
Consistent: f value along a path neverdecreasesAdmissible:^ f
(C
h(
C) +
f^ (A
h(
A) =
Consistent:
f^ (
A) =
<^
f^ (
C)
Claim: If
y^
is expanded due to
x,
f^ (
y)
f^ (
x)
Proof:^ f
(y
g(
x) +
cost
(x
,^ y
h(
y)
ā„^
g(
x) +
h(
x)
ā^
h(
y) +
h(
y) =
g(
x) +
h(
x) =
f^ (
x)
The āestimateā of plan cost keeps rising as you progress.
Optimality of A* Graph Search
Optimality of A* Graph Search
Sketch: consider what A* does with aconsistent heuristic:Fact 1: In tree search, A* expands nodes inincreasing total f value (f-contours)Fact 2: For every state
s, the optimal path is
discovered.Result: A* graph search is optimalFact 1 Proof. Previous slide.
Proof of A* optimality.
v
s
x
g Start (v^ )
g(x
)
pathCost(
v^ ,^ s
)
cost(
x,^ s)
Fact 2: The optimal path is discovered to every state
s.
Proof: Consider first error.State
s^
discovered from
x.
Optimal path is from
y^
6 =^
x.
There is a vertex
v^
in the optimal path to
y^
in fringe.
s^ in fringe with key
f^ (
s) =
g(
x) +
cost
(x
,^ s) +
h(
s).
v^ in fringe with key
f^ (
v^ ) =
g(
v^ ) +
h(
v^ )
h(
v^ )
ā^
h(
s)^
ā¤^
pathCost
(v
,^ s
)^ by induction.
g(
v^ ) +
pathCost
(v^
,^ s)
g(
x) +
cost
(x
,^ s)
f^ (
v^ )
f^ (
s)
But then
v^
would have been expanded before
s!
