Notes

Dijstra’s Algorithm Corrected syllabus

Tree Search Implementation Strategies

Require data structure to model search tree

Tree Node: State (e.g. Sibiu) Parent (e.g. Arad) Action (e.g. GoTo(Sibiu)) Path cost or depth (e.g. 140) Children (e.g. Faragas, Oradea) (optional,

helpful in debugging)

Queue

Methods: Empty(queue)

Returns true if there are no more elements Pop(queue)

Remove and return the first element Insert(queue, element)

Inserts element into the queue InsertFIFO(queue, element) – inserts at the end InsertLIFO(queue, element) – inserts at the front InsertPriority(queue, element, value) – inserts

sorted by value

INFORMED SEARCH

Search

goal

start

Uninformed search Informed search

Informed Search

What if we had an evaluation function h(n) that gave us an estimate of the cost associated with getting from n to the goal h(n) is called a heuristic

Romania with step costs in km

h(n)

Greedy best-first search

Evaluation function f(n) = h(n) (heuristic) e.g., f(n) = hSLD(n) = straight-line distance

from n to Bucharest

Greedy best-first search expands the node that is estimated to be closest to goal

Romania with step costs in km

n h(n)

f(n)

Best-First Algorithm

Performance of greedy best-first search

Complete?

Optimal?

Failure case for best-first search

Performance of greedy best-first search

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

Neamt Iasi Neamt

Optimal? No

Complexity of greedy best first search

Time? O(bm), but a good heuristic can give

dramatic improvement

Space? O(bm) -- keeps all nodes in memory

What can we do better?

A* search

Ideas: Avoid expanding paths that are already

expensive Consider

Cost to get here (known) – g(n) Cost to get to goal (estimate from the heuristic)

– h(n)

A * Evaluation functions

Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path

through n to goal

start goaln

g(n) h(n)

f(n)

n g(n)

h(n)

f(n)

A* Heuristics

A heuristic h(n) is admissible if for every node n,

h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.

An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: hSLD(n) (never overestimates the

actual road distance)

What happens if heuristic is not admissible?

Will still find solution (complete)

But might not find best solution (not optimal)

Properties of A* (w/ admissible heuristic)

Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )

Optimal? Yes

Time? Exponential, approximately O(bd) in the worst case

Space? O(bm) Keeps all nodes in memory

The heuristic h(x) guides the performance of A*

Let d(x) be the actual distance between S and G h(x) = 0 :

A* is equivalent to Uniform-Cost Search h(x) <= d (x) :

guarantee to compute the shortest path; the lower the value h(x), the more node A* expands

h(x) = d (x) : follow the best path; never expand anything else;

difficult to compute h(x) in this way! h(x) > d(x) :

not guarantee to compute a best path; but very fast h(x) >> g(x) :

h(n) dominates -> A* becomes the best first search

Admissible heuristics

Admissible heuristics

E.g., for the 8-puzzle:

Admissible heuristics

E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = summed Manhattan distance for all tiles (i.e., no. of

squares from desired location of each tile)

h1(S) = ? h2(S) = ?

Admissible heuristics

E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from

desired location of each tile)

h1(S) = ? 8 h2(S) = ? 3+1+2+2+2+3+3+2 = 18 Which is better?

Dominance

If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1 h2 is better for search

What does better mean? All searches we’ve discussed are

exponential in time

Comparison of algorithms(number of nodes expanded)

D Iterative deepening

A*(teleporting tiles)

A* (manhattan distance)

2 10 6 6

6 112 13 12

10 680 20 18

12 364035 227 73

14 2896689 539 113

18 1.8 * 108 3056 363

24 8.6 * 1010 39135 1641

Visually

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

1 0 0 0

1 2 3 4 5 6 7 8 9 1 0

Number

of expe

cted exp

ansions

S e a r c h d e p t h

U n i n f o r m e d

H 1

H 2

Where do heuristics come from? From people

Knowledge of the problem

From computers By considering a simpler version of the

problem Called a relaxation

Relaxed problems

8-puzzle If the rules of the 8-puzzle are relaxed so that a

tile can move anywhere, then h1(n) gives the shortest solution

If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution

Consider the example of straight line distance (Romania navigation) Is that a relaxation?

Iterative-Deepening A* (IDA*) Further reduce memory requirements of

A*

Regular Iterative-Deepening: regulated by depth

IDA*: regulated by f(n)=g(n)+h(n)

Questions?