CS 484 – Artificial Intelligence1 Announcements Department Picnic: today, after class Lab 0 due...

CS 484 – Artificial Intelligence 1

Announcements

• Department Picnic: today, after class• Lab 0 due today• Homework 2 due Tuesday, 9/18• Lab 1 due Thursday, 9/20

• Autumn – Current Event Tuesday

Search

Lecture 4


What is search?

• Getting from Arad to Bucharest? ?

?

Arad

SibiuTimisoara

Zerind


Brute Force Search

• exhaustive - examines every node in the search tree.

• Generate and test is simplest:• Generate possible solutions to the problem.• Test if valid solution.• Stop when a valid solution is found.

• The method used to generate possible solutions must be carefully chosen.


Depth-First Search

• DFS - exhaustive search• Follows each path to its deepest node,

before backtracking to try the next path.


Breadth-First Search

• BFS - exhaustive search• Follows each path to a given depth before

moving on to the next depth.


Comparison of Depth-First and Breadth-First Search

• Choose search method wisely• Depth first search may never find the

answer. Why?


Properties of Search Methods

• Complexity• Time and memory

• Completeness• Guaranteed to find a goal if one exists.

• Optimality & Admissibility• Guaranteed to find the best path to the goal.

• Irrevocability• Never back-tracks


Implementations

• DSF implemented using a stack to store states• Recursive function

• BFS implementations are almost identical to depth-first• Difference: use queue rather than a stack.


Depth-First Iterative Deepening

• Combo of DFS and BFS exhaustive search• DFS to depth of 1, then to depth of 2, 3, …

until a goal reached.• Efficient in memory use

• Copes with infinitely long branches.

• Not as inefficient in time as it might appear, • Only examines the largest row (the last one)

once.


Modified Traveling Salesman Problem

• Find shortest path from A to F

• Edges: length of roads

A

B

C

E

F

D

10

3 3

3 46

12

3

A

CB

BCD

DD

D

E

EE

EC

F

FF

F

F

F

F

FB


Heuristics

• Heuristic: a rule used to make search more efficient or effective.

• Use a heuristic evaluation function, f(n):• f(n) is the approximate distance of a node, n, from a

goal node.

• For two node m and n, if f(m) < f(n), than m is more likely to be on an optimal path

• f(n) may not be 100% accurate, but it should give better results than pure guesswork.


Heuristics – how informed?

• The more informed a heuristic is, the better it will perform.

• Heuristic h is more informed than j, if:• h(n) j(n) for all nodes n.

• A search method using h will search more efficiently than one using j.

• A heuristic should reduce the number of nodes that need to be examined.


Choosing Good Heuristics

• Right heuristic can have a significant impact in the time required to solve the problem• Should reduce the number of nodes that need to

be searched• The heuristic cannot take too long to calculate


8-puzzle

• States: location of each of the eight tiles and the blank one

• Initial State: Any state can be designated so• Successor function: Generates legal states that

result from trying 4 actions• Goal test: Checks whether state matches goal state• Path cost: Each step costs 1, total cost == length of

path

7 6

4 3 1

2 5 8

1 2 3

8 4

7 6 5Sta

rt

Sta

te

Goal

Sta

te


8-puzzle

• Game Properties• Typically 20 moves to get from random start to goal

state

• Branching Factor:

• Search tree examines 320 states (3.5 billion)

• Use heuristics to reduce states that are searched• How many states is a given state from the goal?

7 6

4 3 1

2 5 8

1 2 3

8 4

7 6 5Sta

rt

Sta

te

Goal

Sta

te


Admissible Heuristics

• Admissible heuristic:

• Possible 8-puzzle admissible heuristics


Hill Climbing

• Think of it as finding the highest point in a three dimensional search space:

• Check the height one foot away from your current location in each direction; North, South, East and West.

• Find a higher position, move to that location, and restart the algorithm.

• An informed, irrevocable search method.


Foothills

• Cause difficulties for hill-climbing methods.• A foothill is a local maximum.


Plateaus

• Cause difficulties for hill-climbing methods.• Flat areas that make it hard to find where to go

next.


Ridges

• Cause difficulties for hill-climbing methods• B is higher than A.

At C, the hill-climber can’t find a higher point North, South, East or West, so it stops.


Hill Climbing Algorithm

Function hill() {queue = [];state = root_node;while (true) { if is_goal(state) return SUCCESS; else { sort(successor(state)); add_to_front_of_queue (successor(state)); }

if queue == [] return FAILURE; state = queue[0]; remove_first_item

(queue);}

}



• Find shortest path from A to F

• Solve using hill climbing

A

B

C

E

F

D

distance as the crow flies

25

8

206

12

What happens now?


Best-First Search

• Like hill-climbing:• Picks the most likely path (based on heuristic

value) from the partially expanded tree

• Tends to find a shorter1 path than depth-first or breadth-first search, but does not guarantee to find the best path.

1 depends how shorter is defined


Hill Climbing Algorithm

Function hill() {queue = [];state = root_node;while (true) { if is_goal(state) return SUCCESS; else { sort(successor(state)); add_to_front_of_queue (successor(state)); }


(queue);}

}


Best-Search Algorithm

Function best() {queue = [];state = root_node;while (true) { if is_goal(state) return SUCCESS; else { add_to_front_of_queue (successor(state)); sort(queue); }


(queue);}

}


Optimal Paths

• An optimal path - shortest possible path from root to goal (i.e lowest cost)

• not necessarily at the shallowest depth, if edges have costs

• The British Museum procedure finds an optimal path by examining every possible path, and selecting the one with the least cost.

• There are more efficient ways.


A* Algorithms

• Uses the cost function:• f(node) = g(node) + h(node).

• g(node) is the cost of the path so far leading to the node.

• h(node) is an underestimate of how far node is from a goal state.

• f is a path-based evaluation function.• An A* algorithm expands paths from nodes that

have the lowest f value.


A* Algorithms (continued)

• A* algorithms are optimal:• They are guaranteed to find the shortest path• Provided h(node) is always an underestimate. (h is an

admissible heuristic).• A* methods are also optimally efficient – they

expand the fewest possible paths to find the right one.

• If h is not admissible, the method is called A, rather than A*.



• Get from A to F using shortest path

• Solve using A*• g(n) – cost so far

• h(n) – heuristic

A

B

C

E

F

D

h(n) = distance as the crow flies

157

26 5

3 66

11

8

146

11


Uniform Cost Search

• Also known as branch and bound.• Like A*, but uses:

• f(node) = g(node).• g(node) is the cost of the path leading up to

node.• Optimal as long as all paths have a positive

weight


Greedy Search

• Like A*, but uses:• f(node) = h(node).

• Hence, always expands the node that appears to be closest to a goal, regardless of what has gone before.

• Not optimal, and not guaranteed to find a solution at all.

• Can easily be fooled into taking poor paths.


Real World Search Problem

• Route-finding Problem• States: represented by location and current time• Initial State: Specified by problem• Successor function: returns states resulting

from taking any scheduled flight leaving later than current time plus transit time

• Goal test: arrived at destination by prescribed time

• Path cost: monetary cost, waiting time, flight time, seat quality, time of day, etc.

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