HW #1
DUE 29/9/2008

Write Java Applet to solve Goats and Cabbage
“Missionaries and cannibals” problem with the following
search algorithms:
Breadth first
 Iterative deepening

Your applet should have graphical interface to illustrate
the movements of goats and cabbage across the river.
Also, print the following statistics at the end of the
search:
Maximum depth.
 Time complexity (time and number of nodes)
 Space complexity.

1
BLIND SEARCH





Depth-first search and breadth-first search are
examples of blind (or uninformed) search strategies.
Breadth-first search produces an optimal solution
(eventually, and if one exists), but it still searches
blindly through the state-space.
Neither uses any knowledge about the specific domain
in question to search through the state-space in a more
directed manner.
If the search space is big, blind search can simply take
too long to be practical, or can significantly limit how
deep we're able to look into the space.
A search strategy is defined by picking the order of node
2
expansion
INFORMED SEARCH - OVERVIEW

Heuristic Search

Best-First Search Approach
Greedy
 A*



Heuristic Functions
Local Search and Optimization




Hill-climbing
Simulated Annealing
Local Beam
Genetic Algorithms
3
INFORMED SEARCH

Relies on additional knowledge about the problem
or domain


Used to distinguish more promising paths towards a
goal


frequently expressed through heuristics.
may be mislead, depending on the quality of the
heuristic
In general, performs much better than uninformed
search

but frequently still exponential in time and space for
realistic problems
4
INFORMED SEARCH (CONT.)




Find a solution more quickly
Find solutions even when there is limited time
available
Often find a better solution, since more profitable
parts of the state-space can be examined, while
ignoring the unprofitable parts.
A search strategy which is better than another at
identifying the most promising branches of a searchspace is said to be more informed.
5
BEST-FIRST SEARCH

Relies on an evaluation function f(n) that gives an
indication of how useful it would be to expand a node




family of search methods with various evaluation functions
usually gives an estimate of the distance to the goal
often referred to as heuristics in this context
The node with the lowest value is expanded first


Fringe is ordered by f(n)
the name is a little misleading: the node with the lowest value for
the evaluation function is not necessarily one that is on an
optimal path to a goal
function BEST-FIRST-SEARCH(problem, EVAL-FN) returns solution
fringe := queue with nodes ordered by EVAL-FN
return TREE-SEARCH(problem, fringe)
6
GREEDY BEST-FIRST SEARCH

Minimizes the estimated cost to a goal
expand the node that seems to be closest to a goal
 utilizes a heuristic function as evaluation function

f(n) = h(n) = estimated cost from the current node to a goal
 heuristic functions are problem-specific
 e.g., fSLD(n) = straight-line distance
 often straight-line distance for route-finding and similar
problems


often better than depth-first.
function GREEDY-SEARCH(problem) returns solution
return BEST-FIRST-SEARCH(problem, h)
Completeness Time Complexity Space Complexity Optimality
no
bm
bm
no
b: branching factor, d: depth of the solution, m: maximum depth of the search tree, l: depth limit
7
GREEDY BEST-FIRST SEARCH SNAPSHOT
1
7
2
4
6
3
5
Initial
Visited
Fringe
Current
Visible
Goal
9
5
6
7
5
6
7
Heuristics 7
8
7
9
5
10
4
11
3
2
12
13
4
14
5
15
6
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Fringe: [13(4), 7(6), 8(7)] + [24(0), 25(1)]
8
ROUTE FINDING
374
366
253
329
9
GREEDY BEST-FIRST SEARCH EXAMPLE
10
GREEDY BEST-FIRST SEARCH EXAMPLE
11
GREEDY BEST-FIRST SEARCH EXAMPLE
12
GREEDY BEST-FIRST SEARCH EXAMPLE
So is Arad->Sibiu->
Fagaras->Bucharest
optimal?
13
GREEDY SEARCH, EVALUATION

Completeness: NO


Check on repeated states
Minimizing h(n) can result in false starts, e.g. Iasi to Fagaras.
GREEDY SEARCH, EVALUATION
Cost
Max
An obvious problem with greedy search is that it
doesn't take account of the cost so far, so it isn't
optimal, and can wander into dead-ends, like depthfirst search.
local minimum = looping
0

Initial
State
Goal
15
A* SEARCH

Combines greedy and uniform-cost search to find the
(estimated) cheapest path through the current node


f(n) = g(n) + h(n)
= path cost + estimated cost to the goal from n
heuristics must be admissible


h(n) never overestimate the cost to reach the goal
very good search method, but with complexity problems
function A*-SEARCH(problem) returns solution
return BEST-FIRST-SEARCH(problem, g+h)
Completeness Time Complexity Space Complexity Optimality
yes
bd
bd
yes
b: branching factor, d: depth of the solution, m: maximum depth of the search tree, l: depth limit
16
A* SEARCH
Evaluation Function:
F(n) = g(n) + h(n)
Path cost from root
to node n
Estimated cost of
cheapest path from
node n to goal
17
A* SNAPSHOT
1
9
9
4
3
11
10
7
2
3
7
2
7
2
4
10
4
6
5
Initial
Visited
Fringe
Current
Visible
Goal
5
6
13
5
6
7
Edge Cost
2
5
4
4
4
3
11
8
3
7
4
9
7
5
10
2
4
4
11
8
6
3
2
12
5
4
4
13
4
9
9
Heuristics 7
12
3
13
6
14
5
15
2 8
3
6
f-cost
9
10
2
13
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Fringe: [2(4+7), 7(3+4+6), 13(3+2+3+4)] + [24(3+2+4+4+0), 25(3+2+4+3+1)]
18
A* SNAPSHOT WITH ALL F-COSTS
4
3
11
10
7
2
3
7
2
17
5
7
2
11
6
4
Initial
Visited
Fringe
Current
Visible
Goal
9
1
4
10
5
6
13
5
6
7
Edge Cost
2
20
8
3
24 24
4
21
7
4
5
9
7
29 23
4
14
5
10
2
4
18
4
13
4
11
8
19
6
18
3
11
2
12
5
3
4
12
4
13
3
16 13
6
4
14
2
13 14 13
9
9
18
22
5
6
15
8 3
Heuristics 7
f-cost
9
10
2
25 21 31
25
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
19
A* PROPERTIES


The value of f never decreases along any path
starting from the initial node
This property can be used to draw contours
regions where the f-cost is below a certain threshold
 the better the heuristics h, the narrower the contour
around the optimal path

20
A* SNAPSHOT WITH CONTOUR F=11
4
3
11
10
7
2
3
7
2
17
5
7
2
11
6
4
Initial
Visited
Fringe
Current
Visible
Goal
9
1
4
10
5
6
13
5
6
7
Edge Cost
2
20
8
3
24 24
4
21
7
4
5
9
7
29 23
4
14
5
10
2
4
18
4
13
4
11
8
19
6
18
3
11
2
12
5
3
4
12
4
13
3
16 13
6
4
14
2
13 14 13
9
9
18
22
5
6
15
8 3
Heuristics 7
f-cost
9
10
2
25 21 31
25
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Contour
21
A* SNAPSHOT WITH CONTOUR F=13
4
3
11
10
7
2
3
7
2
17
5
7
2
11
6
4
Initial
Visited
Fringe
Current
Visible
Goal
9
1
4
10
5
6
13
5
6
7
Edge Cost
2
20
8
3
24 24
4
21
7
4
5
9
7
29 23
4
14
5
10
2
4
18
4
13
4
11
8
19
6
18
3
11
2
12
5
3
4
12
4
13
3
16 13
6
4
2
13
8
7
6
5
4
3
2
1
0
1
16
17
18
19
20
21
22
23
24
25
14
13
14
9
9
18
22
5
6
15
8 3
Heuristics 7
f-cost
9
10
2
25 21 31
25
3
4
5
6
7
27
28
29
30
31
Contour
2
26
22
A* SEARCH EXAMPLE
23
A* SEARCH EXAMPLE
24
A* SEARCH EXAMPLE
25
A* SEARCH EXAMPLE
26
A* SEARCH EXAMPLE
27
A* SEARCH EXAMPLE
28
A* SEARCH

A* expands nodes in increasing f value

Gradually adds f-contours of nodes
29
OPTIMALITY OF A*

A* will find the optimal solution


A* is optimally efficient


the first solution found is the optimal one
no other algorithm is guaranteed to expand fewer nodes
than A*
A* is not always “the best” algorithm

optimality refers to the expansion of nodes


other criteria might be more relevant
it generates and keeps all nodes in memory

improved in variations of A*
30
COMPLEXITY OF A*

The number of nodes within the goal contour search
space is still exponential
with respect to the length of the solution
 better than other algorithms, but still problematic


Frequently, space complexity is more severe than
time complexity

A* keeps all generated nodes in memory
31
MEMORY-BOUNDED SEARCH

Search algorithms that try to conserve memory

Most are modifications of A*
iterative deepening A* (IDA*)
 simplified memory-bounded A* (SMA*)


recursive best-first search (RBFS)
32
ITERATIVE DEEPENING A* (IDA*)

Explores paths within a given contour (f-cost limit) in a
depth-first manner

this saves memory space because depth-first keeps only the
current path in memory

but it results in repeated computation of earlier contours since it
doesn’t remember its history

was the “best” search algorithm for many practical problems
for some time

does have problems with difficult domains
33
SIMPLIFIED MEMORY-BOUNDED A* (SMA*)

Uses all available memory for the search

drops nodes from the queue when it runs out of space

those with the highest f-costs

If there is a tie (equal f-values) we first delete the oldest
nodes first.

complete if there is enough memory for the shortest
solution path

often better than A* and IDA*
but some problems are still too tough
 trade-off between time and space requirements

34
HEURISTICS FOR SEARCHING

For many tasks, a good heuristic is the key to
finding a solution
prune the search space
 move towards the goal

37
HEURISTIC FUNCTIONS 8-PUZZLE

Example: 8-Puzzle
 Average solution cost for a random puzzle
is 22 moves

Branching factor is about 3
Empty tile in the middle -> four moves
 Empty tile on the edge -> three moves
 Empty tile in corner -> two moves


322 is approx 3.1e10
Get rid of repeated states
 181440 distinct states

38
HEURISTIC FUNCTIONS 8-PUZZLE
h1=the number of misplaced tiles
 h2=the sum of the Manhattan distances of the tiles
from their goal positions

h1 = 7
 h2 = 4+0+3+3+1+0+2+1+2 = 16

39
HEURISTIC FUNCTIONS 8-PUZZLE
h1=0
 h2=0

 h1=2
 h1=2
 h2=8
 h2=2
40