State Space Analysis

1
State Space Analysis

• State a suitable representation for the problem
  state
• State what the initial and final states are in
  this representation
• State the available operators for getting from
  one state to another, giving any conditions on
  when they may be applied

2
Room Navigation

• Goal Find a path from Room A to H

A
A B C D E F G H I J K L
E
B
I
C
F
J
D
K
H
G
L
3
Example Blocks World

• Put ltblock name 1gt on ltblock name 2gt
• Constraints
• can only move 1 block at a time
• blocks must be clear to be moved

Goal Put C on E
A
B
D
C
E
4
Blocks World Operators

• MOVE moves a block from one location to another
  by moving the hand
• preconditions both blocks must be clear
• CLEAR moves the block above a block to the table
• preconditions block above current block must be
  clear

5
Reasoning Process

• Each operator has preconditions
• Preconditions can be met by other operators

6
Example
Goal Put C on E
A
B
D
C
E

• Move(C, E)
• Clear(C)
• Move(B, Table)
• Clear(B)
• Move(A, Table)
• finish Move(B, Table)
• Clear(E)
• Move(D, Table)
• finish Move(C, E)

7
Space/Time requirements (Search)

• branching factor b 10, depth d
• Assume 1000 nodes can be goal-checked and
  expanded per second, and a node needs 100 bytes
  of storage.
• Breadth-first search
• Depth-first search for depth 12 space
  12kilobytes

8
More Blind Search methods

• Depth-limited Search
• Iterative Deepening Search
• Bidirectional Search

9
Depth-limited search

• Impose a cutoff on the maximum depth of a path
• if depth set appropriately (e.g. of cities - 1)
• guaranteed to find a solution, but not to find
  the shortest solution
• if depth too small
• wont find a solution at all

10
Iterative deepening search

• How to pick a good limit?
• pick it by solving the problem at all depths
• Explore the space doing limited depth-first
  search at depth 0, then 1, then 2, . . .
• Wasteful?
• space used is like depth-first search
• time is less of a problem and most of the nodes
  are at the bottom of the tree, so actual number
  re-opened is relatively small

11
Bidirectional search

• Simultaneously search forward from the start and
  backward from the goal, stopping when they meet.
• Properties
• if branching factor is b in both directions, only
  need to explore half the depth
• b 10, d 6
• breadth-first 1,111,111 nodes
• bidirectional 2,222 nodes
• needs reversible operators or easy way to
  calculate predecessors
• works best with 1 goal state, but can work with
  multiple if it is a finite set
• needs efficient way to check to see if a node
  already exists
• what kind of search does each half do?

12
Heuristic

• from the Greek verb heuriskein
• to find to discover
• Archmides supposedly said Heureka (I have found
  it) after discover the principle of flotation in
  his bath
• History
• 1957 used to refer to the study of
  problem-solving techniques (particularly for
  mathematics)
• 1963 used to be something that is not an
  algorithm (A process that may solve a given
  problem, but offers no guarantee of doing so, is
  called a heuristic for that problem
• rule of thumb
• now refers to any technique that improves the
  average-case performance on a problem-solving task

13
Optimal Searches (Ch. 4)

• Goal find optimal solution
• Methods
• branch and bound
• greedy search
• A

14
Branch-and-Bound (Uniform Cost)

• Keep track of all partial paths extend shortest
  until goal is found
• Continue expanding partial paths whose length is
  less than the complete path already found
• Why? If always expand just the shortest, isnt
  the first one found certain to be the shortest
  overall?

15
Sample problem city path

• Salesman must get from S to G

4
4
3
5
5
4
3
4
2

• S-A-D-E-F-G costs 35243 17
• S-D-E-F-G costs 4243 13
• S-A-B-E-F-G costs 34543 19

16
S
D 4
A 3
B 7
D 8
A 9
E 6
S
D 4
A 3
B 7
D 8
A 9
E 6
C 11
E 12
B 11
F 10
17
S
D 4
A 3
B 7
D 8
A 9
E 6
C 11
E 12
E 10
B 13
B 11
F 10
18
S
D 4
A 3
B 7
D 8
A 9
E 6
C 11
E 12
E 10
B 13
B 11
F 10
B 15
F 14
G 13
19
S
D 4
A 3
B 7
D 8
A 9
E 6
C 11 deadend
E 12
E 10
B 13
B 11
F 10
B 15
F 14
C 15
A 15
G 13
D 14
F 16
20
Greedy Search

• There is a function to estimate of distance to
  goal
• Expand node with lowest estimate
• Note doesnt take into account cost of path so
  far
• Can have problems if redundancies arent detected

21
Combine previous 2 methods

• Know cost of path already constructed
• Have a function to estimate dist. to goal
• characteristics of estimate important
• if perfect, great
• if overestimate, may put searcher off the path
  altogether
• if underestimate, may cause some waste but wont
  eliminate best solution
• stop when all partial paths have estimates longer
  than solution already found

22
Sample problem city path

• Salesman must get from S to G

4
4
3
5
5
4
3
4
2

• S-A-D-E-F-G costs 35243 17
• S-D-E-F-G costs 4243 13
• S-A-B-E-F-G costs 34543 19

23
Example

• Route planning
• straight line distance on map is always an
  underestimate (or a perfect estimate)

S
D 4 8.9 12.9
A 3 10.4 13.4
A 910.4 19.4
E 66.9 12.9
B 11 6.717.7
F 103 13
24
S
D 4 8.9 12.9
A 3 10.4 13.4
A 910.4 19.4
E 66.9 12.9
B 11 6.717.7
F 103 13
G 13
25
Redundant Partial Paths

• The best way through a particular, intermediate
  place is the best way to it from the starting
  place, followed by the best way from it to the
  goal. There is no neeed to look at any other
  paths to or from the intermediate
  place.
• Dynamic Programming Principle

S
D 4
A 3
B 7
D 8
No reason to expand this
26
A Algorithm

• Combination of methods
• Branch and Bound
• Cost estimate (underestimate)
• Properties
• optimal
• complete
• number of nodes exponential in length of solution
  (space problem, also time)

27
Admissible Heuristic

• heuristic function that never overestimates the
  cost to reach the goal
• result estimated cost of a path at any point
  deeper in the tree never increases - Monotonicity

28
Example 8-puzzle

• Slide tiles horizontally or vertically into space
  until tiles are in desired order
• Typical solution 20 steps
• Branching factor about 3
• nodes (exhaustive) 320 3.5x109
• different states 9! 362,880

29
Possible Heuristics

• h1 - the number of tiles that are in the wrong
  position
• in example, h1 7
• admissible since each out of place tile must be
  moved at least once
• h2 - the sum of the distances (horizontal and
  vertical) of the tiles from their goal positions
• city block distance, Manhattan distance
• admissible since any move can only move one tile
  one step closer to the goal
• in example, h2 232420218

30
Which heuristic is better

• h2
• for every node n in a configuration c, h2(n) gt
  h1(n) therefore h2(c) gt h1(c)
• h2 is said to DOMINATE h1
• A using h2 will expand fewer nodes on average
  than A using h1

31
How to find heuristics?

• Find an actual estimate to a relaxed problem
• If a tile could move anywhere, then h1 would be
  accurate
• If a tile could move into an adjacent square
  whether occupied or not, then h2 would be accurate

32
Memory Bounded Search

• A can still take lots of memory
• Try iterative deepening A (IDA)
• each search is a depth-first search using an
  f-cost limit (rather than a depth limit)
• Properties
• complete, optimal (like A)
• space requirement is proportional to the longest
  path (e.g. bd)
• time depends on heuristic
• if values f can take on is small, then good
• if values f can take on is large, N2 where N is
  the number of iterations
• restrict values of f differences of a fixed-cost
  amount c
• solutions can be worse than optimal by at most c

33
SMA (Simplified Memory-Bounded A)

• IDA only remembers current f-cost and doesnt
  retain any other history so SMA
• utilize whatever memory is available
• avoid repeated states as often as possible
• Properties
• complete if there is enough memory available to
  store the shallowest solution path
• optimal if enough memory available to store
  shallowest optimal solution path
• optimally efficient if enough memory available
  for entire search tree

34
SMA process

• When expanding
• if no memory available drop the least-promising
  node
• remember f-cost in ancestor of node dropped so
  that node isnt regenerated unnecessarily