Transcript The n queens problem

The n queens problem
Many solutions to a classic problem:
On an n x n chess board, place n queens so
no queen threatens another
Minimal example – 4 queens
4 x 4 Board
Queen threatens next
piece in any row, column
or diagonal
Problem: Place 4 queens so no queen is threatened
What kind of problem?
Design state space for search
nodes – how to represent state
edges – what transformations to
neighbouring state
e.g.,
state: positions of four queens
edge: move any queen to new position
Possible state spaces
Every state has four queens:
Neighbour state has
one queen in
different position
formulation
Start state is undecided:
 Random?
 Guess based on knowledge?
Form of graph – all states are potential
solutions
Edges for neighbour relationship are
undirected
formulation
How many states?
 Any queen anywhere:
164 = 65536
 Queens on different squares:
16x15x14x13 = 43680
 Queens in separate columns:
44 = 256
 Queens in separate cols, rows:
4x3x2x1 = 24
(n2)n
n2!/(n2-n)!
nn
n!
1. Complete state formulation
What actions, branching factor
 Move a queen anywhere:
4x16 = 64
 Move queen to open space:
4x12 = 48
 Move queen in column:
4x3 = 12
 Exchange rows of two queens:
4x3/2 = 6
n x n2
n x (n2-n)
n x (n-1)
n(n-1)/2
Fitness function:
no queen threatened
Operationalizing
Heuristics for evaluating states: how
(relatively) bad is the threat situation?
 Number of unthreatened queens
 Total pairwise threats
Depends on the representation;
e.g., need to count column threats?
Finding solution: local search
repeat
create random state
find local optimum based on fitness
function heuristic (e.g.,max number
of unthreatened queens)
until fitness function satisfied
Partial-solution spaces
Every state has 0,1,2,3 or 4 queens
Edges lead to states with one more queen
Partial state formulation
Start state is fixed:
 Empty board
Form of graph – hierarchical, directed,
multi-partite
Actions/changes are directed edges that
add one more queen into destination
state
Partial state formulation
How many states?
 Any queen anywhere:
160+161+162+163+164
= 69905 (c/w 65536)
i=0,n (n2)i
 Any queen on empty square:
47296 (43680)
 Queens in separate columns:
341 (256)
 Queens in separate rows/cols:
40 (24)
i=0,n n2!/(n2-n)!
i=0,n ni
i=1,n n!/(n-i)!
Partial state formulation
How many neighbours: branching factor
 Place a queen anywhere:
4x4 = 16
n2
 Place queen on open space:
12 < b ≤ 16
n(n-1)<b≤n2
 Place queen in column:
4
n
 Place queen in col, row:
0<b≤4
0<b≤n
Partial state formulation
Heuristics for evaluating states: is there
a threat?
e.g.,
Total pairwise threats:
Number of
unthreatened spaces
0
0
1
2
Partial formulation
Search is globally controlled – spanning
tree
- Better for ‘optimizing’ - finding multiple
solutions and choosing best
Example algorithm from Peter Alfeld
<http://www.apl.jhu.edu/~hall/NQueens.html>
Analyze this algorithm
•Complete / incremental?
•How many states?
•Branching factor?
•Heuristic evaluation of fitness?
Example algorithm from Peter Alfeld
<http://www.apl.jhu.edu/~hall/NQueens.html>
•Complete / incremental?
•Incremental
•How many states?
•O(Σi=0,n ni)
•Branching factor?
•n
•Heuristic evaluation?
•No conflict
An O(n) algorithmic solution
ACM SIGART Bulletin, 2(2), page 22-24,
1991
-finds one arrangement of queens
solution by Marty Hall based on this
algorithm:
O(n) algorithm