1. 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

2. Minimal example – 4 queens
Queen threatens next piece in any row, column or diagonal
4 x 4 Board
Problem: Place 4 queens so no queen is threatened

3. 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

4. Possible state spaces Every state has four queens: Neighbour state has
one queen in
different position

5. formulation Start state is undecided:
Random?
Guess based on knowledge?
Form of graph – all states are potential solutions
Edges for neighbour relationship are undirected

6. formulation How many states? Any queen anywhere: 164 = 65536 (n2)n
Queens on different squares: 16x15x14x13 = n2!/(n2-n)!
Queens in separate columns: = nn
Queens in separate cols, rows: 4x3x2x1 = n!

7. 1. Complete state formulation
What actions, branching factor
Move a queen anywhere: 4x16 = n x n2
Move queen to open space: 4x12 = n x (n2-n)
Move queen in column: x3 = n x (n-1)
Exchange rows of two queens: 4x3/2 = n(n-1)/2

8. 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?

9. 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

10. Partial-solution spaces
Every state has 0,1,2,3 or 4 queens
Edges lead to states with one more queen

11. 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

12. Partial state formulation
How many states?
Any queen anywhere: i=0,n (n2)i
= (c/w 65536)
Any queen on empty square:
47296 (43680) i=0,n n2!/(n2-n)!
Queens in separate columns:
341 (256) i=0,n ni
Queens in separate rows/cols:
40 (24) i=1,n n!/(n-i)!

13. Partial state formulation
How many neighbours: branching factor
Place a queen anywhere: 4x4 = n2
Place queen on open space: < b ≤ 16 n(n-1)<b≤n2
Place queen in column: n
Place queen in col, row:
0 < b ≤ < b ≤ n

14. Partial state formulation
Heuristics for evaluating states: is there a threat?
e.g.,
Total pairwise threats:
Number of
unthreatened spaces

15. Partial formulation Search is globally controlled – spanning tree
- Better for ‘optimizing’ - finding multiple solutions and choosing best

16. Example algorithm from Peter Alfeld <http://www. apl. jhu
Analyze this algorithm
Complete / incremental?
How many states?
Branching factor?
Heuristic evaluation of fitness?

17. Example algorithm from Peter Alfeld <http://www. apl. jhu
Complete / incremental?
How many states?
Branching factor?
Heuristic evaluation?
Incremental
O(Σi=0,n ni) n
No conflict

18. 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