Backtracking

1
Backtracking
Eight queens problem 1 try all possible C64 8
4,426,165,368 2 never put more than one queen on
a given row, vector representation each row
specify which column (3,1,6,2,8,6,4,7)
2
1 2 3 4 5 6 7 8
X
X
X
X
X
X
X
X
(3,1,6,2,8,6,4,7)
3
Queen1 for i1 1 to 8 do for i2 1 to 8 do
. . . . for i3 1 to 8 do
sol i1, i2, . . . i8 if
solution ( sol ) then write sol stop write
there is no solution
Num. Of positions 8 8 16,777,216
(first soln after 1,299,852 )
4
3 Never put queen on the same row (different
numbers on soln vector)
Queen2 sol initial-permutation while sol !
final-permutation and not solution(sol) do
sol next-permutation if solution(sol) then
write sol else write
there is no solution
5
Permutation T1 . . n is a global array
initialize to 1,2,. . n initial call perm(1)
Perm(i) if i n then use T else for
j i to n do exchange Ti and Tj
perm(i1)
exchange
Ti and Tj
Number of positions 8! 40,320 (first soln
after 2830)
6
8-queen as tree search a vector V1. .k of
integers between 1 and 8 is k-promising, if none
of the k queens threatens any of the others. A
vector V is k-promising if, for every pair of
integers i and j between 1 and k with i ! j, we
have Vi - Vj is-not-in i-j, 0, j-i.
Solutions to the 8-queen correspond to vectors
that are 8-promising.
7

• Let N be the set of k-promising vectors, k 0 ..
  8. Let G (N,A) be the directed graph such
  that (U,V) is-in A iff there exists an integer k,
  k0..8 , such that
• U is k-promising
• V is (k1)-promising, and
• Ui Vi for every i in 1..k

k 0
. . .
Number of node lt 8! (node 2057, first soln after
114 )
k 8
8
General Template for backtracking
Backtrack ( v1..k ) // v is k-promising
vector if solution ( v ) then write v else for
each (k1)-promising vector w such
that w1..k v1..k do
backtrack( w1.. k1 )