1. A backtrack data structure and algorithm
CS420: Dancing Links
A backtrack data structure and algorithm
by Donald Knuth
(wim bohm cs.colostate.edu)
Except as otherwise noted, the content of this presentation is licensed under the Creative Commons Attribution 2.5 license.

2. Dancing Links Doubly linked list nodes have referencesx
Doubly linked list
nodes have references
L to left node L and
R to right node

3. Dancing Links Doubly linked list Remove x R[L[x]]=R[x] L[R[x]]=L[x]
notice that x still, via its L and R,
points at left and right nodes,
and thus we can easily ... x

4. Dancing Links Doubly linked list Remove x R[L[x]]=R[x] L[R[x]]=L[x]
Put x back
R[L[x]]=x
L[R[x]]=x x x

5. Using Dancing Links O(1) put x back operation
Works in following BackTrack scenario
. we have created a “space to be searched” as a global
doubly linked data structure
. we search this space by DFS, selecting: taking out
certain options and putting them back in reverse order
What does not work
. adding completely new options
. putting options back in other than reverse order

6. Why reverse order? x y x y
remove x

7. Why reverse order? x y x y
remove x x y
remove y

8. What happens if x is put back first?y x y
remove x x y
remove y

9. What happens if x is put back first?y
remove y x y
put back x
x points at y
and y at x,
but y is not in the chain!!

10. Exact sub-set / cover problem
Given a matrix of 0-s and 1-s, find a subset of rows with exactly one 1 in each column of A
Backtrack approach
Pick a column c, pick a row r with 1 in c, remove columns c and j with 1 in r and rows with a 1 coinciding with 1-s in r
a b c d e f g reducing the problem
eg c in row 1
selects subset {c,e,f}
so remove columns c,e,f
and row 1 and row 3
because it overlaps with row 1

11. Exact sub-set / cover problem
Backtrack approach
Pick column c, row 1 subset {c,e,f}
which rows and cols disappear?
a b c d e f g

12. Exact sub-set / cover problem
Backtrack approach
Pick column c, row 1 {c,e,f} rows 1,3 and cols c,e,f disappear
Pick column a, row 2 {a,d,g} which rows, cols disappear?
a b d g

13. Exact sub-set / cover problem
Backtrack approach
Pick column c, row 1 {c,e,f}
Pick column a, row 2 {a,d,g} all rows/cols disappear
what is the problem?

14. Exact sub-set / cover problem
Backtrack approach
Pick column c, row 1 {c,e,f}
Pick column a, row 2 {a,d,g} all rows/cols disappear
b is not covered, so backtrack

15. Exact sub-set / cover problem
Backtrack
Pick column c, row 1 {c,e,f} rows 1,3 and cols c,e,f disappear
Pick column a, row 4 {a,d} which rows, cols disappear now?
a b d g

16. Exact sub-set / cover problem
Backtrack
Pick column c, row 1 {c,e,f} rows 1,3 and cols c,e,f disappear
Pick column a, row 4 {a,d} rows 2,4 cols a,d disappear
b g

17. Exact sub-set / cover problem
Backtrack
Pick column c, row 1 {c,e,f} rows 1,3 and cols c,e,f disappear
Pick column a, row 4 {a,d} rows 2,4 cols a,d
Pick column b, row 5 {b,g} row 5 disappears
Exact cover accomplished!
matrix is empty

18. Exact sub-set / cover problem
Subsets / rows 1,4,5 provide exact cover! a b c d e f g

19. Exact set cover The columns: elements of a universe
The rows: subsets of elements in the universe
Find a set of subsets that has each element exactly once
Union of the set is the Universe
Intersection of any two subsets is empty
Finding an exact cover is “tough”
NP-Complete e.g. when each subset has three elements
Great candidate for backtrack search
Representation: row column doubly linked sparse matrix
only containing 1-s
Use dancing links to remove and put back candidates

20. Pentominoes
A pentomino is a size 5 n-omino composed of n congruent squares connected orthogonally by side (not point wise)
There are 12 different pentominoes when rotation, and mirroring are allowed
There are 18 pentominoes when only rotation is allowed

21. The 12 pentominoes
F I L N P T
U V W X Y Z

22. Early implementation
Dana Scott wrote a backtrack program in 1958 for the Maniac (4000 instructions / sec) tiling a chessboard with a 2x2 hole in the middle with the 12 pentominoes, using each pentomino exactly once
The code ran in ~3.5 hours: (50 million instructions)

23. One of the 65 solutions

24. Board shapes Tiling different board shapes
Chessboard with 2x2 hole in middle
or with 4 holes in arbitrary places
Rectangles
6 x 10
5 x 12
4 x 15
3 x 20
3D boxes

25. Pentomino problems are exact cover problems
Matrix with 72 columns
12 pentominoes and 60 positions in the board’s grid
Each row has 6 1-s
1 for the pentomino, 5 for its positions
Each row is a description of a pentomino in a certain position
There are 1568 such rows

26. Translating puzzle to set cover
Pentominoes matrix: 72 columns 1568 rows
Let’s do a simpler puzzle
4 triominoes
I: L: ┐: :
No rotation or flip
Rectangle 3 x 4
# possible solutions?

27. Translating puzzle to set cover
Pentominoes matrix: 72 columns 1568 rows
Let’s do a simpler puzzle
4 triominoes
I: L: ┐: :
No rotation or flip
Rectangle 3 x 4
Possible solution:

28. Setcover matrix for simple puzzle
16 columns:
4 columns for the pieces
12 columns for the positions
Rows:
4 I placements
I L ┐ - (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2) (3,3) (3,4)

29. Setcover matrix cont’ Rows 6 L placements
I L ┐ - (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2) (3,3) (3,4)

30. Setcover matrix cont’
6 ┐placements
6 – placements

31. Backtrack algorithm X sketch
// A is the exact cover matrix if (empty(A)) return solution else { choose a column c choose a row r with Ar,c = 1 include r in solution for each column j with Ar,j = 1 { delete column j from A for each row i with Ai,j = 1 delete row i from A } repeat recursively on reduced A

32. algorithm X The subsets form a search tree Root: original problem
Level k node: k subsets have been chosen and all columns (elements) in the subsets and all overlapping subsets (rows) have been removed
Any systematic rule for choosing columns will find all solutions

33. Choosing c One way Better ways
Pick pentominoes in alphabetical order
F I L N P T U V W X Y Z
Not good: F has 192 possible places, I then has ~32, very big search space ~2x212
Better ways
Choose lexicographically first uncovered position, starting with (1,1), search space ~107
Scott realized that X has essentially three possible places centered at (2,3), (2,4) and (3,3). The rest leads to symmetrical solutions. Search space ~350,000
Knuth has a more general solution: pick column with minimal number of 1-s

34. Let’s dance Represent each 1 in A by a node with 5 links
L(x), R(x) (left right)
U(x), D(x) (up down)
C(x) column header
Each row is a doubly (L,R) linked circular list
Each column is a doubly linked (U,D) circular list
Each column has a column header node c with additional name N(c) and size S(c).
The column headers form a circular row with a header h
Each node points at its column header with link C

35. Our example h
a 2
b 2
c 2
d 2
e 1
f 2
g 2
a b c d e f g

36. Search algorithm DLX
search(int k){ // search is called with k=0 from outside if(R[h]=h) print O else { // O is the set of currently picked rows choose column c cover column c // pick element c for each row r = D[c], D[D[c]]… while r!=c { O[k]=r // pick a row with element c for each j = R[r], R[R[r]]… while j!=r cover column C[j] // all elements in the row are now covered search(k+1) for each j = L[r], L[L[r]]… while j!=r uncover column C[j] // uncover in reverse order } uncover column c and return; } }

37. cover column c
// remove c from headers L[R[c]]= L[c]; R[L[c]]=R[c] // remove all rows in c’s column for each row r = D[c], D[D[c]]… while r!=c for each j = R[r], R[R[r]]… while j!=r { U[D[j]]=U[j]; D[U[j]]=D[j]; S[C[j]]--; }

38. uncover c
// put rows back IN REVERSE ORDER // last out first back in for each row r = U[c], U[U[c]]… while r!=c for each j = L[r], L[L[r]]… while j!=r { S[C[i]]++; U[D[j]]=j; D[U[j]]=j; } // put header back L[R[c]]=c; R[L[c]]=c;

39. cover a: remove header h a 2 b 2 c 2 d 2 e 1 f 2 g 2 a b c d e f g

40. cover a: remove row 2 h a 2 b 2 c 2 d 1 e 1 f 2 g 1 b c d e f g

41. cover a; remove row 2 and 4 h a 2 b 2 c 2 d 0 e 1 f 2 g 1 b c d e f g
Notice the asymmetry in D2
(element D in row 2) versus D4
(element D in row 4).
Putting back D2 first would
change D4’s Links, but D4 is
not in the set!
This is why we need to
uncover in exactly the
reverse order, so we know
that the element that is put
back refers to elements in the
set h
a 2
b 2
c 2
d 0
e 1
f 2
g 1

42. cover a; last step: remove row 5
b c d e f g

43. Picking subset 2 leads to failure
a b c d e f g
b c e f
nothing left for e
select a, row 2
cover a,d,g
remove rows 2,4,5
select b, row 3
cover b,c,f
remove rows 1,3

44. Picking subset 2 leads to failure
a b c d e f g
b c e f
select a, row 2
cover a,d,g
select c, row 1
remoce rows 1,3
now what?

45. Picking subset 2 leads to failure
a b c d e f g
b c e f
nothing left for b
select a, row 2
cover a,d,g
select c, row 1
cover c,e,f

46. Picking subset 4 succeeds
c e f
a b c d e f g
b c e f g
select c
pick subset 1
empties A
select a
pick subset 4
cover a,d
remove rows 2,4
select b
picking subset 3
remove rows 1,3,5 fails again
pick subset 5
cover b,g
remove rows 3,5
Exact cover: subsets 4 {a,d}, 5 {b,g} and 1 {c,e,f}