A new combinatorial Gray code
for balanced combinations
Torsten Mütze
joint work with
Christoph Standke, Veit Wiechert (TU Berlin)
Balanced combinations
• balanced combination := a -element subset of
• Example:
• bitstring representation:
• Dyck path representation:
flaws = 3
Balanced combinations
flaws = 0
Balanced combinations
flaws =
Balanced combinations
total number =
flaws
the k-th Catalan number
The Chung-Feller theorem
:= set of all Dyck paths with 2k steps and e flaws
Theorem [Chung, Feller 49]: For any
we have
.
Various alternative proofs (combinatorial, analytic, …) and
generalizations appeared over the years: [Hodges 55], [Narayana 67],
[Woan 01], [Eu, Fu, Yeh 05], [Chen 08], [Ma, Yeh 09], [Ruckavicka 11], …
Proving Chung-Feller
Establish a bijection
[Hodges 55], [Chen 08]
(#flaws increases by 1)
Note: and
may
differ in many positions
Is there a simpler proof?
Our results
Theorem: There is a bijection
such that and
differ in only two positions.
Our results
Theorem: There is a bijection
such that and
differ in only two positions.
flaws
Our results
‚simple and explicit‘
Theorem: There is a bijection
such that and
differ in only two positions.
‚simple‘
Theorem: There is an algorithm which for given
computes each Dyck path in
in time
.
Applications
• Hamilton cycles in certain vertex-transitive graphs:
the odd graph
and the middle levels graph
• More generally: For which values of do they have a -factor?
• Only one general positive answer known:
‚middle levels conjecture‘ [M. 14]
• This work: two more positive answers
Theorem: The odd graph has a
-factor.
Theorem: The middle levels graph has a
-factor.
# cycles =
(Catalan number)
Combinatorial Gray codes
• Goal: Generate all objects in a combinatorial class
(permutations, subsets, strings, trees, Dyck paths, etc.)
such that consecutive objects differ only ‚a little bit‘
• Ideally: Generate each new object in time
• Fundamental task in combinatorial algorithms
[Nijenhuis, Wilf 75] [Knuth TAOCP Vol. 4, 11]
Combinatorial Gray codes
• Examples:
• Generate all permutations of
such that
consecutive permutations differ only in one transposition
[Johnson 63], [Trotter 62], [Sedgewick 77]
• Generate all
subsets of
such that
consecutive sets differ in adding/removing one element
[Gray 53], [Tootill 56], [Bitner, Ehrlich, Reingold 76], [Wagner, West 91],
[Savage, Winkler 95], [Bhat, Savage 96]
• Generate all
many -element subsets of
s.t. consecutive sets differ in exchanging one element
[Tang, Liu 73], [Bitner, Ehrlich, Reingold 76], [Eades, Hickey, Read 84],
[Eades, McKay 84], [Ruskey 88], [Chase 89], [Jenkyns, McCarthy 95],
[Ruskey, Williams 09]
Combinatorial Gray codes
• Examples:
Special case
: balanced combinations
{1,
4,5 }
(1,0,0,1,1,0)
• Constant-time
Generate all generation
many -element
[Ehrlich 73],
subsets
[Bitner,ofEhrlich, Reingold 76]
1,0,0,0,0,0
s.t. consecutive
sets
differ
in exchanging
one
element
• Only
swaps of the
form
[Eades,
McKay
84]
0,0,0,0,0,1
1,0,0 1,0
• Only distance-≤2 swaps [Chase 89], [Jenkyns, McCarthy 95]
0,0,1 0,1
• Only distance-1 swaps [Eades, Hickey, Read 84], [Ruskey 88]
(possible iff even and odd, or
)
Combinatorial Gray codes
• Examples:
• Only distance-1 swaps [Eades, Hickey, Read 84], [Ruskey 88]
(possible iff even and odd, or
)
1,0
0,1
Combinatorial Gray codes
• Examples:
• Generate only
i
((()))
(=parenthesis expressions, rooted trees,
triangulations, binary trees…
see [Stanley 15])
iii
ii
(()())
(())()
v
iv
()()()
()(())
• Constant-time generation [Ruskey, Proskurowski 90], [Walsh 98]
• Only distance-1 swaps [Bultena, Ruskey 98]
(possible iff even or less than 5)
Combinatorial Gray codes
i
ii
iii
iv
v
Combinatorial Gray codes
• Our minimum-change bijection f is ‚orthogonal‘ to
Gray codes from before (01-20 and i-v)
i
ii
iii
iv
v
Definition of
• Count downsteps starting at
→
• Flip
-th downstep touching the line
• Flip first upstep to the left of d touching the line
2
1
4
3
5
u
d
Definition of
• Count downsteps starting at
→
• Flip
-th downstep touching the line
• Flip first upstep to the left of d touching the line
u
d
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u
u
d
d
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u
d
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u u dd
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u
d
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u d
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u
d
Definition of
• Count downsteps starting at
→
• Flip
-th downstep touching the line
• Flip first upstep to the left of d touching the line
Definition of
• Count downsteps starting at
→
• Flip
-th downstep touching the line
• Flip first upstep to the right of d touching the line
Definition of
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
Definition of
Repeat:
• Flip first downstep to the left of u touching the line
• Flip first upstep to the right of d touching the line
Efficient computation
Repeat:
• Flip first downstep to the right of u touching the line
• Flip first upstep to the left of d touching the line
u
d
Idea: bidirectional pointers
below hills and above valleys
 Fast navigation along Dyck path to reposition u and d
 Compute each application of in time
Thank you!