How to count leaves in the trees from Tetris?
Transcript of How to count leaves in the trees from Tetris?
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How to count leaves in the trees from Tetris?
Elise VandommePostdoc at the LaCIM (UQAM)
Colloque panquebecois des etudiants de l’ISMTrois-Rivieres – May 2017
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Trees in Tetris
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Trees in Tetris
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Trees in Tetris
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Polyomino
A polyomino is a connected union of unit squares, called cells.
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Solomon W. Golomb (1932 - 2016)
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?Unknown for n ≥ 57
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?Unknown for n ≥ 57
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?Unknown for n ≥ 57
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?Unknown for n ≥ 57
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?
Unknown for n ≥ 57
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Combinatorial Problems
Monomino:
Domino:
Tromino:
Tetromino:
How many polyominoes with n cells?Unknown for n ≥ 57
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Combinatorial Problems
Can we pave the plane with a finite set of polyominoes?
Examples:
Polyominoes:
Yes, we can!
Polyominoes:
No, we can’t!
In general, the problem is undecidable.
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Combinatorial Problems
Can we pave the plane with a finite set of polyominoes?
Examples:
Polyominoes:
Yes, we can!
Polyominoes:
No, we can’t!
In general, the problem is undecidable.
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Combinatorial Problems
Can we pave the plane with a finite set of polyominoes?
Examples:
Polyominoes:
Yes, we can!
Polyominoes:
No, we can’t!
In general, the problem is undecidable.
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Combinatorial Problems
Can we pave the plane with a finite set of polyominoes?
Examples:
Polyominoes:
Yes, we can!
Polyominoes:
No, we can’t!
In general, the problem is undecidable.
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Combinatorial Problems
Can we pave the plane with a finite set of polyominoes?
Examples:
Polyominoes:
Yes, we can!
Polyominoes:
No, we can’t!
In general, the problem is undecidable.
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: 45× 36 rectangle
Polyominoes: all pentominoes
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?
http://www.knowltonmosaics.com
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: a checkerboard
without two opposite corners
Polyominoes: a domino
No, we can’t!
In general, the problem is NP-complete.
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: a checkerboard
without two opposite corners
Polyominoes: a domino
Yes, we can!
No, we can’t!
In general, the problem is NP-complete.
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: a checkerboard without two opposite corners
Polyominoes: a domino
No, we can’t!
In general, the problem is NP-complete.
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: a checkerboard without two opposite corners
Polyominoes: a domino
No, we can’t!
In general, the problem is NP-complete.
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Combinatorial Problems
Can we pave a finite area with a finite set of polyominoes?Example:
Area: a checkerboard without two opposite corners
Polyominoes: a domino
No, we can’t!
In general, the problem is NP-complete.
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Behind each polyomino hides a graph
Vertices are the cells.
Two vertices are adjacent if their corresponding cells have acommon side.
Such graph can contain cycles.
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Behind each polyomino hides a graph
Vertices are the cells.
Two vertices are adjacent if their corresponding cells have acommon side.
Such graph can contain cycles.
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Behind each polyomino hides a graph
Vertices are the cells.
Two vertices are adjacent if their corresponding cells have acommon side.
Such graph can contain cycles.
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Behind each polyomino hides a graph
Vertices are the cells.
Two vertices are adjacent if their corresponding cells have acommon side.
Such graph can contain cycles.
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Tree-like Polyominoes
If there are no cycles, we call it a tree-like polyomino.
This tree-like polyomino has 9 cells and 4 leaves.
For a given n, what is the maximal number of leaves realizedby a tree-like polyomino with n cells?
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Tree-like Polyominoes
If there are no cycles, we call it a tree-like polyomino.
This tree-like polyomino has 9 cells and 4 leaves.
For a given n, what is the maximal number of leaves realizedby a tree-like polyomino with n cells?
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Tree-like Polyominoes
If there are no cycles, we call it a tree-like polyomino.
This tree-like polyomino has 9 cells and 4 leaves.
For a given n, what is the maximal number of leaves realizedby a tree-like polyomino with n cells?
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Maximal number of leaves?
For n ≥ 1, the maximal number of leaves realized by a tree-like polyomino with n cells is given by
`(n) =
0 if n = 1
2 if n = 2
n − 1 if n = 3, 4, 5
`(n − 4) + 2 if n ≥ 6
Theorem (Blondin Masse, de Carufel, Goupil, Samson 2017)
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Trees in Minecraft
tree-like
polyominoes ⇒
tree-like
polycubes
The maximal number of leaves realized by a tree-like polycubesatisfies a linear recurrence.
Theorem (Blondin Masse, de Carufel, Goupil, Samson 2017)
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Trees in Minecraft
tree-like polyominoes ⇒ tree-like polycubes
The maximal number of leaves realized by a tree-like polycubesatisfies a linear recurrence.
Theorem (Blondin Masse, de Carufel, Goupil, Samson 2017)
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Trees in Minecraft
tree-like polyominoes ⇒ tree-like polycubes
The maximal number of leaves realized by a tree-like polycubesatisfies a linear recurrence.
Theorem (Blondin Masse, de Carufel, Goupil, Samson 2017)
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Tree-like polyominoes are induced subtrees of Z2.
Tree-like polycubes are induced subtrees of Z3.
Can we study this problem in other graphs?
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Tree-like polyominoes are induced subtrees of Z2.
Tree-like polycubes are induced subtrees of Z3.
Can we study this problem in other graphs?
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Tree-like polyominoes are induced subtrees of Z2.
Tree-like polycubes are induced subtrees of Z3.
Can we study this problem in other graphs?
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2
2 3 4 4 5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2
3 4 4 5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3
4 4 5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3 4
4 5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3 4 4
5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3 4 4
5 −∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3 4 4 5
−∞
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Definition
For a graph G = (V ,E ) and n ≥ 2
Tn = set of induced subtrees with n vertices
LG (n) = max{# leaves in T |T ∈ Tn}Leafed sequence of G : LG (n)n∈{2,...,|V |}
8 7
54
1 2 3
6
n 2 3 4 5 6 7 8
LG (n) 2 2 3 4 4 5 −∞
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Particular cases
K6 LKp(n) =
{2 if n = 2 et p ≥ 2
−∞ if 3 ≤ n ≤ p
C6 LCp(n) =
{2 if 2 ≤ n < p
−∞ if n = p
R6 LRp(n) =
2 if 2 = n ou bp2c+ 1 < n < p
n − 1 if 3 ≤ n ≤ bp2c+ 1
−∞ if p ≤ n ≤ p + 1
K3,2 LKp,q(n) =
2 if n = 2
n − 1 if 3 ≤ n ≤ max(p, q) + 1
−∞ if max(p, q) + 1 < n ≤ p + q
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Particular cases
K6 LKp(n) =
{2 if n = 2 et p ≥ 2
−∞ if 3 ≤ n ≤ p
C6 LCp(n) =
{2 if 2 ≤ n < p
−∞ if n = p
R6 LRp(n) =
2 if 2 = n ou bp2c+ 1 < n < p
n − 1 if 3 ≤ n ≤ bp2c+ 1
−∞ if p ≤ n ≤ p + 1
K3,2 LKp,q(n) =
2 if n = 2
n − 1 if 3 ≤ n ≤ max(p, q) + 1
−∞ if max(p, q) + 1 < n ≤ p + q
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Particular cases
K6 LKp(n) =
{2 if n = 2 et p ≥ 2
−∞ if 3 ≤ n ≤ p
C6 LCp(n) =
{2 if 2 ≤ n < p
−∞ if n = p
R6 LRp(n) =
2 if 2 = n ou bp2c+ 1 < n < p
n − 1 if 3 ≤ n ≤ bp2c+ 1
−∞ if p ≤ n ≤ p + 1
K3,2 LKp,q(n) =
2 if n = 2
n − 1 if 3 ≤ n ≤ max(p, q) + 1
−∞ if max(p, q) + 1 < n ≤ p + q
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Particular cases
K6 LKp(n) =
{2 if n = 2 et p ≥ 2
−∞ if 3 ≤ n ≤ p
C6 LCp(n) =
{2 if 2 ≤ n < p
−∞ if n = p
R6 LRp(n) =
2 if 2 = n ou bp2c+ 1 < n < p
n − 1 if 3 ≤ n ≤ bp2c+ 1
−∞ if p ≤ n ≤ p + 1
K3,2 LKp,q(n) =
2 if n = 2
n − 1 if 3 ≤ n ≤ max(p, q) + 1
−∞ if max(p, q) + 1 < n ≤ p + q
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Particular cases: hypercubes
Only partial results:
LH3(n) =
2 if 2 ≤ n ≤ 3
3 if n = 4
−∞ if 5 ≤ n ≤ 8
LH4(n) =
2 if n ∈ {2, 3}3 if n ∈ {4, 6, 8}4 if n ∈ {5, 7, 9}−∞ if 10 ≤ n ≤ 16
Observations:
The leafed sequence can increase by at most 1 each step.The leafed sequence is not always non-decreasing.
The leafed sequence LG (n)n∈{2,...,|V |} is non-decreasingiff G is a tree.
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Particular cases: hypercubes
Only partial results:
LH3(n) =
2 if 2 ≤ n ≤ 3
3 if n = 4
−∞ if 5 ≤ n ≤ 8
LH4(n) =
2 if n ∈ {2, 3}3 if n ∈ {4, 6, 8}4 if n ∈ {5, 7, 9}−∞ if 10 ≤ n ≤ 16
Observations:
The leafed sequence can increase by at most 1 each step.
The leafed sequence is not always non-decreasing.
The leafed sequence LG (n)n∈{2,...,|V |} is non-decreasingiff G is a tree.
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Complexity
Instance: a graph G and two integers n, k ≥ 1
Question: Is there an induced subtree of G with nvertices and k leaves?
Problem LIS
The LIS problem is NP-complete.
Theorem (Blondin Masse, de Carufel, Goupil, V. 2017)
Reduction to the problem:
Instance: a graph G and an integer n ≥ 1
Question: Is there an induced subtree of G with more than nvertices?
which is NP-complet. [Erdos, Saks, Sos 1986]
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Complexity
Instance: a graph G and two integers n, k ≥ 1
Question: Is there an induced subtree of G with nvertices and k leaves?
Problem LIS
The LIS problem is NP-complete.
Theorem (Blondin Masse, de Carufel, Goupil, V. 2017)
Reduction to the problem:
Instance: a graph G and an integer n ≥ 1
Question: Is there an induced subtree of G with more than nvertices?
which is NP-complet. [Erdos, Saks, Sos 1986]
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Complexity
Instance: a graph G and two integers n, k ≥ 1
Question: Is there an induced subtree of G with nvertices and k leaves?
Problem LIS
The LIS problem is NP-complete.
Theorem (Blondin Masse, de Carufel, Goupil, V. 2017)
Reduction to the problem:
Instance: a graph G and an integer n ≥ 1
Question: Is there an induced subtree of G with more than nvertices?
which is NP-complet. [Erdos, Saks, Sos 1986]
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If G has an induced subtree with > n vertices
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If G has an induced subtree with > n vertices
⇒ ∃T induced subtree with n + 1 vertices {v1, . . . , vn+1}
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If G has an induced subtree with > n vertices
⇒ ∃T induced subtree with n + 1 vertices {v1, . . . , vn+1}⇒ H has a subtree induced by {v1, . . . , vn+1} ∪ {v ′1, . . . , v ′n+1}
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If G has an induced subtree with > n vertices
⇒ ∃T induced subtree with n + 1 vertices {v1, . . . , vn+1}⇒ H has a subtree induced by {v1, . . . , vn+1} ∪ {v ′1, . . . , v ′n+1}
with 2(n + 1) vertices and n + 1 leaves
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If H has an induced subtree T with 2(n + 1) vertices andn + 1 leaves
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If H has an induced subtree T with 2(n + 1) vertices andn + 1 leaves
⇒ T has ≥ n + 1 vertices in V
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LIS is NP-complete
Polynomial transformation f : (G , n) 7→ (H, 2(n + 1), n + 1)
G H
a b
cd
e
f
a b
cd
e
a′ b′
c ′d ′
e ′
If H has an induced subtree T with 2(n + 1) vertices andn + 1 leaves
⇒ T has ≥ n + 1 vertices in V
⇒ G has an induced subtree with > n vertices
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What about trees?
For a tree T with m vertices, the leafed sequence LT (n) iscomputed in polynomial time and space.
Theorem (Blondin Masse, de Carufel, Goupil, V. 2017)
Algorithm based on the dynamic programming paradigm
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Idea of the algorithm
u v
LTu
=?
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Idea of the algorithm
u v
LTu
=?
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Idea of the algorithm
u v
LTu
=?
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Idea of the algorithm
u v
LTu
=?
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Idea of the algorithm
u v
LTu
=?
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Idea of the algorithm
For a directed tree Tu rooted in u
f (Tu) = #{x ∈ Tu | deg+(x) = 0}LTu
(n) = max{f (T ′u) : T ′u ⊆ Tu, |T ′u| = n}
Generalization to a directed forest F with k connectedcomponents Fi :
LF
(n) = max
{k∑
i=1
LFi
(λ(i))
∣∣∣∣∣ λ ∈ C (n, k)
}
If F is the forest of the subtrees rooted in the children of u,
LTu
(n) =
{n if n = 0, 1
LF
(n − 1) otherwise
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Idea of the algorithm
u v
01 01 01
01
01 01
0112
011230112334556
LTu
=?
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Idea of the algorithm
u v
01 01 01
01
01 01
0112
011230112334556L
Tu=?
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Idea of the algorithm
u v
01 01 01
01
01 01
0112
011230112334556L
Tu=?
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Idea of the algorithm
u v
01 01 01
01
01 01
0112
01123
0112334556LTu
=?
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Idea of the algorithm
u v
01 01 01
01
01 01
0112
011230112334556
LTu
=?
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Idea of the algorithm
Assuming LFi
are known,
Naive computation of LF
via
LF
(n) = max
{k∑
i=1
LFi
(λ(i))
∣∣∣∣∣ λ ∈ C (n, k)
}
is not polynomial
Computation of LF
via
LF
(n) = max{LF−F1
(i) + LF1
(n − i) | 0 ≤ i ≤ n}
in time Θ(k|F |2)
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Idea of the algorithm
Assuming LFi
are known,
Naive computation of LF
via
LF
(n) = max
{k∑
i=1
LFi
(λ(i))
∣∣∣∣∣ λ ∈ C (n, k)
}
is not polynomial
Computation of LF
via
LF
(n) = max{LF−F1
(i) + LF1
(n − i) | 0 ≤ i ≤ n}
in time Θ(k|F |2)
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Idea of the algorithm
01123
0123012
u v
01 01 01
01
01 01
0112
011230112334556
01
0112
01
01011223
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Idea of the algorithm
01123
0123
012
u v
01 01 01
01
01 01
0112
011230112334556
01
0112
01
01011223
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Idea of the algorithm
01123
0123012
u v
01 01 01
01
01 01
0112
011230112334556
01
0112
01
01011223
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Idea of the algorithm
01123
0123012
u v
01 01 01
01
01 01
0112
011230112334556
01
0112
01
01011223
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Idea of the algorithm
01123
0123012
u v
01 01 01
01
01 01
0112
011230112334556
01
0112
01
01011223
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Idea of the algorithm
Maximal number of leaves in the subtrees with n vertices,containing the edge {u, v} :
L{u,v}(n) = max{LTu
(i) + LTv
(n − i)∣∣∣ 1 ≤ i ≤ n − 1
}
u v
0112334556
011223
L{u,v} = 012234456677889
LT (n) = max{L{u,v}(n)
∣∣ {u, v} ∈ E}
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Idea of the algorithm
Maximal number of leaves in the subtrees with n vertices,containing the edge {u, v} :
L{u,v}(n) = max{LTu
(i) + LTv
(n − i)∣∣∣ 1 ≤ i ≤ n − 1
}
u v
0112334556
011223
L{u,v} = 012234456677889
LT (n) = max{L{u,v}(n)
∣∣ {u, v} ∈ E}
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Could we improve the time and space complexity?
We can not hope to obtain a procedure computing LT (n)which deletes leaves successively.
Counter-example :
LT (7) = 5 et LT (9) = 6
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From graph theory to combinatorics on words
Each tree has a non-decreasing leaf sequence L = (Li )i≥2.
For any graph, Li+1 − Li ≤ 1.
The difference word ∆L := (Li+1 − Li )i≥3 is a binary word.
Leafed sequence: L = 223445567789
Difference word: ∆L =
0
1101011011
The language of a class of graphs is the set of all their differencewords.
Language of Trees =???
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From graph theory to combinatorics on words
Each tree has a non-decreasing leaf sequence L = (Li )i≥2.
For any graph, Li+1 − Li ≤ 1.
The difference word ∆L := (Li+1 − Li )i≥3 is a binary word.
Leafed sequence: L = 223445567789
Difference word: ∆L = 01101011011
The language of a class of graphs is the set of all their differencewords.
Language of Trees =???
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From graph theory to combinatorics on words
Each tree has a non-decreasing leaf sequence L = (Li )i≥2.
For any graph, Li+1 − Li ≤ 1.
The difference word ∆L := (Li+1 − Li )i≥3 is a binary word.
Leafed sequence: L = 223445567789
Difference word: ∆L =
0
1101011011
The language of a class of graphs is the set of all their differencewords.
Language of Trees =???
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From graph theory to combinatorics on words
Each tree has a non-decreasing leaf sequence L = (Li )i≥2.
For any graph, Li+1 − Li ≤ 1.
The difference word ∆L := (Li+1 − Li )i≥3 is a binary word.
Leafed sequence: L = 223445567789
Difference word: ∆L =
0
1101011011
The language of a class of graphs is the set of all their differencewords.
Language of Trees =???
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From graph theory to combinatorics on words
Each tree has a non-decreasing leaf sequence L = (Li )i≥2.
For any graph, Li+1 − Li ≤ 1.
The difference word ∆L := (Li+1 − Li )i≥3 is a binary word.
Leafed sequence: L = 223445567789
Difference word: ∆L =
0
1101011011
The language of a class of graphs is the set of all their differencewords.
Language of Trees =???
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS
It is A194850!
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Caterpillars
L = 223455667778 and ∆L = 1110101001
What is the language of caterpillars ?
Language of caterpillars ( Language of trees
Enumerate caterpillars
Get their associated difference words
ε, 0, 1, 00, 10, 11, 0000, 1000, 1001, 1010, 1100, 1101, . . .
Count them by length
n 0 1 2 3 4 5 6 7 8 9
1 2 3 5 8 14 23 41 70 125
Check OEIS It is A194850!29/31
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Prefix normal word
A prefix normal word w1 · · ·wn is a binary word such that∀` ≤ n, i ≤ n − `,
|w1 · · ·w`|1 ≥ |wi · · ·wi+`|1.
L = 223455667778 and ∆L = 1110101001
Conjecture:
Language of Caterpillars = {prefix normal words}
Language of Caterpillars ⊇ {prefix normal words}
Proposition (Blondin Masse, de Carufel, Goupil, Lapointe, V.)
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Prefix normal word
A prefix normal word w1 · · ·wn is a binary word such that∀` ≤ n, i ≤ n − `,
|w1 · · ·w`|1 ≥ |wi · · ·wi+`|1.
L = 223455667778 and ∆L = 1110101001
Conjecture:
Language of Caterpillars = {prefix normal words}
Language of Caterpillars ⊇ {prefix normal words}
Proposition (Blondin Masse, de Carufel, Goupil, Lapointe, V.)
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Prefix normal word
A prefix normal word w1 · · ·wn is a binary word such that∀` ≤ n, i ≤ n − `,
|w1 · · ·w`|1 ≥ |wi · · ·wi+`|1.
L = 223455667778 and ∆L = 1110101001
Conjecture:
Language of Caterpillars = {prefix normal words}
Language of Caterpillars ⊇ {prefix normal words}
Proposition (Blondin Masse, de Carufel, Goupil, Lapointe, V.)
30/31
![Page 104: How to count leaves in the trees from Tetris?](https://reader036.fdocuments.in/reader036/viewer/2022090905/613c9009a9aa48668d4a4d3c/html5/thumbnails/104.jpg)
Perspectives
Determine when an integer sequence corresponds to a tree orto a graph
Find an application in Chemistry or Biology
Extend the results obtained in Z2 to the infinite triangular grid(polyiamonds) and to the infinite hexagonal grid (polyhexes)
31/31
![Page 105: How to count leaves in the trees from Tetris?](https://reader036.fdocuments.in/reader036/viewer/2022090905/613c9009a9aa48668d4a4d3c/html5/thumbnails/105.jpg)
Perspectives
Determine when an integer sequence corresponds to a tree orto a graph
Find an application in Chemistry or Biology
Extend the results obtained in Z2 to the infinite triangular grid(polyiamonds) and to the infinite hexagonal grid (polyhexes)
31/31
![Page 106: How to count leaves in the trees from Tetris?](https://reader036.fdocuments.in/reader036/viewer/2022090905/613c9009a9aa48668d4a4d3c/html5/thumbnails/106.jpg)
Perspectives
Determine when an integer sequence corresponds to a tree orto a graph
Find an application in Chemistry or Biology
Extend the results obtained in Z2 to the infinite triangular grid(polyiamonds) and to the infinite hexagonal grid (polyhexes)
31/31
![Page 107: How to count leaves in the trees from Tetris?](https://reader036.fdocuments.in/reader036/viewer/2022090905/613c9009a9aa48668d4a4d3c/html5/thumbnails/107.jpg)
Perspectives
Determine when an integer sequence corresponds to a tree orto a graph
Find an application in Chemistry or Biology
Extend the results obtained in Z2 to the infinite triangular grid(polyiamonds) and to the infinite hexagonal grid (polyhexes)
31/31