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Parking in trees Marie-Louise Bruner Ongoing work with Alois Panholzer CanaDAM 2013, June 10th Marie-Louise Bruner Parking in trees June 10th, 2013 1 Table of Contents Parking functions 1 Generalization to trees 2 Enumeration 3

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Parking in trees Marie-Louise Bruner

Ongoing work with Alois Panholzer CanaDAM 2013, June 10th

Marie-Louise Bruner Parking in trees June 10th, 2013 1

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Table of Contents

1

Parking functions

2

Generalization to trees

3

Enumeration

4

Further generalizations

5

Outlook

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

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Parking functions

What are parking functions?

3, 1, 1, 5, 2 is a parking function 3, 1, 1, 5, 5 is not

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Parking functions

What are parking functions?

3, 1, 1, 5, 2 is a parking function 3, 1, 1, 5, 5 is not Alternative characterization A sequence p = p1, p2, . . . , pn ∈ {1, 2, . . . , n}n is a parking function if and

  • nly if it is a major function, i.e.:

If q = q1, q2, . . . , qn is the increasing rearrangement of p then it holds that: qi ≤ i for all i ∈ {1, 2, . . . , n} .

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Parking functions

Why parking functions?

Konheim and Weiss, 1966: An Occupancy Discipline and Applications Linear probing hashing: First non-trivial algorithm to be analysed by Knuth in 1962.

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Parking functions

How many parking functions are there?

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Parking functions

How many parking functions are there?

n cars and (n + 1) spaces:

  • ne space remains empty.

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Parking functions

How many parking functions are there?

n cars and (n + 1) spaces:

  • ne space remains empty.

Pn = (n + 1)n ·

1 n+1

= (n + 1)n−1 (Proof by Pollak, 1974)

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Parking functions

What if there are more spaces than cars?

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Parking functions

What if there are more spaces than cars?

m < n cars and (n + 1) spaces: (n + 1 − m) spaces remain empty.

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Parking functions

What if there are more spaces than cars?

m < n cars and (n + 1) spaces: (n + 1 − m) spaces remain empty. Pn,m = (n + 1)m · n+1−m

n+1

= (n + 1)m−1 · (n + 1 − m)

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Generalization to trees

From one way streets to trees

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Generalization to trees

From one way streets to trees

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Generalization to trees

What kind of trees?

rooted trees Edges are directed towards the root. Trees of size n are labelled with the integers 1, 2, . . . , n. The order of the children of a given node is of no relevance.

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Generalization to trees

What kind of trees?

rooted trees Edges are directed towards the root. Trees of size n are labelled with the integers 1, 2, . . . , n. The order of the children of a given node is of no relevance. − → Cayley trees

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Generalization to trees Marie-Louise Bruner Parking in trees June 10th, 2013 10

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Enumeration

How many parking functions on trees are there?

How many pairs (T, F) are there? T ... Cayley tree of size n F ... sequence of integers from {1, . . . , n} that is a parking function for T Fn ... number of pairs (T, F)

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Enumeration

Decomposition idea

We decompose a tree accompanied by a parking function by the last node that is filled. We have to consider two different cases: The last node to be filled is the root node. The last node to be filled is not the root node.

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Enumeration

Recurrence, first case

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Enumeration

Recurrence, first case

  • r≥1

1 r!

  • r

i=1 ki=n−1

ki≥1

Fk1 · Fk2 · . . . · Fkr

  • n

k1, k2, . . . , kr, 1

  • n − 1

k1, k2, . . . , kr

  • n

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Enumeration

Recurrence, second case

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Enumeration

Recurrence, second case

  • r≥0

1 r!

  • r

i=1 ki=n−k−1

k,ki≥1

Fk ·Fk1 ·. . .·Fkr ·

  • n

k, k1, . . . , kr, 1

  • n − 1

k, k1, . . . , kr

  • k(n−k)

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Enumeration

Technicalities: From recurrences to functional equations

With ˜ Fn := Fn/(n!)2 this decomposition leads to: n˜ Fn =

  • r≥1

1 r!

  • r

i=1 ki =n−1 ki ≥1

˜ Fk1 · . . . · ˜ Fkr · n +

  • r≥0

1 r!

  • r

i=1 ki =n−k−1 k,ki ≥1

˜ Fk · ˜ Fk1 · . . . · ˜ Fkr · k(n − k), for n ≥ 2,˜ F1 = 1.

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Enumeration

Technicalities: From recurrences to functional equations

With ˜ Fn := Fn/(n!)2 this decomposition leads to: n˜ Fn =

  • r≥1

1 r!

  • r

i=1 ki =n−1 ki ≥1

˜ Fk1 · . . . · ˜ Fkr · n +

  • r≥0

1 r!

  • r

i=1 ki =n−k−1 k,ki ≥1

˜ Fk · ˜ Fk1 · . . . · ˜ Fkr · k(n − k), for n ≥ 2,˜ F1 = 1. The generating function ˜ F(z) :=

n≥1 ˜

Fnzn fulfils the following differential equation: ˜ F ′(z) = exp(˜ F(z)) ·

  • 1 + z ˜

F ′(z) 2 , ˜ F(0) = 0,

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Enumeration

Technicalities: From recurrences to functional equations

With ˜ Fn := Fn/(n!)2 this decomposition leads to: n˜ Fn =

  • r≥1

1 r!

  • r

i=1 ki =n−1 ki ≥1

˜ Fk1 · . . . · ˜ Fkr · n +

  • r≥0

1 r!

  • r

i=1 ki =n−k−1 k,ki ≥1

˜ Fk · ˜ Fk1 · . . . · ˜ Fkr · k(n − k), for n ≥ 2,˜ F1 = 1. The generating function ˜ F(z) :=

n≥1 ˜

Fnzn fulfils the following differential equation: ˜ F ′(z) = exp(˜ F(z)) ·

  • 1 + z ˜

F ′(z) 2 , ˜ F(0) = 0, which has the following solution: ˜ F(z) = T(2z) + ln

  • 1 − T(2z)

2

  • ,

where T(z) is the Cayley tree function fulfilling T(z) = z exp(T(z)).

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Enumeration

Enumeration result

Theorem The total number Fn of n-parking functions of Cayley-trees of size n is given as follows: Fn = n! · (n − 1)! ·

n−1

  • k=0

(n − k) · (2n)k n · k! .

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Enumeration

Enumeration result

Theorem The total number Fn of n-parking functions of Cayley-trees of size n is given as follows: Fn = n! · (n − 1)! ·

n−1

  • k=0

(n − k) · (2n)k n · k! . Theorem The numbers Fn are asymptotically given as follows: Fn ∼ √ 2√π2n+1n2n n

3 2 en

.

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Further generalizations

Further generalizations

Consider the case where there are more parking spaces than drivers: If there are m < n drivers ˜ m := n − m parking spaces remain empty. − → decomposition of the tree according to the empty node, which has the largest label amongst all ˜ m empty nodes − → exact enumeration formula for Tn,m

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Further generalizations

Further generalizations

Consider the case where there are more parking spaces than drivers: If there are m < n drivers ˜ m := n − m parking spaces remain empty. − → decomposition of the tree according to the empty node, which has the largest label amongst all ˜ m empty nodes − → exact enumeration formula for Tn,m Consider parking functions on n-mappings instead of trees.

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Further generalizations

Mappings and their graph representation

Definition An n-mapping M is a function from the set [n] := {1, 2, . . . , n} to itself.

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Further generalizations

Mappings and their graph representation

Definition An n-mapping M is a function from the set [n] := {1, 2, . . . , n} to itself. M can also be represented by its functional graph: VM = {1, 2, . . . , n} and EM = {(i, M(i)) : 1 ≤ i ≤ n} .

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Further generalizations

Mappings and their graph representation

Definition An n-mapping M is a function from the set [n] := {1, 2, . . . , n} to itself. M can also be represented by its functional graph: VM = {1, 2, . . . , n} and EM = {(i, M(i)) : 1 ≤ i ≤ n} .

1 2 3 8 10 4 5 6 7 9 11 12 13 14 15 16 19 18 17

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Further generalizations

Mappings and their graph representation

Definition An n-mapping M is a function from the set [n] := {1, 2, . . . , n} to itself. M can also be represented by its functional graph: VM = {1, 2, . . . , n} and EM = {(i, M(i)) : 1 ≤ i ≤ n} .

1 2 3 8 10 4 5 6 7 9 11 12 13 14 15 16 19 18 17

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Further generalizations

Decomposition idea for connected mappings

We consider three different cases. The last node to be filled can be: the root node of the

  • nly tree constituting

the cycle, the root node of a tree in a cycle of length ≥ 2

  • r

a non-cyclic node.

...

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Further generalizations

Enumeration result for parking functions in mappings

Theorem Mn = n · Fn

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Further generalizations

Enumeration result for parking functions in mappings

Theorem Mn = n · Fn and Mn,m = n · Fn,m for all n ∈ N and 1 ≤ m ≤ n.

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Further generalizations

From parking in mappings back to hashing

Linear probing hashing h(i) ... hash value attributed to entry i in case of collision: try h(i) + 1 mod n j-th collision: try h(i) + j mod n

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Further generalizations

From parking in mappings back to hashing

Linear probing hashing h(i) ... hash value attributed to entry i in case of collision: try h(i) + 1 mod n j-th collision: try h(i) + j mod n “Random mapping” hashing h(i) = Fi ... i-th entry in parking function F in case of collision: try M(h(i)) j-th collision: try Mj(h(i))

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Outlook

Outlook

Defective parking functions Total displacement: total driving distance of the drivers Other tree families? Ordered trees, binary trees...

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