Parking in trees Marie-Louise Bruner Ongoing work with Alois - PowerPoint PPT Presentation

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 Further generalizations 4 Outlook 5 Marie-Louise Bruner Parking in trees June 10th, 2013 2

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? Marie-Louise Bruner Parking in trees June 10th, 2013 3

Parking functions What are parking functions? 3 , 1 , 1 , 5 , 2 is a parking function 3 , 1 , 1 , 5 , 5 is not Marie-Louise Bruner Parking in trees June 10th, 2013 4

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 = p 1 , p 2 , . . . , p n ∈ { 1 , 2 , . . . , n } n is a parking function if and only if it is a major function, i.e.: If q = q 1 , q 2 , . . . , q n is the increasing rearrangement of p then it holds that: q i ≤ i for all i ∈ { 1 , 2 , . . . , n } . Marie-Louise Bruner Parking in trees June 10th, 2013 4

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. Marie-Louise Bruner Parking in trees June 10th, 2013 5

Parking functions How many parking functions are there? Marie-Louise Bruner Parking in trees June 10th, 2013 6

Parking functions How many parking functions are there? n cars and ( n + 1) spaces: one space remains empty. Marie-Louise Bruner Parking in trees June 10th, 2013 6

Parking functions How many parking functions are there? n cars and ( n + 1) spaces: one space remains empty. ( n + 1) n · 1 P n = n +1 ( n + 1) n − 1 = (Proof by Pollak, 1974) Marie-Louise Bruner Parking in trees June 10th, 2013 6

Parking functions What if there are more spaces than cars? Marie-Louise Bruner Parking in trees June 10th, 2013 7

Parking functions What if there are more spaces than cars? m < n cars and ( n + 1) spaces: ( n + 1 − m ) spaces remain empty. Marie-Louise Bruner Parking in trees June 10th, 2013 7

Parking functions What if there are more spaces than cars? m < n cars and ( n + 1) spaces: ( n + 1 − m ) spaces remain empty. ( n + 1) m · n +1 − m P n , m = n +1 ( n + 1) m − 1 · ( n + 1 − m ) = Marie-Louise Bruner Parking in trees June 10th, 2013 7

Generalization to trees From one way streets to trees Marie-Louise Bruner Parking in trees June 10th, 2013 8

Generalization to trees From one way streets to trees Marie-Louise Bruner Parking in trees June 10th, 2013 8

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. Marie-Louise Bruner Parking in trees June 10th, 2013 9

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 − Marie-Louise Bruner Parking in trees June 10th, 2013 9

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

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

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

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

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

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

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

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

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

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

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

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 F n ... number of pairs ( T , F ) Marie-Louise Bruner Parking in trees June 10th, 2013 11

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. Marie-Louise Bruner Parking in trees June 10th, 2013 12

Enumeration Recurrence, first case Marie-Louise Bruner Parking in trees June 10th, 2013 13

Enumeration Recurrence, first case 1 � �� n − 1 � n � � F k 1 · F k 2 · . . . · F k r n r ! k 1 , k 2 , . . . , k r , 1 k 1 , k 2 , . . . , k r � r r ≥ 1 i =1 k i = n − 1 k i ≥ 1 Marie-Louise Bruner Parking in trees June 10th, 2013 13

Enumeration Recurrence, second case Marie-Louise Bruner Parking in trees June 10th, 2013 14

Enumeration Recurrence, second case 1 � �� n − 1 � n � � F k · F k 1 · . . . · F k r · k ( n − k ) r ! k , k 1 , . . . , k r , 1 k , k 1 , . . . , k r � r r ≥ 0 i =1 k i = n − k − 1 k , k i ≥ 1 Marie-Louise Bruner Parking in trees June 10th, 2013 14

Enumeration Technicalities: From recurrences to functional equations F n := F n / ( n !) 2 this decomposition leads to: With ˜ 1 1 n ˜ � � F k 1 · . . . · ˜ ˜ � � F k · ˜ ˜ F k 1 · . . . · ˜ F n = F k r · n + F k r · k ( n − k ) , r ! r ! � r � r r ≥ 1 r ≥ 0 i =1 ki = n − 1 i =1 ki = n − k − 1 ki ≥ 1 k , ki ≥ 1 for n ≥ 2,˜ F 1 = 1. Marie-Louise Bruner Parking in trees June 10th, 2013 15

Enumeration Technicalities: From recurrences to functional equations F n := F n / ( n !) 2 this decomposition leads to: With ˜ 1 1 n ˜ � � F k 1 · . . . · ˜ ˜ � � F k · ˜ ˜ F k 1 · . . . · ˜ F n = F k r · n + F k r · k ( n − k ) , r ! r ! � r � r r ≥ 1 r ≥ 0 i =1 ki = n − 1 i =1 ki = n − k − 1 ki ≥ 1 k , ki ≥ 1 F n z n fulfils the for n ≥ 2,˜ F 1 = 1. The generating function ˜ n ≥ 1 ˜ F ( z ) := � following differential equation: � 2 � F ′ ( z ) = exp(˜ ˜ 1 + z ˜ ˜ F ′ ( z ) F ( z )) · , F (0) = 0 , Marie-Louise Bruner Parking in trees June 10th, 2013 15