the node profile of symmetric digital search trees
play

The Node Profile of Symmetric Digital Search Trees (joint with M. - PowerPoint PPT Presentation

Nov 30, 2023 •257 likes •1.06k views

The Node Profile of Symmetric Digital Search Trees (joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University June 8th, 2015 Michael Fuchs (NCTU) Node Profile of DSTs

  1. The Node Profile of Symmetric Digital Search Trees (joint with M. Drmota, H.-K. Hwang and R. Neininger) Michael Fuchs Department of Applied Mathematics National Chiao Tung University June 8th, 2015 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 1 / 28

  2. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  3. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  4. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: B 5 , 0 = 0 , B 5 , 1 = 0 , B 5 , 2 = 2 , B 5 , 3 = 4 , Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  5. Node Profile of (Rooted) Trees B n,k = number of external nodes at level k ; I n,k = number of internal nodes at level k . Example: B 5 , 0 = 0 , I 5 , 0 = 1 ; B 5 , 1 = 0 , I 5 , 1 = 2 ; B 5 , 2 = 2 , I 5 , 2 = 2 ; B 5 , 3 = 4 , I 5 , 3 = 0 . Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 2 / 28

  6. Relations to Other Shape Parameters Many shape parameters can by analyzed through the profile. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

  7. Relations to Other Shape Parameters Many shape parameters can by analyzed through the profile. Depth: P ( D n = k ) = B n,k / ( n + 1) ; Width: max { B n,k : k ≥ 0 } ; Total Path Length: � k kB n,k ; Height: max { k : B n,k > 0 } ; Shortest Path: min { k : B n,k > 0 } ; Fill-up Level: max { k : I n,k = 2 k } ; Etc. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 3 / 28

  8. Profile of Random Trees √ n -Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

  9. Profile of Random Trees √ n -Trees: Aldous (1991); Drmota and Gittenberger (1997); Kersting (1998); Pitman (1999); etc. log n -Trees: Binary Search Trees: Chauvin, Drmota, Jabbour-Hattab (2001); Drmota and Hwang (2005); F., Hwang, Neininger (2006). Recursive Trees: Drmota and Hwang (2005); F., Hwang, Neininger (2006). Plane-oriented Recursive Trees: Hwang (2007). m -ary Seach Trees: Drmota, Janson, Neininger (2008). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 4 / 28

  10. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  11. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  12. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  13. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 011011 010101 1 101110 010000 0 1 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  14. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 101110 010000 0 1 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  15. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 101110 010000 0 1 101010 001100 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  16. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 1 0 101110 010000 0 1 1 101010 001100 0 1 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  17. Tries Ren´ e de la Briandais (1959) Name from data retrieval (suggested by Fredkin). Example: 0 1 011011 010101 0 1 0 101110 010000 0 1 1 101010 001100 0 1 0 1 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 5 / 28

  18. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  19. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  20. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  21. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  22. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  23. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  24. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  25. Digital Search Trees (DSTs) Edward G. Coffman & James Eve (1970) Closely related to Lempel-Ziv compression scheme. Example: 011011 0 1 010101 101110 0 1 0 1 010000 101010 0 1 0 1 0 1 001100 Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 6 / 28

  26. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  27. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  28. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Question: What can be said about the profile? Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  29. Random Model Bits generated by iid Bernoulli random variables with mean p − → Bernoulli model Two types: p = 1 / 2 : symmetric digital trees; p � = 1 / 2 : asymmetric digital trees. Question: What can be said about the profile? In this talk, we are interested in mean, variance and limit laws of the profile for symmetric DSTs. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 7 / 28

  30. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  31. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  32. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws. Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  33. Profile of Digital Trees Tries: Mean, variance, limit laws: Hwang, Nicod´ eme, Park and Szpankowski (2009). PATRICIA tries: Mean: Magner, Knessl, Szpankowski (2014); Variance & limit laws: Szpankowkski & Magner ( → Thursday). Asymmetric DSTs: Mean: Drmota and Szpankowski (2011); Variance: Kazemi and Vahidi-Asl (2011); so far no limit laws. Symmetric DSTs: Variance & limit laws: Drmota, F., Hwang, Neininger ( → this talk). Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 8 / 28

  34. Profile of Tries Hwang, Nicod´ eme, Park, Szpankowski (2009) Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 9 / 28

  35. Plot of Mean Profile of Symmetric Tries Hwang, Nicod´ eme, Park, Szpankowski (2009): Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 10 / 28

  36. Symmetric Tries: Mean We have, � n (1 − 2 − k ) n − 1 , if 2 − k n → ∞ ; µ n,k := E ( B n,k ) ∼ ˜ if 4 − k n → 0 , M k, 1 ( n ) , where M k, 1 ( z ) = z ( e − z/ 2 k − e − z/ 2 k − 1 ) . ˜ Michael Fuchs (NCTU) Node Profile of DSTs June 8th, 2015 11 / 28

Recommend

Multi-way Search Trees Each internal node of a multi-way search tree T : - has at least two

Multi-way Search Trees Each internal node of a multi-way search tree T : - has at least two

(2,4) T REES Search Trees (but not binary) also known as 2-4, 2-3-4 trees very important as basis for Red-Black trees (so pay attention!) (2,4) Trees 1 Multi-way Search Trees Each internal node of a multi-way search tree T :

639 views • 12 slides
Binary Search Trees A binary search tree is a binary tree T such that - each internal node

Binary Search Trees A binary search tree is a binary tree T such that - each internal node

AVL T REES Binary Search Trees AVL Trees AVL Trees 1 Binary Search Trees A binary search tree is a binary tree T such that - each internal node stores an item (k, e) of a dictionary. - keys stored at nodes in the left subtree of

547 views • 26 slides
Tirgul 5 AVL trees Binary search trees (reminder) Each tree node contains a value.

Tirgul 5 AVL trees Binary search trees (reminder) Each tree node contains a value.

Tirgul 5 AVL trees Binary search trees (reminder) Each tree node contains a value. For every node, its left subtree contains smaller values and its right subtree contains larger values. The time complexity of a search

353 views • 8 slides
search for 8\h is 779 262 727 L 97 < 27 179 , 62 - Search is a generalization of BST search

search for 8\h is 779 262 727 L 97 < 27 179 , 62 - Search is a generalization of BST search

m-arysearchtree.sn rooted trees where each node may have up to m children K children has k -1 keys . a node with a natural generalization of BST search : search is ' Exi An order - 5 ( 5- ar y ) search tree . search for 8\h is

385 views • 20 slides
( ( ) ) ( ) ( ) = = Work = h log t n B- B -Trees Trees B B- -Trees

( ( ) ) ( ) ( ) = = Work = h log t n B- B -Trees Trees B B- -Trees

B- B -Trees Trees B B- -Trees Trees Search for key R ( ( ) ) ( ) ( ) = = Work = h log t n B- B -Trees Trees B B- -Trees Trees Each Disk-Read or Disk-Write = one Basic unit of work O(1) Typical Node x

287 views • 4 slides
, iaie . with - . . EAYI.tw/newArynee.n.ndersoni99s example : / . - tree : Red - Black and

, iaie . with - . . EAYI.tw/newArynee.n.ndersoni99s example : / . - tree : Red - Black and

r - Some history : simpler to code as binary 3- node AA - Trees Encoding : - No null pointers : Bayer 1972 : create tree node 2-3 Trees a ' Black node Sentinel node , nil , and all : Guibasa Red - black Trees - i . Red node 1978 ( a binary

825 views • 3 slides
Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees

Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees

Advanced Implementations of Tables: Balanced Search Trees and Hashing Balanced Search Trees Binary search tree operations such as insert, delete, retrieve, etc. depend on the length of the path to the desired node The path length can

721 views • 59 slides
10/17/2016 Review: Binary Search Trees (BST) Structure property (binary tree) Each node

10/17/2016 Review: Binary Search Trees (BST) Structure property (binary tree) Each node

10/17/2016 Review: Binary Search Trees (BST) Structure property (binary tree) Each node has 2 children 8 Result: keeps operations simple CSE373: Data Structures and Algorithms Order property 5 11 AVL Trees All keys

202 views • 7 slides
How efficient are binary search trees? Balanced search trees Balancing a tree means to keep the

How efficient are binary search trees? Balanced search trees Balancing a tree means to keep the

CS206 CS206 How efficient are binary search trees? Balanced search trees Balancing a tree means to keep the left and right subtree of Binary search tree operations take time O ( h ) , where h is the every node of roughly equal size.

200 views • 3 slides
Informed Search (Ch. 3.5-3.6) Announcements HW1 graded Heuristics However, for A* to be optimal

Informed Search (Ch. 3.5-3.6) Announcements HW1 graded Heuristics However, for A* to be optimal

Informed Search (Ch. 3.5-3.6) Announcements HW1 graded Heuristics However, for A* to be optimal the heuristic h(node) needs to be... For trees: admissible which means: h(node) < optimal path from h to goal (i.e. h(node) is an

508 views • 23 slides
Trees Binary Trees A binary tree is composed of zero or more nodes in which no node can have

Trees Binary Trees A binary tree is composed of zero or more nodes in which no node can have

Trees Binary Trees A binary tree is composed of zero or more nodes in which no node can have more than two children. Each node contains: A value (some sort of data item). A reference or pointer to a left child (may be null), and

609 views • 41 slides
61A Lecture 23 Friday, October 19 Trees with Internal Node Values 2 Trees with Internal Node

61A Lecture 23 Friday, October 19 Trees with Internal Node Values 2 Trees with Internal Node

Last day of Midterm 2 Material 61A Lecture 23 Friday, October 19 Trees with Internal Node Values 2 Trees with Internal Node Values Trees can have values at their roots as well as their leaves. 2 Trees with Internal Node Values Trees can

1.04k views • 89 slides
in 97 . , - , CF B. Tree restructuring ( Parent lost one key Rotation ( Adoption )

in 97 . , - , CF B. Tree restructuring ( Parent lost one key Rotation ( Adoption )

- , A- Secondary Memory Multi way Search Trees : Node Structure : constant intm : . : - Most large data structures class B. Tree Node It 9 on disk storage Kno . of children reside nchild int ( / \ - organized in blocks

344 views • 3 slides
Balanced Search Trees 2-3-4 trees red-black trees References: Algorithms in Java (handout)

Balanced Search Trees 2-3-4 trees red-black trees References: Algorithms in Java (handout)

Balanced Search Trees 2-3-4 trees red-black trees References: Algorithms in Java (handout) Balanced search trees Dynamic sets Search Insert Delete Maximum Minimum Successor(x) (find minimum element x)

1.01k views • 68 slides
AVL Trees AVL Tree Definition (9.2) AVL trees are 4 44 balanced. 2 3 An AVL Tree is a 17

AVL Trees AVL Tree Definition (9.2) AVL trees are 4 44 balanced. 2 3 An AVL Tree is a 17

1 8 6 z 4 3 v AVL Trees AVL Trees AVL Tree Definition (9.2) AVL trees are 4 44 balanced. 2 3 An AVL Tree is a 17 78 binary search tree 1 2 1 88 32 50 such that for every 1 1 internal node v of T, 48 62 the heights of

453 views • 9 slides
Binary Search Trees and Balanced Binary Search Trees using AVL Trees Mark Redekopp David Kempe

Binary Search Trees and Balanced Binary Search Trees using AVL Trees Mark Redekopp David Kempe

1 CSCI 104 Binary Search Trees and Balanced Binary Search Trees using AVL Trees Mark Redekopp David Kempe Sandra Batista 2 Properties, Insertion and Removal BINARY SEARCH TREES 3 Binary Search Tree Binary search tree = binary tree

630 views • 61 slides
The problem with binary search trees Height-balanced trees AVL trees Search time average case:

The problem with binary search trees Height-balanced trees AVL trees Search time average case:

The problem with binary search trees Height-balanced trees AVL trees Search time average case: lg( n ) Tyler Moore Search time worst case: n Can you construct such a tree? CS 2123, The University of Tulsa Solution: height-balanced binary trees

212 views • 4 slides
iwi Final Focus : Worst-case ' or randomized expected Howtodesignlanalyze ? -

iwi Final Focus : Worst-case ' or randomized expected Howtodesignlanalyze ? -

' -- T .-nys-n- Recap / Other Better criteria ? Lots of search trees / : Different combinations Lesson : : Some keys - unbalanced BSI of rotations Expected case can : - bring given node to root more popular than others - AVL Trees -

250 views • 3 slides
Informed Search (Ch. 3.5-3.6) Informed search In uninformed search, we only had the node

Informed Search (Ch. 3.5-3.6) Informed search In uninformed search, we only had the node

Informed Search (Ch. 3.5-3.6) Informed search In uninformed search, we only had the node information (parent, children, cost of actions) Now we will assume there is some additional information, we will call a heuristic that estimates the

594 views • 22 slides
General Trees children that any node may have. Chapter 7 Well, non-binary trees anyway.

General Trees children that any node may have. Chapter 7 Well, non-binary trees anyway.

General Trees General trees are similar to binary trees, except that there is no restriction on the number of General Trees children that any node may have. Chapter 7 Well, non-binary trees anyway. General Trees More formally A

283 views • 6 slides
Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the

Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the

Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the Indexed AVL trees number of elements in the tree Indexed red-black trees AVL (Adelson-Velsky and Landis) Indexed operations also

253 views • 6 slides
Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the

Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the

Balanced Binary Search Trees Balanced Binary Search Trees height is O(log n), where n is the Indexed AVL trees number of elements in the tree Indexed red-black trees AVL (Adelson-Velsky and Landis) Indexed operations also

297 views • 8 slides
Trees, Binary Search Trees, Lab 7, Project 2 Bryce Boe

Trees, Binary Search Trees, Lab 7, Project 2 Bryce Boe

Trees, Binary Search Trees, Lab 7, Project 2 Bryce Boe 2013/11/13 CS24, Fall 2013 Outline Stack/Queue Review Trees Binary Search Trees

557 views • 31 slides
Hsien-Kuei Hwang Academia Sinica, Taiwan (joint with M. Drmota, M. Fuchs, R. Neininger) April

Hsien-Kuei Hwang Academia Sinica, Taiwan (joint with M. Drmota, M. Fuchs, R. Neininger) April

PROFILE OF RANDOM RECURSIVE TREES AND RANDOM BINARY SEARCH TREES Hsien-Kuei Hwang Academia Sinica, Taiwan (joint with M. Drmota, M. Fuchs, R. Neininger) April 26, 2004 Profile of random recursive trees and random binary search trees, INRIA,

1.95k views • 160 slides

More recommend