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Advanced Tree Data Structures

Advanced Tree Data Structures. Nelson Padua-Perez Chau-Wen Tseng Department of Computer Science University of Maryland, College Park. Overview. Binary trees Balance Rotation Multi-way trees Search Insert. Degenerate Worst case Search in O(n) time. Balanced Average case

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Advanced Tree Data Structures

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  1. Advanced Tree Data Structures Nelson Padua-Perez Chau-Wen Tseng Department of Computer Science University of Maryland, College Park

  2. Overview • Binary trees • Balance • Rotation • Multi-way trees • Search • Insert

  3. Degenerate Worst case Search in O(n) time Balanced Average case Search in O( log(n) ) time Tree Balance Degenerate binary tree Balanced binary tree

  4. Tree Balance • Question • Can we keep tree (mostly) balanced? • Self-balancing binary search trees • AVL trees • Red-black trees • Approach • Select invariant (that keeps tree balanced) • Fix tree after each insertion / deletion • Maintain invariant using rotations • Provides operations with O( log(n) ) worst case

  5. 4 44 3 2 17 78 1 2 1 88 32 50 1 Heights of children shown in red 1 48 62 AVL Trees • Properties • Binary search tree • Heights of children for node differ by at most 1 • Example

  6. AVL Trees • History • Discovered in 1962 by two Russian mathematicians, Adelson-Velskii & Landis • Algorithm • Find / insert / delete as a binary search tree • After each insertion / deletion • If height of children differ by more than 1 • Rotate children until subtrees are balanced • Repeat check for parent (until root reached)

  7. Red-black Trees • Properties • Binary search tree • Every node is red or black • The root is black • Every leaf is black • All children of red nodes are black • For each leaf, same # of black nodes on path to root • Characteristics • Properties ensures no leaf is twice as far from root as another leaf

  8. Red-black Trees • Example

  9. Red-black Trees • History • Discovered in 1972 by Rudolf Bayer • Algorithm • Insert / delete may require complicated bookkeeping & rotations • Java collections • TreeMap, TreeSet use red-black trees

  10. Tree Rotations • Changes shape of tree • Move nodes • Change edges • Types • Single rotation • Left • Right • Double rotation • Left-right • Right-left

  11. Tree Rotation Example • Single right rotation 2 3 3 1 2 1

  12. Tree Rotation Example • Single right rotation 3 5 2 5 3 6 4 6 1 4 2 1 Node 4 attached to new parent

  13. Example – Single Rotations 1 2 2 1 single left rotation 3 3 T T 0 3 T T T T T 1 0 1 2 3 T 2 3 2 single right rotation 2 1 3 1 T T 3 0 T T T T T 0 2 1 2 3 T 1

  14. T T 3 1 T T T T T 0 0 2 3 1 T 2 Example – Double Rotations 2 1 right-left double rotation 3 1 3 2 T T 0 2 T T T T T 3 0 1 3 2 T 1 3 2 left-right double rotation 1 1 3 2

  15. Multi-way Search Trees • Properties • Generalization of binary search tree • Node contains 1…k keys (in sorted order) • Node contains 2…k+1 children • Keys in jth child < jth key < keys in (j+1)th child • Examples 5 12 5 8 15 33 2 8 17 44 1 3 7 9 19 21

  16. Types of Multi-way Search Trees 5 12 • 2-3 tree • Internal nodes have 2 or 3 children • Index search trie • Internal nodes have up to 26 children (for strings) • B-tree • T = minimum degree • Non-root internal nodes have T-1 to 2T-1 children • All leaves have same depth 2 8 17 c a o s T-1 … 2T-1 1 2 … 2T

  17. Multi-way Search Trees • Search algorithm • Compare key x to 1…k keys in node • If x = some key then return node • Else if (x < key j)search child j • Else if (x > all keys) search child k+1 • Example • Search(17) 25 5 12 30 40 1 2 8 17 27 36 44

  18. Multi-way Search Trees • Insert algorithm • Search key x to find node n • If ( n not full ) insert x in n • Else if ( n is full ) • Split n into two nodes • Move middle key from n to n’s parent • Insert x in n • Recursively split n’s parent(s) if necessary

  19. Multi-way Search Trees • Insert Example (for 2-3 tree) • Insert( 4 ) 5 12 5 12 2 8 17 2 4 8 17

  20. Multi-way Search Trees • Insert Example (for 2-3 tree) • Insert( 1 ) 5 5 12 2 12 1 2 4 8 17 1 4 8 17 Split node 2 5 12 Split parent 1 4 8 17

  21. B-Trees • Characteristics • Height of tree is O( logT(n) ) • Reduces number of nodes accessed • Wasted space for non-full nodes • Popular for large databases • 1 node = 1 disk block • Reduces number of disk blocks read

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