Balanced Binary Search Trees

1. Balanced Binary Search Trees height is O(log n), where n is the number of elements in the tree AVL (Adelson-Velsky and Landis) trees red-black trees find, insert, and erase take O(log n) time

2. Balanced Binary Search Trees Indexed AVL trees Indexed red-black trees Indexed operations alsotake O(log n) time

3. Balanced Search Trees weight balanced binary search trees 2-3 & 2-3-4 trees AA trees B-trees BBST etc.

4. AVL Tree • A binary tree • More specifically it’s a “self-balancing binary search tree” • for every node x, define its balance factor: balance factor of x = height of left subtree of x - height of right subtree of x • balance factor of every node x must be -1, 0, or 1 • rebalancing if necessary

5. Balance Factors -1 1 1 -1 0 1 This is an AVL tree. 0 0 -1 0 0 0 0

6. Height The height of an AVL tree that has n nodes is at most1.44 log2 (n+2). The height of every n node binary tree is at least log2 (n+1).

7. -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 60 35 1 20 5 0 25 AVL Search Tree

8. AVL Tree • Find(key) • Same as BST find, but worst-case O(logn) • Insert(key, value) • Same as BST insert, then check balance and rebalance if necessary • Erase(key) • Similar to insertion of AVL, but must continue to trace the path towards the root

9. insert(9) -1 10 0 1 1 7 40 -1 0 1 -1 0 45 3 8 30 0 -1 0 0 0 0 60 35 1 9 20 5 0 25

10. insert(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 -1 0 0 0 -2 60 35 1 20 5 0 -1 RR imbalance => new node is in right subtree of right subtree of blue node (node with bf = -2) 25 0 29

11. insert(29) -1 10 1 1 7 40 -1 0 1 0 45 3 8 30 0 0 0 0 0 60 35 1 25 5 0 0 20 29 RR rotation.

12. AVL Rotations • Four cases to consider, cases 1 to 4: • RR (right-right)  Single rotation to left • LL (left-left)  Single rotation to right • RL (right-left)  Double rotation • One rotation to get to case 1 • Then case 1 • LR (left-right)  Double rotation • One rotation to get to case 2 • Then case 2 Insertion sequence: RR) 1, 2, 3 LL) 3, 2, 1 RL) 1, 3, 2 LR) 3, 1, 2

14. Red Black Trees Colored Nodes Definition • Binary search tree. • Each node is colored red or black. • Root and all external nodes are black. • No root-to-external-node path has two consecutive red nodes. • All root-to-external-node paths have the same number of black nodes

15. Red Black Tree Properties • Remember these 4 important properties: • Every node is either red or black • The root and external nodes are black • If a node is red, then its parent is black • All simple paths from any node x to a descendent leaf have the same number of black nodes = black-height(x)

16. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red Black Tree

17. RB Tree unbalanced cases • 8 unbalanced cases • CLRS reduces that to 6 • More details are taught in: • COT5405 Analysis of Algorithms course

18. Red Black Trees Colored Edges Definition • Binary search tree. • Child pointers are colored red or black. • Pointer to an external node is black. • No root to external node path has two consecutive red pointers. • Every root to external node path has the same number of black pointers.

19. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red Black Tree

20. Red Black Tree • The height of a red black tree that has n (internal) nodes is between log2(n+1) and 2log2(n+1). • RB vs AVL: • AVL may require more rotations in insert/delete • RB may have higher height • C++ STL Class map => red black tree • Java standard library TreeMap => red-black tree

21. u v Graphs • G = (V,E) • V is the vertex set. • Vertices are also called nodes and points. • E is the edge set. • Each edge connects two different vertices. • Edges are also called arcs and lines. • Directed edge has an orientation (u,v).

22. u v Graphs • Undirected edge has no orientation (u,v). • Undirected graph => no oriented edge. • Directed graph => every edge has an orientation.

23. 2 3 8 1 10 4 5 9 11 6 7 Undirected Graph

24. 2 3 8 1 10 4 5 9 11 6 7 Directed Graph (Digraph)

25. 2 3 8 1 10 4 5 9 11 6 7 Applications—Communication Network • Vertex = city, edge = communication link.

26. 2 4 3 8 8 1 6 10 2 4 5 4 4 3 5 9 11 5 6 7 6 7 Driving Distance/Time Map • Vertex = city, edge weight = driving distance/time.

27. 2 3 8 1 10 4 5 9 11 6 7 Street Map • Some streets are one way.

28. n = 4 n = 1 n = 2 n = 3 Complete Undirected Graph Has all possible edges.

29. Number Of Edges—Undirected Graph • Each edge is of the form (u,v), u != v. • Number of such pairs in an n vertex graph is n(n-1). • Since edge (u,v) is the same as edge (v,u), the number of edges in a complete undirected graph is n(n-1)/2. • Number of edges in an undirected graph is <= n(n-1)/2.

30. Number Of Edges—Directed Graph • Each edge is of the form (u,v), u != v. • Number of such pairs in an n vertex graph is n(n-1). • Since edge (u,v) is not the same as edge (v,u), the number of edges in a complete directed graph is n(n-1). • Number of edges in a directed graph is <= n(n-1).

31. 2 3 8 1 10 4 5 9 11 6 7 Vertex Degree Number of edges incident to vertex. degree(2) = 2, degree(5) = 3, degree(3) = 1

32. 8 10 9 11 Sum Of Vertex Degrees Sum of degrees = 2e (e is number of edges)

33. 2 3 8 1 10 4 5 9 11 6 7 In-Degree Of A Vertex in-degree is number of incoming edges indegree(2) = 1, indegree(8) = 0

34. 2 3 8 1 10 4 5 9 11 6 7 Out-Degree Of A Vertex out-degree is number of outbound edges outdegree(2) = 1, outdegree(8) = 2

35. Sum Of In- And Out-Degrees each edge contributes 1 to the in-degree of some vertex and 1 to the out-degree of some other vertex sum of in-degrees = sum of out-degrees = e, where e is the number of edges in the digraph