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Balanced Binary Search Trees

Learn about the benefits and operations of AVL and Red-Black trees in implementing efficient balanced search trees. AVL rotations and Red-Black tree features are also explained.

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Balanced Binary Search Trees

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  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 get, put, and remove 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 • binary 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 is -1, 0, or 1

  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 most 1.44 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. put(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

  9. put(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

  10. put(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.

  11. AVL Rotations • RR • LL • RL • LR

  12. 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

  13. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red Black Tree

  14. 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.

  15. 10 7 40 45 3 8 30 60 35 1 20 5 25 Example Red Black Tree

  16. Red Black Tree • The height of a red black tree that has n (internal) nodes is between log2(n+1) and 2log2(n+1). • java.util.TreeMap => red black tree

  17. 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).

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

  19. 2 3 8 1 10 4 5 9 11 6 7 Undirected Graph

  20. 2 3 8 1 10 4 5 9 11 6 7 Directed Graph (Digraph)

  21. 2 3 8 1 10 4 5 9 11 6 7 Applications—Communication Network • Vertex = city, edge = communication link.

  22. 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.

  23. 2 3 8 1 10 4 5 9 11 6 7 Street Map • Some streets are one way.

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

  25. 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.

  26. 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).

  27. 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

  28. 8 10 9 11 Sum Of Vertex Degrees Sum of degrees = 2e (e is number of edges)

  29. 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

  30. 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

  31. 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

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