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CS 225 Data Structures Ma March 9 AV AVL Applications G G Carl - PowerPoint PPT Presentation

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CS 225 Data Structures Ma March 9 AV AVL Applications G G Carl Evans AV AVL Runtime Proof On Friday, we proved an upper-bound on the height of an AVL tree is 2 lg(n) or O( lg(n) ) : N(h) := Minimum # of nodes in an AVL tree of height

  1. CS 225 Data Structures Ma March 9 – AV AVL Applications G G Carl Evans

  2. AV AVL Runtime Proof On Friday, we proved an upper-bound on the height of an AVL tree is 2 × lg(n) or O( lg(n) ) : N(h) := Minimum # of nodes in an AVL tree of height h N(h) = 1 + N(h-1) + N(h-2) > 1 + 2 h-1/2 + 2 h-2/2 > 2 × 2 h-2/2 = 2 h-2/2+1 = 2 h/2 Theorem #1: Every AVL tree of height h has at least 2 h/2 nodes.

  3. AV AVL Runtime Proof On Friday, we proved an upper-bound on the height of an AVL tree is 2 × lg(n) or O( lg(n) ) : # of nodes (n) ≥ N(h) > 2 h/2 n > 2 h/2 lg(n) > h/2 2 × lg(n) > h h < 2 × lg(n) , for h ≥ 1 Proved: The maximum number of nodes in an AVL tree of height h is less than 2 × lg(n).

  4. Su Summary of of Ba Balance ced BS BST AVL Trees - Max height: 1.44 * lg(n) - Rotations:

  5. Su Summary of of Ba Balance ced BS BST AVL Trees - Max height: 1.44 * lg(n) - Rotations: Zero rotations on find One rotation on insert O( h ) == O( lg(n) ) rotations on remove Red-Black Trees - Max height: 2 * lg(n) - Constant number of rotations on insert (max 2), remove (max 3).

  6. Wh Why Ba Balanced BS BST?

  7. Su Summary of of Ba Balance ced BS BST Pros: - Running Time: - Improvement Over: - Great for specific applications:

  8. Su Summary of of Ba Balance ced BS BST Cons: - Running Time: - In-memory Requirement:

  9. Re Red-Bl Black T Trees i in C+ C++ C++ provides us a balanced BST as part of the standard library: std::map<K, V> map;

  10. Re Red-Bl Black T Trees i in C+ C++ V & std::map<K, V>::operator[]( const K & )

  11. Re Red-Bl Black T Trees i in C+ C++ V & std::map<K, V>::operator[]( const K & ) std::map<K, V>::erase( const K & )

  12. Re Red-Bl Black T Trees i in C+ C++ iterator std::map<K, V>::lower_bound( const K & ); iterator std::map<K, V>::upper_bound( const K & );

  13. CS 225 CS 225 -- -- Cou Course U Update Your grades can now be viewed on moodle (https://learn.illinois.edu/) We will discuss the grades for the course as a whole (ex: average, etc) in lecture on Wednesday.

  14. It Iter erators Why do we care? 1 DFS dfs(...); 2 for ( ImageTraversal::Iterator it = dfs.begin(); it != dfs.end(); ++it ) { 3 std::cout << (*it) << std::endl; 4 }

  15. Iter It erators Why do we care? 1 DFS dfs(...); 2 for ( ImageTraversal::Iterator it = dfs.begin(); it != dfs.end(); ++it ) { 3 std::cout << (*it) << std::endl; 4 } 1 DFS dfs(...); 2 for ( const Point & p : dfs ) { 3 std::cout << p << std::endl; 4 }

  16. It Iter erators Why do we care? 1 DFS dfs(...); 2 for ( ImageTraversal::Iterator it = dfs.begin(); it != dfs.end(); ++it ) { 3 std::cout << (*it) << std::endl; 4 } 1 DFS dfs(...); 2 for ( const Point & p : dfs ) { 3 std::cout << p << std::endl; 4 } 1 ImageTraversal & traversal = /* ... */; 2 for ( const Point & p : traversal ) { 3 std::cout << p << std::endl; 4 }

  17. Ev Every Data Structure So Far Unsorted Sorted Unsorted Sorted Binary Tree BST AVL Array Array List List Find Insert Remove Traverse

  18. Ra Range-ba base sed d Searche hes Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? Tree construction:

  19. Ra Range-ba base sed d Searche hes Balanced BSTs are useful structures for range-based and nearest-neighbor searches. Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? Ex: 6 11 41 44 3 33 55

  20. Ra Range-ba base sed d Searche hes Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? Ex: 6 11 41 44 3 33 55

  21. Ra Range-ba base sed d Searche hes Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? Tree construction:

  22. Ra Range-ba base sed d Searche hes

  23. Ra Range-ba base sed d Searche hes 33 6 44 3 11 41 55 3 6 11 33 41 44

  24. Ra Range-ba base sed d Searche hes Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? 33 6 44 3 11 41 55 3 6 11 33 41 44

  25. Ra Range-ba base sed d Searche hes 33 6 44 3 11 41 55 3 6 11 33 41 44

  26. Ru Running T Time 33 6 44 3 11 41 55 3 6 11 33 41 44

  27. Ra Range-ba base sed d Searche hes Q: Consider points in 1D: p = {p 1 , p 2 , …, p n }. …what points fall in [11, 42]? Ex: 6 11 41 44 3 33 55

  29. Ra Range-ba base sed d Searche hes Consider points in 2D: p = {p 1 , p 2 , …, p n }. Q: What points are in the rectangle: p 2 [ (x 1 , y 1 ), (x 2 , y 2 ) ]? p 5 p 6 p 1 Q: What is the nearest point to (x 1 , y 1 ) ? p 3 p 4 p 7

  30. Ra Range-ba base sed d Searche hes Consider points in 2D: p = {p 1 , p 2 , …, p n }. Tree construction: p 2 p 5 p 6 p 1 p 3 p 4 p 7

  31. Ra Range-ba base sed d Searche hes p 2 p 5 p 6 p 1 p 7 p 3 p 1 p 2 p 3 p 4 p 5 p 6 p 4 p 7

  32. kD kD-Tr Trees p 2 p 5 p 6 p 1 p 7 p 3 p 1 p 2 p 3 p 4 p 5 p 6 p 4 p 7

  33. kD kD-Tr Trees p 2 p 5 p 6 p 1 p 7 p 3 p 1 p 2 p 3 p 4 p 5 p 6 p 4 p 7

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