AVL: Balanced Search Tree

A presentation I created for CS 513 Fall 2012

Transcript

1. AVL: BALANCED SEARCH TREE Jim Counts | CSUSM Fall 2012 | count009@cougars.csusm.edu
2. BINARY SEARCH TREES CAN BECOME UNBALANCED ► We all love binary search trees ► Search, Insert, Delete are all in Θ(log ) ► But only on average ► Worst tree is completely unbalanced ► Unbalanced tree is the same as an array ► Θ() 0 1 2 0 1 2
3. SELF-BALANCING TREES ► We want to preserve the good properties of a Binary Search Tree ► Efficiency ► Sorted Elements ► But Avoid Degeneration ► AVL Tree is one such implementation
4. WHAT IS AN AVL TREE? ► Invented by these guys: AdelsonVelskii, M., & Landis, E. M. (1963). An algorithm for the organization of information. Defense Technical Information Center. ► AVL is a binary search tree ► Each node includes a “balance factor” ► Height balanced tree
5. AN AVL • Balance Factor is the difference in heights of the sub trees • Empty tree = -1 • -1, 0, or +1
6. NOT AN AVL This tree is a valid binary search tree, but it is not an AVL. The root does not have a valid value 2 ∉ {−1,0,1}
7. UNBALANCED TREES ARE BALANCED THROUGH ROTATION
8. R-ROTATION
9. L-ROTATION
10. LR-ROTATION
11. RL-ROTATION
12. WORKED EXAMPLE 5(0) 5 (-1) 6(0)
13. WORKED EXAMPLE 5 (-2) 6(-1) 8(0) 5 (0) 6(0) 8(0)
14. EFFICIENCY DEPENDS ON HEIGHT ► And Height depends on : ► log2 ≤ ℎ < 1.4405 log2 ( + 2) − 1.3277 ► Something to do with Fibonacci and the golden ratio… ► This inequality suggested that search, insert and delete* operations would be in Θ(log ) ► Exact formula is unknown. ► Experiments show an approximation to 1.01 log2 + 0.1 ► So, on average, efficiency is comparable to BST
15. DISCUSSION ►Are there any down sides? ►Why don’t we all just use AVL trees instead of plain old BST?