AVL Trees. Version of September 6, AVL Trees Version of September 6, / 22

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1 VL Trees Version of September 6, 6 VL Trees Version of September 6, 6 /

2 inary Search Trees x < x > x inary-search-tree property For every node x ll eys in its left subtree are smaller than the ey value in x ll eys in its right subtree are larger than the ey value in x VL Trees Version of September 6, 6 /

3 Height The height of a node in a tree is the number of edges on the longest downward path from the node to a leaf Node height = max(children height) + Leaves: height = Tree height = root height Empty tree: height = Tree operations typically tae O(height) time 3 Question Let n be the size of a binary search tree. How can we eep its height O(log n) under insertion and deletion? VL Trees Version of September 6, 6 3 /

4 alanced inary Search Tree: VL Tree n VL [delson-velsii & Landis, 96] tree is a binary search tree in which for every node in the tree, heights of its left and right subtrees differ by at most non-vl Tree VL Tree 6 The balance factor of a node is the height of its right subtree minus the height of its left subtree. node with balance factor, or is considered balanced. VL Trees Version of September 6, 6 4 /

5 VL Trees with Minimum Number of Nodes Let n h be the minimum number of nodes in an VL tree of height h n = n = n = n + n + = 4 n 3 = n + n + = 7 VL Trees Version of September 6, 6 5 /

6 Height of VL Tree Let n h denote the minimum number of nodes in an VL tree of height h n =, n = (base) n h = n h + n h + For any VL tree of height h and size n, n n h = n h + n h + > n h > 4n h 4 > > i n h i If h is even, let i = h/. n > h/ n = h/ h = O(log n) If h is odd, let i = (h )/. n > (h)/ n = (h)/ h = O(log n) Thus, many operations (e.g., insertion, deletion, and search) on an VL tree will tae O(logn) time VL Trees Version of September 6, 6 6 /

7 VL Trees and Fibonacci Numbers We saw that n h = n h + n h +. Recall Fibonacci numbers satisfy f h = f h + f h. Now compare h n h f h Lemma: n h = f h+ Proof: by induction n h+ = +n h +n h = +f h+ +n h+ = f h+3 Since f h cφ h for golden ratio φ = + 5, this also immediately provides alternative derivation that h = O(log n). VL Trees Version of September 6, 6 7 /

8 Insertion asically follows insertion strategy of binary search tree ut may cause violation of VL tree property Restore the destroyed height balance if needed Insert Restore height balance VL Trees Version of September 6, 6 8 /

9 Insertion: Observation fter an insertion, only nodes that are on the path from the insertion node to the root might have their balance altered ecause only those nodes have their subtrees altered Insert VL Trees Version of September 6, 6 9 /

10 Insertion: Four Cases Let denote the lowest node violating VL tree property Case (Left-Left case) insert into the left subtree of the left child of Case (Left-Right case) insert into the right subtree of the left child of Case 3 (Right-Left case) insert into the left subtree of the right child of Case 4 (Right-Right case) insert into the right subtree of the right child of Cases and 4 are mirror image symmetries with respect to, as are cases and 3 VL Trees Version of September 6, 6 /

11 Insertion: Left-Left Case Right rotation with as the pivot The new subtree rooted at has height +, exactly the same height before the insertion The rest of the tree (if any) that was originally above node always remains balanced + + VL Trees Version of September 6, 6 /

12 Insertion: Right-Right Case Left rotation with as the pivot The new subtree rooted at has height +, exactly the same height before the insertion The rest of the tree (if any) that was originally above node always remains balanced + + VL Trees Version of September 6, 6 /

13 Insertion: Left-Right Case Single rotation fails to fix it Subtree is too tall VL Trees Version of September 6, 6 3 /

14 Left-Right Case: Special Case When subtree, and are empty, =. Insert C: C C C Left rotation and then right rotation with C as the pivot. Done! VL Trees Version of September 6, 6 4 /

15 Left-Right Case: General Case X δ X δ X + 3 δ X + δ VL Trees Version of September 6, 6 5 /

16 Left-Right Case: General Case X δ X δ + 3 X δ X + δ VL Trees Version of September 6, 6 6 /

17 Right-Left Case: Special Case When subtree, and are empty, =. Insert C: C C C Right rotation and then left rotation with C as the pivot. Done! VL Trees Version of September 6, 6 7 /

18 Insertion: Summary Left-Left case: single rotation Right rotation Left-Right case: double rotation Left rotation and then right rotation Right-Left case: double rotation Right rotation and then left rotation Right-Right case: single rotation Left rotation For each insertion, at most two rotations are needed to restore the height balance of the entire tree. Note that in all cases, height of rebalanced subtree is unchanged! This means no further tree modifications are needed. VL Trees Version of September 6, 6 8 /

19 Deletion Delete a node as in ordinary binary search tree If the node is a leaf, remove it If not, replace it with either the largest in its left subtree or the smallest in its right subtree, and remove that node Note: The replacement node has at most one subtree Trace the path from the parent of removed node to root For each node along path, restore the height balance if needed by doing single or double rotations VL Trees Version of September 6, 6 9 /

20 Deletion (continued) For each node along the path, restore the height balance if needed by doing single or double rotations Very similar to insertion with one major caveat In insertion a (single/double) rotation restored balance and ept height of rebalanced subtree unchanged. Only one rotation needed. In deletion, rotation restores balance but final height of rotated subtree might decrease by one. If this occurs, need to continue waling up path towards root, continuing to restoring balance by rotations when necessary. Since path has length O(h) this might require O(h) = O(log n) rotations. Deletion can also be done in O(log n) time. VL Trees Version of September 6, 6 /

21 Deletion Example Diagram below illustrates example in which subtree rooted at has height + 3. n item is deleted from subtree, reducing its height from + to, leading to an imbalance. fter a single rotation, the subtree is now rooted at with no imbalance. ut, has height +. This might cause an imbalance further up the tree, so the algorithm might need to continue waling upwards, correcting that imbalance VL Trees Version of September 6, 6 /

22 Going Further VL trees are one particular type of alanced Search trees, yielding O(log n) behavior for dictionary operations. There are many other types of alanced Search Trees, e.g. red-blac trees -trees (a, b) trees (, 3) and (, 3, 4) trees are special cases treaps (randomized STs) splay Trees (only O(log n) in amortized sense) VL Trees Version of September 6, 6 /