Search Balanced Trees
Binary search trees keep keys ordered, with efficient lookup Insert, Delete, Find, all are O(log n) in average case Worst case is bad Compared to hashing? Advantages? Balanced trees are guaranteed O(log n) in the worst case Fundamental operation is a rotation: keep tree roughly balanced AVL tree was the first one, still studied since simple conceptually Red-Black tree uses rotations, harder to code, better in practice B-trees are used when data is stored on disk rather than in memory Splay trees rebalance sometimes, amortized good performance, simple (relatively) to implement
9.1
CPS 100
Rotations and balanced trees
Height-balanced trees For every node, left and right subtree heights differ by at most 1 After insertion/deletion need to rebalance Every operation leaves tree in a balanced state: invariant property of tree Find deepest node thats unbalanced On path from root to inserted/deleted node Rebalance at this unbalanced point only
Are these trees heightbalanced?
CPS 100
9.2
Tree Balancing example
Tree with AVL property What happens if we insert a node with a value of 1?
12 8 4 2 6 10 14 16
Need to rotate tree to maintain AVL property
12 4 2 1 6 8 10 14 16
Which nodes in the tree are unbalanced?
What would have happened if we inserted a node with a value of 5? What are the four ways an insertion can unbalance a tree?
9.3
CPS 100
Rotation to rebalance
N N
C A B
A B C
When a node N is unbalanced height differs by 2 (must be more than one) Change N->left->left
doLeft
Change N->left->right
doLeftRight
Change N->right->left Tree * doLeft(Tree * root) doRightLeft { Tree * newRoot = root->left; Change N->right->right root->left newRoot->right; = doRight newRoot->right = root; First/last cases are symmetric return newRoot; Middle cases require two rotations } First of the two puts tree into doLeft or doRight
CPS 100
9.4
Rotation to rebalance
N
?????
N
C A B
A B C
Suppose we add a new node in right subtree of left child of root Single rotation cant fix Need to rotate twice First stage is shown at bottom Rotate blue node right This is left child of unbalanced
C A B1 B2 A
B1
B2
Tree * doRight(Tree * root) { Tree * newRoot = root->right; root->right = newRoot->left; newRoot->left = root; return newRoot; }
9.5
CPS 100
Double rotation complete
Calculate where to rotate and what case, do the rotations
Tree * doRight(Tree * root) { Tree * newRoot = root->right; root->right = newRoot->left; newRoot->left = root; return newRoot; } Tree * doLeft(Tree * root) { Tree * newRoot = root->left; root->left = newRoot->right; newRoot->right = root; return newRoot; }
C A B1 B2 A B1
B2
C
B1 A
B2 C
CPS 100...