3.1 Height-Balanced Trees 3.2 Weight-Balanced Trees

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Presentation transcript:

3.1 Height-Balanced Trees 3.2 Weight-Balanced Trees Chen Song

Height-Balanced Trees

Height-Balanced Trees Height – The maximum length of any path from the root to a leaf. Height-Balanced Tree – In each interior node, the height of the right subtree and left subtree differ by at most one.

Height-Balanced Trees Example:

Height-Balanced Trees Theorem A height-balanced tree of height h has at least ( 3+ 5 2 5 ) ( 1+ 5 2 ) ℎ − ( 3− 5 2 5 ) ( 1− 5 2 ) ℎ leaves.

Height-Balanced Trees Tree T Tree T has at least leaves: leaves(h)=leaves(h-1)+leaves(h-2) leaves(0)=1 leaves(1)=2 Characteristic equation: xh=xh-1+xh-2 => x2-x-1=0

Height-Balanced Trees Insert and Delete After insert or delete: Case 1: Tree is still balanced Case 2: Tree is not balanced at node n |n->left-height − n->right-height|=2

Height-Balanced Trees 1. n->left->height = n->right->height+2 and n->left->left->height=n->right->height+1 Right rotation on node n 2. n->left->height = n->right->height+2 and n->left->left->height=n->right-height Left rotation on node n->left, and follow right rotation on node n

Height-Balanced Trees 3. n-> right->height = n->left->height+2 and n->right->right->height=n->left->height+1 Left rotation on node n->left 4. n->right->height = n->left->height+2 and n->right->right->height=n->left-height Right rotation on node n->right, and follow left rotation on node n

Height-Balanced Trees Height-Balanced Tree structure supports search, insert, and delete in O(logn) time Search => O(logn) Insert => search + insert + rebalance => O(logn) O(logn) O(1) O(logn) Delete => search + delete + rebalance => O(logn) O(logn) O(1) O(logn)

Weight-Balanced Trees

Weight-Balanced Trees Weight – The number of leaves of a tree. Weight-Balanced Tree – The weight of the right and left subtree in each node differ by at most one.

Weight-Balanced Trees For each subtree, the left and right sub-subtrees has each at least a fraction of α of total weight of the subtree. αWT≤WT1≤(1-α)WT αWT≤WT2≤(1-α)WT Tree T: T1 T2

Weight-Balanced Trees Theorem An α-weight-balanced tree of height h≥2 has at least ( 1 1−α ) ℎ leaves.

Weight-Balanced Trees Rebalance n is current node α∈[ 2 7 , 1− 1 2 ] Case 1: n->left->weight ≥ α*n->weight and n->right->weight ≥ α*n->weight No rebalancing

Weight-Balanced Trees Case 2: n->right->weight ≥ α*n->weight If n->left->left->weight > (α+ε)n->weight, do right rotation on node n Else left rotation on n->left, followed right rotation on node n. ε ≤ α2-2α+ 1 2

Weight-Balanced Trees Case 3: n->left->weight ≥ α*n->weight If n->right->right->weight > (α+ε)n->weight, do left rotation on node n Else right rotation on n->right, followed left rotation on node n.

Weight-Balanced Trees Theorem The weight-balanced tree structure supports search, insert, and delete in O(logn) time. Search => O(logn) Insert => search + insert + rebalance => O(logn) O(logn) O(1) O(logn) Delete => search + delete + rebalance => O(logn) O(logn) O(1) O(logn)

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