AVL/Splay Trees and Amortized analysis¶
约 350 个字 预计阅读时间 1 分钟
BST¶
The height of BST can be as bad as \(O(N)\)
AVL Trees¶
- Balance factor \(h_L - h_R \in \{-1, 0, 1\}\)
- Trouble finder & Trouble maker (important)
- Single rotation (LL/RR rotation)
- Double rotation (LR/RL rotation) (Parent first & Grandparent next !!!)
- Height (BF) of the trouble finder doesn't change after insert+rotation
- Maximum height of trees? The minimum nodes of height k is \(n_k\), then we have \(\mathbf{n_k = n_{k-1} + n_{k-2} + 1} = F_{k+2} - 1\). The nodes is exponential to the height.
- Height of empty tree = \(-1\)
Splay Trees¶
- Zig-Zag (Double Rotation)
- Zig-Zig (Single Rotation on GrandParent & Single Rotation on Parent)
| Tree | LL/RR | LR/RL |
|---|---|---|
| AVL Tree | Single Rotation | Rotate Parent, Rotate Grandparent |
| Splay Tree | Rotate Grandparent, Rotate parent | Rotate Parent, Rotate Grandparent |
Amortized Analysis¶
Average-case bound \(\leq\) Amortized bound \(\leq\) Worst case bound
- Aggregate analysis
- Accounting method (different operation can have different amortized cost)
-
Potential method
Potential function \(\Phi(D)\)
\(\Phi_0(D)\) must be the minimum
\(\tilde c_i = c_i + \Phi(D_i) - \Phi(D_{i-1})\)
\(\tilde c\) is the amortized cost and \(c\) is the real cost of an operation.
If \(c_i\) is large, it means we pay in advance in this operation, so the potential should decrease to balance the amortized cost (\(\Phi(D_i) \le \Phi(D_{i-1})\)). (Potential function design principal)
If \(c_i\) is large, we needs to borrow money from \(\Phi\), which will be paid later by the low-cost \(c_i\)
- For Splay Trees, \(\Phi = \sum_{i \in T} log(S(i))\), where \(S(i)\) is the rank of node i. (Size of the subtree \(i\))
| Data Structure | Potential function | Time complexity |
|---|---|---|
| Splay Trees | \(\Phi = \sum_{i \in T} log(S(i))\), where \(S(i)\) is the rank of node i. (Size of the subtree \(i\)) | \(O(logn)\) |
| Skew Heaps | \(\Phi = \) number of heavy nodes. Heavy: size of \(t_r \geq \frac{1}{2} t_{root}\) | \(O(logn)\) |
| Binomial Heap (Insertion) | \(\Phi = \) number of heap ordered trees | \(O(1)\) |
| Fibonacci Heap | \(\Phi =T(H) + 2m(H)\) | \(O(1)\) insertion, \(O(logn)\) deletion |