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RB-Tree and B+ Tree

约 143 个字

Ex.

Suppose: \(bh(Tree) < h(Tree) / 2\).

We choose the longest simple path down to leaf. Then the length of the path is \(h(Tree)\).

The number of red nodes is \(n(red) = h(Tree) - bh(Tree) \geq bh(Tree) + 1\), which is greater than the number of black nodes.

Plus the root is black, there are only \(bh(Tree)\) number of spaces for red nodes. There must be two continuous red nodes. This violates the rule (No continuous red nodes along a path).

Therefore, \(bh(Tree) \geq h(Tree) / 2\)

\(bh(Tree) = bh(child) + 1 \ge h(child)/2 + 1 = (h(Tree) - 1)/2 + 1 = h(Tree)/2 + 1/2\)

Some Corollaries

  • In every simple path, red nodes \(\leq\) black nodes.
  • Minimum RB tree: All blacks
  • Maximum RB tree: Red nodes between all black nodes
  • \(H \leq 2*bh\)

  • A RB tree with \(N\) internal nodes has \(H \leq 2ln(N+1)\)

    \(2N \geq 2^{bh} - 1, H/2 \leq bh \leq log(2N+1), \)

RB-Tree Insertion

Always insert a red

Check uncle.

Same color: push up

Else: change(P, GP) Rotate

RB-Tree Deletion

If we delete a red node, everything is fine.

If we delete a black node, we temporarily add this black property to node \(x\) (which occupies the deleted node's position, if null, we can preserve the deleted node of convenience, and delete it after maintain the r-b property), Then node \(x\) is double-black.

There are two ways to eliminate double-black:

  • Change the color of a black node to red, on the branch not containing \(x\), So \(bh( Path(\not x) ) = bh(Path(x)) - 1\). Double black is eliminated. (Case 2)
  • Change the color of a red node to black, on the branch not containing \(x\), and rotate it the branch containing \(x\). So \(bh( Path(x) ) = bh(Path(\not x)) + 1\) Double black is eliminated. (Case 4)
  • Case 3 -> Case 4, Case1 -> Case ⅔/4

Cheat sheet:

Make sibling black. (Case 1) F&S ex color, rotate (Why ex? Make new root black. In case new root causes new error)

Nephew all black. (Case 2) Make sibling red. Check parent.

Near Nephew red. (Case 3) N&S ex color. rotate

Far Nephew red. (Case 4) Far nephew black. F&S ex color. rotate

AVL RB tree
Insertion \(\leq2\) \(\leq2\)
Deletion \(O(logn)\) \(\leq3\)

B+ Tree

  • The root is either a leaf or has between 2 and M children
  • All nonleaf nodes (except the root) have between \(\lceil M/2 \rceil\) and \(M\) children.
  • Leafs has between \(\lceil M/2 \rceil\) and \(M\) data!
  • image-20240305112407332
  • Depth = \(log_{\lceil M/2\rceil}(N)\)
  • Find = \(log_{\lceil M/2\rceil}(N) * log_2M = logN\)
  • Insert = \(log_{\lceil M/2\rceil}(N) * M = (M/logM)*logN\)