问题
i know how to build it with n insertions ( each with O(log(n)) efficiency ) (n*log(n)) overall ,i also know that the equivalent structure of 2-3-4 tree can also be build with linear time from sorted array. can anyone please provide a simple explanation about the red-black version?
回答1:
No matter what kind of BST you are going to build. The algorithm will be the same. Only need to build balanced binary tree.
- Place middle element to the current position
- Place [begin; middle) elements to the left subtree.
- Place (middle; end) elements to the right subtree.
This is O(N) algorithm. It can be shown, that the result tree will be balanced.
We have balanced tree, so by definition:
length(longest path) - length(shortest path) <= 1
So you need to mark all nodes as black, except the deepest nodes in the tree (mark them as red).
回答2:
A complete binary tree of height H has 2^H-1 nodes.
To make a red black tree from a sorted list:
- Remove every second item from the list until you have 2^H-1 items remaining for some H. Note that you will always have enough.
- Build a complete tree out of the remaining items, all black.
- Now attach all the items you removed to the tree. Each item will be a red node, attached to whichever of the black nodes around its proper position is a leaf.
The easiest way to do step (3) is just to do a pre-order traversal of the tree, attaching new red nodes to every black leaf until you run out of items.
NOTE: Sasha's algorithm also works, but this one obviously works.
回答3:
From a functional data structure perspective: there is a paper for Constructing Red-Black Trees, which discovered the pattern of continuous insertion and related it to 1-2 number system.
It's a fun read.
来源：https://stackoverflow.com/questions/34970804/building-red-black-tree-from-sorted-array-in-linear-time