Binary search trees do not have a specific element order, but an inorder traversal can always be obtained. The ordering of the keys, which is by default achieved with sorting, determines the order of the elements in the tree.

Question:

Based on multiple sources, it is evident that the implementation of

std::map

involves utilizing a

red-black tree

. It is my understanding that such

data structures

do not arrange their elements in any specific order, but rather focus on maintaining the BST property and meeting height balancing requirements.

How is it possible that map::begin takes constant time, enabling us to iterate over an ordered sequence?

Solution 1:

Assuming that

std::map

operates on an internally maintained Binary Search Tree (BST), although it is not strictly mandated by the standard, it is likely that most libraries utilize this data structure, such as a

red-black

tree.

In a binary search tree (BST), the process of finding the smallest element involves traversing the left branches until reaching a leaf, which has a time complexity of O(log(N)). Nevertheless, if the aim is to provide the “begin()” iterator in constant time, one can easily achieve this by internally keeping track of the smallest element. Whenever an insertion results in a change to the smallest element, it can be updated accordingly. This approach incurs memory overhead, but it is a trade-off that can be made.

There may be alternative methods to identify the smallest element (such as intentionally keeping the root node imbalanced). Regardless of the approach, it is not a difficult task.

To traverse the “ordered” sequence, you can achieve this by performing an in-order traversal of the tree. Commencing from the leftmost leaf node, follow the pattern of going up, then right, then further up, then right, and so on. These rules are straightforward and can be easily implemented. Take a look at a previous implementation of a basic BST inorder iterator that i wrote some time ago. By conducting the in-order traversal, you will systematically visit each node in the correct order, from the smallest to the largest. Essentially, it creates the illusion that the “array” is sorted, when in fact it is the traversal that gives this appearance of being sorted.

Solution 2:

A red-

black tree

tree possesses the ability to insert a node at any location in the tree with a cost of O(log N). In typical

std::map

implementations, the container retains the

tree sorted

and ensures that when a new node is inserted, it is placed in the correct location to maintain the

tree sort

and then rebalances the tree to preserve the red-black property.

Therefore, it can be concluded that

red-black trees

do not possess an innate sorting order.

Solution 3:

RB trees are a type of self-balancing binary search trees. Unlike binary search trees, they don’t have a specific order for storing their elements, but you can always obtain an inorder traversal. I’m uncertain about how map::begin achieves constant time, but I presume it involves keeping track of the path to the smallest element (which would typically have a time complexity of O(log(n))).