Why is std::map implemented as a red-black tree?

Question

Probably the two most common self balancing tree algorithms are Red-Black trees and AVL trees. To balance the tree after an insertion/update both algorithms use the notion of rotations where the nodes of the tree are rotated to perform the re-balancing.

While in both algorithms the insert/delete operations are O(log n), in the case of Red-Black tree re-balancing rotation is an O(1) operation while with AVL this is a O(log n) operation, making the Red-Black tree more efficient in this aspect of the re-balancing stage and one of the possible reasons that it is more commonly used.

Red-Black trees are used in most collection libraries, including the offerings from Java and Microsoft .NET Framework.

It really depends on the usage. AVL tree usually has more rotations of rebalancing. So if your application doesn't have too many insertion and deletion operations, but weights heavily on searching, then AVL tree probably is a good choice.

std::map uses Red-Black tree as it gets a reasonable trade-off between the speed of node insertion/deletion and searching.

The previous answers only address tree alternatives and red black probably only remains for historical reasons.

Why not a hash table?

A type only requires < operator (comparison) to be used as a key in a tree. However, hash tables require that each key type has a hash function defined. Keeping type requirements to a minimum is very important for generic programming so you can use it with a wide variety of types and algorithms.

Designing a good hash table requires intimate knowledge of the context it which it will be used. Should it use open addressing, or linked chaining? What levels of load should it accept before resizing? Should it use an expensive hash that avoids collisions, or one that is rough and fast?

Since the STL can't anticipate which is the best choice for your application, the default needs to be more flexible. Trees "just work" and scale nicely.

(C++11 did add hash tables with unordered_map. You can see from the documentation it requires setting policies to configure many of these options.)

What about other trees?

Red Black trees offer fast lookup and are self balancing, unlike BSTs. Another user pointed out its advantages over the self-balancing AVL tree.

Alexander Stepanov (The creator of STL) said that he would use a B* Tree instead of a Red-Black tree if he wrote std::map again, because it is more friendly for modern memory caches.

One of the biggest changes since then has been the growth of caches. Cache misses are very costly, so locality of reference is much more important now. Node-based data structures, which have low locality of reference, make much less sense. If I were designing STL today, I would have a different set of containers. For example, an in-memory B*-tree is a far better choice than a red-black tree for implementing an associative container. - Alexander Stepanov

Should maps always use trees?

Another possible maps implementation would be a sorted vector (insertion sort) and binary search. This would work well for containers which aren't modified often but are queried frequently. I often do this in C as qsort and bsearch are built in.

Do I even need to use map?

Cache considerations mean it rarely makes sense to use std::list or std::deque over std:vector even for those situations we were taught in school (such as removing an element from the middle of the list). Applying that same reasoning, using a for loop to linear search a list is often more efficient and cleaner than building a map for a few lookups.

Of course choosing a readable container is usually more important than performance.

AVL trees have a maximum height of 1.44logn, while RB trees have a maximum of 2logn. Inserting an element in a AVL may imply a rebalance at one point in the tree. The rebalancing finishes the insertion. After insertion of a new leaf, updating the ancestors of that leaf has to be done up to the root, or up to a point where the two subtrees are of equal depth. The probability of having to update k nodes is 1/3^k. Rebalancing is O(1). Removing an element may imply more than one rebalancing (up to half the depth of the tree).

RB-trees are B-trees of order 4 represented as binary search trees. A 4-node in the B-tree results in two levels in the equivalent BST. In the worst case, all the nodes of the tree are 2-nodes, with only one chain of 3-nodes down to a leaf. That leaf will be at a distance of 2logn from the root.

Going down from the root to the insertion point, one has to change 4-nodes into 2-nodes, to make sure any insertion will not saturate a leaf. Coming back from the insertion, all these nodes have to be analysed to make sure they correctly represent 4-nodes. This can also be done going down in the tree. The global cost will be the same. There is no free lunch! Removing an element from the tree is of the same order.

All these trees require that nodes carry information on height, weight, color, etc. Only Splay trees are free from such additional info. But most people are afraid of Splay trees, because of the ramdomness of their structure!

Finally, trees can also carry weight information in the nodes, permitting weight balancing. Various schemes can be applied. One should rebalance when a subtree contains more than 3 times the number of elements of the other subtree. Rebalancing is again done either throuh a single or double rotation. This means a worst case of 2.4logn. One can get away with 2 times instead of 3, a much better ratio, but it may mean leaving a little less thant 1% of the subtrees unbalanced here and there. Tricky!

Which type of tree is the best? AVL for sure. They are the simplest to code, and have their worst height nearest to logn. For a tree of 1000000 elements, an AVL will be at most of height 29, a RB 40, and a weight balanced 36 or 50 depending on the ratio.

There are a lot of other variables: randomness, ratio of adds, deletes, searches, etc.