Optimizing AVLTree with B-tree

Question

PREMISE

So lately i have been thinking of a problem that is common to databases: trying to optimize insertion, search, deletion and update of data. Usually i have seen that most of the databases nowadays use the BTree or B+Tree to solve such a problem, but they are usually used to store data inside the disk and i wanted to work with in-memory data, so i thought about using the AVLTree (the difference should be minimal because the purpose of the BTrees is kind of the same of the AVLTree but the implementation is different and so are the effects). Before continuing with the reasoning behind this i would like to get in a deeper level of what i am trying to solve. So in a modern database data stored in a table with a PRIMARY KEY which tends to be INDEXED (i am not very experienced in indexing so what i will say is basic reasoning i put into this problem), usually the PRIMARY KEY is an increasing number (even though nowadays is a bad practice) starting from 1. Using normally an AVLTree should be more then enough to solve the problem cause this particular tree is always balanced and offers O(log2(n)) operations, BUT i wanted to reach this on a deeper level trying to optimize it even more then needed.

THEORY

So as the title of the question suggests i am trying to optimize the AVLTree merging it with a Btree. Basically every node of this new Tree is lets say an array of ten elements every node as also the corresponding height in the tree and every element of the array is ordered ascending.

INSERTION

The insertion initally fills the array of the root node when the root node is full it generates the left and right children which also contains an array of 10 elements. Whenever a new node is added the Tree autorebalances the nodes based on the first key of the vectors of the left and right child using also their height (note that this is actually how the AVLTree behaves but the AVLTree only has 2 nodes and no vector just the values).

SEARCH

Searching an element works this way: staring from the root we compare the value we are searching K with the first and last key of the array of the current node if the value is in between, we know that it surely will be in the array of the current node so we can start using a binarySearch with O(log2(n)) complexity into this array of ten elements, otherise we go on the left if the key we are searcing is smaller then the first key or we go to the right if it is bigger.

DELETION

The same of the searching but we delete the value.

UPDATE

The same of the searching but we update the value.

CONCLUSION

If i am not wrong this should have a complexity of O(log10(log2(10))) which is always logarithmic so we shouldn't care about this optimization, but in my opinion this could make the height of the tree so much smaller while providing also quick time on the search.

Answer 1

B tree and B+ tree are indeed used for disk storage because of the block design. But there is no reason why they could not be used also as in-memory data structure.

The advantages of a B tree include its use of arrays inside a single node. Look-up in a limited vector of maybe 10 entries can be very fast.

Your idea of a compromise between B tree and AVL would certainly work, but be aware that:

• You need to perform tree rotations like in AVL in order to keep the tree balanced. In B trees you work with redistributions, merges and splits, but no rotations.
• Like with AVL, the tree will not always be perfectly balanced.
• You need to describe what will be done when a vector is full and a value needs to be added to it: the node will have to split, and one half will have to be reinjected as a leaf.
• You need to describe what will be done when a vector gets a very low fill-factor (due to deletions). If you leave it like that, the tree could degenerate into an AVL tree where every vector only has 1 value, and then the additional vector overhead will make it less efficient than a genuine AVL tree. To keep the fill-factor of a vector above a minimum you cannot easily apply the redistribution mechanism with a sibling node, as would be done in B-trees. It would work with leaf nodes, but not with internal nodes. So this needs to be clarified...
• You need to describe what will be done when a value in a vector is updated. Of course, you would insert it in its sorted position: but if it becomes the first or last value in that vector, this may violate the order with regards to left and right children, and so also there you may need to define more precisely the algorithm.
• Binary search in a vector of 10 may be overkill: a simple left-to-right scan may be faster, as CPUs are optimised to read consecutive memory. This does not impact the time complexity, since we set that the vector size is limited to 10. So we are talking about doing either at most 4 comparisons (3-4 on average depending on binary search implementation) or at most 10 comparisons (5 on average).

If I am not wrong this should have a complexity of O(log10(log2(n))) which is always logarithmic

Actually, if that were true, it would be sub-logarithmic, i.e. O(loglogn). But there is a mistake here. The binary search in a vector is not related to n, but to 10. Also, this work comes in addition to finding the node with that vector. So it is not a logarithm of a logarithm, but a sum of logarithms:

O(log₁₀n + log₂10) = O(log n)

Therefore the time complexity is no different than the one for AVL or B-tree -- provided that the algorithm is completed with the missing details, keeping within the logarithmic complexity.

You should maybe also consider to implement a pure B tree or B+ tree: that way you also benefit from some of the advantages that neither the AVL, nor the in-between structure has:

• The leaves of the tree are all at the same level
• No rotations are needed
• The tree height only changes at one spot: the root.
• B+ trees provide a very fast mean for iterating all values in their order.