In what order should you insert a set of known keys into a B-Tree to get minimal height?

Question

Given a fixed number of keys or values(stored either in array or in some data structure) and order of b-tree, can we determine the sequence of inserting keys that would gene

Answer 1

This is how I would add elements to b-tree.

Thanks to Steve314, for giving me the start with binary representation,

Given are n elements to add, in order. We have to add it to m-order b-tree. Take their indexes (1...n) and convert it to radix m. The main idea of this insertion is to insert number with highest m-radix bit currently and keep it above the lesser m-radix numbers added in the tree despite splitting of nodes.

1,2,3.. are indexes so you actually insert the numbers they point to.

For example, order-4 tree
      4       8         12           highest radix bit numbers
1,2,3   5,6,7   9,10,11    13,14,15  

Now depending on order median can be:

• order is even -> number of keys are odd -> median is middle (mid median)
• order is odd -> number of keys are even -> left median or right median

The choice of median (left/right) to be promoted will decide the order in which I should insert elements. This has to be fixed for the b-tree.

I add elements to trees in buckets. First I add bucket elements then on completion next bucket in order. Buckets can be easily created if median is known, bucket size is order m.

I take left median for promotion. Choosing bucket for insertion.
    |  4     |  8      |   12       |    
1,2,|3   5,6,|7   9,10,|11    13,14,|15  
        3       2          1             Order to insert buckets.

• For left-median choice I insert buckets to the tree starting from right side, for right median choice I insert buckets from left side. Choosing left-median we insert median first, then elements to left of it first then rest of the numbers in the bucket.

Example

Bucket median first
12,
Add elements to left
11,12,
Then after all elements inserted it looks like,
|   12       | 
|11    13,14,| 
Then I choose the bucket left to it. And repeat the same process.
Median
     12        
8,11    13,14, 
Add elements to left first
       12        
7,8,11    13,14, 
Adding rest
  8      |   12        
7   9,10,|11    13,14, 
Similarly keep adding all the numbers,
  4     |  8      |   12        
3   5,6,|7   9,10,|11    13,14, 
At the end add numbers left out from buckets.
    |  4     |  8      |   12       |   
1,2,|3   5,6,|7   9,10,|11    13,14,|15 

• For mid-median (even order b-trees) you simply insert the median and then all the numbers in the bucket.

• For right-median I add buckets from the left. For elements within the bucket I first insert median then right elements and then left elements.

Here we are adding the highest m-radix numbers, and in the process I added numbers with immediate lesser m-radix bit, making sure the highest m-radix numbers stay at top. Here I have only two levels, for more levels I repeat the same process in descending order of radix bits.

Last case is when remaining elements are of same radix-bit and there is no numbers with lesser radix-bit, then simply insert them and finish the procedure.

I would give an example for 3 levels, but it is too long to show. So please try with other parameters and tell if it works.