ENIGMA09 - Editorial

editorial
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PROBLEM LINK:

Bots

Author and Editorialist: Shubham Chauhan

DIFFICULTY:

Medium

PREREQUISITES:

Modulo Arithmetic , Tree

EXPLANATION:

Problem Naturally can be transformed to a more formal way : How many vertices will a trie contain if we add all possible strings with length 2 * N with equal number of zeroes and ones to it.
So first of all, it is obvious that upper half of this tree would be a full binary tree. Lets take a look on N = 3: level (0 - 1) vertex, level (1 - 2) vertices , level (2 - 4) vertices, level (3 - 8) vertices.

Starting from Nth level not every vertex will duplicate: only those that haven’t spent their 0’s or 1’s will.

So here is how to calculate how may vertices will be there on level i + 1 :

  1. Lets assign Number of non duplicating vertices from level to PD(i) .

  2. Count(i + 1) = PD(i) + (Count(i) - PD(i))*2.

  3. And PD can be calculated pretty easily with binomial coefficients: PD(i) = 2 * C(i , N)

  4. Everything else is implementation techniques: inverse modulo arithmetics + some fast way to calculate C(i , N) and sum counts

SOLUTIONS:

To be Update Soon