# Solving Combinatorial Problems with PySpark

Partitioning combinatorial problems using binary representation.Let us consider the problem statement. Given n real numbers x1, x2, xn, choose any set of distinct numbers such that function f on those chosen numbers gives maximum value. function f can take any number of inputs so one can choose any number of numbers.

Let us consider the problem statement. Given n real numbers x1, x2, xn, choose any set of distinct numbers such that function f on those chosen numbers gives maximum value. function f can take any number of inputs so one can choose any number of numbers.

Expected output: A set of numbers S = {xi, xj,…, xm} or the maximum value for f(xi, xj,…, xm).

A brute force strategy for solving this problem would be to permute over the numbers so first, one selects 1 out of n numbers, then 2, then 3, and so on. If we don’t know function f then we wouldn’t know if the order of the input parameters is important or not. Hence we need permutation vs combination. If the order is not important then we can use the combination.

Sum of Combinations The number of k-combinations for all k is the number of subsets of a set of n elements. There are several ways to see that this number is 2^n. Source: https://en.wikipedia.org/wiki/Combination It is easy to see if there are n numbers possible way to select different k numbers for all k is very close to the binary representation of 2^n. An n bit binary vector can represent numbers from 0 to 2^n-1. If each index of this vector represents one of the n numbers, we can select that number if the bit for that index is on. This way we can iterate through numbers from 0 to 2^n-1, get their representation as a binary vector of n bit, and then select the numbers based on which bits are on.

Method get_data converts num into binary representation and based on which bits are on that index from data is returned.

Output: 000 [] 001 [1] 010 [4] 011 [4, 1] 100 [3] 101 [3, 1] 110 [3, 4] 111 [3, 4, 1] As one can see from the above output, a binary string 010 will get the value 4 at index 1 while 011 will get [4, 1] at index 1 and 2. Once we have the values, we can easily calculate the value of function f and eventually maximum.

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