TheMartian0x48's blog

Bitwise sum
By TheMartian0x48, history, 4 years ago, In English

You are given an integer K.
A function F is defined over non-negative integers as the following:

$$$ F(x) = \sum((x | i) - (x \& i)) $$$

for i such that the given conditions hold true:

$$$ 0 <= i <= x \text{ and } ((x | i) - (x \& i)) = x - i $$$

Determine the number of non-negative integers x such that

$$$ F(x) \leq K $$$

| = bitwise or
& = bitwise and

Example for K = 6, possible x is 0, 1, 2, 3 and 4 as F(0) = 0
F(1) = 1
F(2) = 2
F(3) = 6
F(4) = 4
So, the answer is 5.

Constraints

$$$ 1 \leq T \leq 10^{5} \\ 1 \leq K \leq 10^{18} \\ \text{Time Limit: 5.0 sec(s) for each input file.} \\ \text{Memory Limit: 256 MB} $$$

It was asked in nokia coding hackthon.
I wasn't able to find any pattern to approach any optimal solution. How would you approach this problem?

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4 years ago, # |
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not sure but $$$ x \triangle i = x - i$$$ seems like $$$ i \subseteq x$$$, so I guess now that you know the contribution of each bit, the growth of sum is very fast, that is it will reach to $$$ K $$$ in very less number of operations, so precalculate and then answer for each test case from your precalculated stuff.

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