number of subsets where GCD equals to X

Question

The original statement for this problem can be found here

This is a question from IEEExtream 2014. There is an array of integers given. Input is X, so output is the number of subsets where there GCD equals to X.

Eg:

OriginalSet = {2, 3, 5, 6, 6}
numberOfSubsets(2) => 3 // there are 3 subsets where their GCD equals to 2
i.e. {2, 6}, {2, 6}, {2, 6}

Limits:

OriginalSet.length < pow(10, 5),
element_of_an_array < pow(10, 4),
X < pow(10,4)

I bruteforced the solution and it gives me TLE. I am stuck on this question for few days now. Could you explain me how to solve this efficiently.

Edit

the subset size is not limited to 2. It is from the power-set.
It guarantees that for a given query there is only 100 unique numbers.

I'm sure we had an almost identical question in the last couple of weeks but I can't find it right now. Anyone? — David Richerby, Oct 22 '15 at 11:28
Last week there was a question about calculating the multiplication of the GCD of every possible subset. It was removed — jjohn, Oct 22 '15 at 11:34
@jjohn That sounds like it's the question I was thinking of, yes. Thanks! — David Richerby, Oct 22 '15 at 11:37
Could you please identify constraints, i.e., X and the set size? — beginner1010, Oct 22 '15 at 17:32
What's a "TLE", and why is it bad? What size of parameters? What have you tried? — D.W., Oct 22 '15 at 21:07
TLE: Time limit exceeded. Note this mean that his solution wasn't fast enough to pass all test on the online judge — jjohn, Oct 22 '15 at 21:09
I think there was something with setwise gcd's a few months ago. Solution seemed to involve incrementally building/unioning sets and gcd'ing their gcd's. — KWillets, Oct 22 '15 at 21:11
Found it -- find minimal subset where gcd=1: http://cs.stackexchange.com/questions/10249/finding-the-size-of-the-smallest-subset-with-gcd-1/45539#45539 — KWillets, Oct 22 '15 at 21:24
You can take hint from problem code AMR15B asked in ACM-ICPC 2015 in first round on codechef . — shuboy2014, Apr 20 '16 at 15:31

score 2 · Accepted Answer · answered Oct 24 '15 at 13:42

2

See the solution for "585E - Present for Vitalik the Philatelist" here where is says "Let's calculate the number of subsets with gcd equal to 1". I hope you can extend the logic from there to solve the problem for X.

answered Oct 24 '15 at 13:42

e_noether

1,289
2
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19

score 1 · Answer 2 · edited Apr 13 '17 at 12:48

1

You can use dynamic programming approach to solve this problem; however, you have not yet specified what the limitation on X is. This is too similar to the question Algorithm for finding maximum mutually coprime subset of a multiset of integers In that way, you could also use GCD instead of LCM to solve your problem since the question I gave is a special case of your question.

edited Apr 13 '17 at 12:48

Community

1

answered Oct 23 '15 at 19:00

beginner1010

168
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number of subsets where GCD equals to X

2 Answers2