hossainzarif's blog

A problem(inspired from round 707 div1-A)
By hossainzarif, history, 4 years ago, In English

In 1500A - Going Home, how can we construct a sequence of $$$n$$$ integers such that the answer is NO.

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4 years ago, # |
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We can construct an increasing sequence such that each element is strictly greater than sum of previous numbers . But that will increase exponentially.

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    4 years ago, # ^ |
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    Well, that won't fulfill the constraint as $$$a_i <= 2.5*10^6$$$ needs to satisfy

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      4 years ago, # ^ |
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      largest $$$n$$$ value test case in problem with answer "NO" was $$$1572$$$ . $$$n$$$ in worst case could be $$$2*10^5$$$ . So at an average numbers are around $$$12.5$$$ distance apart (for worst case $$$n$$$) which increases difficulty of finding such sequence satisfying the constraint.

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4 years ago, # |
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Not completely sure but I don't think we can, it is guaranteed to have an answer if n is greater than some limit (you can calculate that), because of the pigeon hole principle.

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    4 years ago, # ^ |
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    Yeah, I know. But, for N <= 1572, we can have NO. And we need to find a sequence having the answer No for those N

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4 years ago, # |
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The sequence of alternate prime nos. (2, 5, 11, 17...) works! Edit: Changed 19 to 17

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4 years ago, # |
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You're looking for a Golomb ruler. There are many ways of constructing such a ruler (apart from the one mentioned in the link I just gave), for instance:

  1. http://www.cs.toronto.edu/~apostol/golomb/presentation-en-combinatorics.pdf
  2. https://www.researchgate.net/publication/239285940_A_review_of_the_available_construction_methods_for_Golomb_rulers
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4 years ago, # |
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n = 4. a = {1, 10, 100, 1000} Clearly works.

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    4 years ago, # ^ |
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    I mean, satisfying the problem constraints($$$a_i <= 2.5*10^6$$$)is needed

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