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POCounting unique sets?
 

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  1. POCounting unique sets?
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    <p>I was solving this problem :<a href="http://uva.onlinejudge.org/index.php?option=com_onlinejudge&amp;Itemid=8&amp;category=286&amp;page=show_problem&amp;problem=3268" rel="nofollow">http://uva.onlinejudge.org/index.php?option=com_onlinejudge&amp;Itemid=8&amp;category=286&amp;page=show_problem&amp;problem=3268</a><br> and I am stuck and can't find any hints.<br> The question:</p> <pre><code> You will be given an integer n ( n&lt;=10^9 ) now you have to tell how many distinct sets of integers are there such that each number from 1 to n can be generated uniquely from a set. Also sum of set should be n. eg for n=5 , one such set is: {1,2,2} as 1 can be generated only by { 1 } 2 by { 2 } 3 by {1,2} ( note the two 2's are indistinguishable) 4 by {2,2} 5 by {1,2,2} for generating a number each number of a set can be used only once. ie for above set we can't do {1,1} to generate 2 as only one 1 is there. Also the set {1,2,2} is equivalent to {2,1,2} ie sets are unordered. </code></pre> <p>My approach:</p> <pre><code> The conclusion I came to was. Let F(S,k) denote number desired sets of sum S whose largest element is k.Then to construct a valid set we can take two paths from this state.Either to F(S+k,k) or to F(2*S+1,S+1).I keep a count of how many times I come to state where S=n(the desired sum) and do not go further if S becomes &gt; n.This is clearly bruteforce which I just wrote to see if my logic was correct(which is correct) .But this will give time limit exceed . How do I improve my approach??I have a feeling it is done by dp/memoization. </code></pre>
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    1. USsasha sami
    1. USsasha sami
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    1. COAs you state the problem, the answer is always "infinitely many". The problem is that you've omitted one of the conditions that was put on the original problem of your source: all weights have to add up to `n`. If you omit that condition, you can always add weights to your set that are larger than `n` - for example, for `n` = 5, the sets `{1,2,2,6}`, `{1,2,2,7}`, `{1,2,2,8}` etc. all work as well. A set that always works is taking `n` copies of the element `1`, so we always have a base case we can extend from.
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      1. POCounting unique sets?
      1. USErik P.
    2. CO(Note that this problem does not actually deal with sets, but rather with objects traditionally known as a multiset or bag - a set traditionally does not allow for multiple copies of a single element.)
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    3. COthks I made the edit
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      1. USsasha sami
 

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