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Here is a nearest neighbor problem.

Given reals $a_1, \ldots, a_n$ (very large $n$!), plus target real $p$, find $a_i$ and $a_j$ whose SUM is closest to $p$. We allow reasonable pre-processing/indexing of $a_1, \ldots, a_n$ (up to $O(n \log n)$), but at query time (given $p$), the result should be returned very fast (e.g., $O(\log n)$ time).

(Simpler example: if we only wanted the SINGLE $a_i$ that is closest to $p$, we would sort $a_1, \ldots, a_n$ offline, $O(n \log n)$, then do binary search at query time, $O(\log n)$).

Solutions that don't work:

1) Sort $a_1, \ldots, a_n$ offline, then at query time, start from both ends & move two pointers inward (http://bit.ly/1eKHHDy). Not good, because of $O(n)$ query time.

2) Sort $a_1, \ldots, a_n$ offline, then at query time, take each $a_i$ and perform binary search for a "buddy" that helps it sum to something close to $p$. Not good, because of $O(n \log n)$ query time.

3) Sort all pairs $(a_{1}, \ldots, a_{n})$ offline, then do binary search. Not good, because of $O(n^2)$ pre-processing.

Thanks!

ps. Further generalizations needed for practice: (1) $a_1, \ldots, a_n$ and $p$ to be 50-dimensional vectors, (2) "close" to be vector cosine distance, and (3) $k$-best closest pairs-that-sum, not just 1-best.

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  • $\begingroup$ Is there a limit on the pre-processing time or the amount of space we can use after pre-processing? If we're limited to $O(n)$ space, do you have any reason to believe it can be solved in, say, $O(\lg n)$ time? That seems awfully unlikely to me. $\endgroup$ – D.W. Aug 17 '13 at 20:13
  • $\begingroup$ Pre-processing is limited to O($n$ log $n$). I updated the problem statement. $\endgroup$ – Kevin Aug 17 '13 at 21:39
  • $\begingroup$ I don't have any reason to believe that querying can be fast, but many useful results for nearest-neighbors (kd-trees, locality-sensitive hashing, etc) seem counter-intuitively good to me. An approximate solution using locality-sensitive hashing would be fine for practical use. $\endgroup$ – Kevin Aug 17 '13 at 22:00
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This is almost certainly impossible.

Suppose you could solve your problem with preprocessing time $P(n)$ and query time $Q(n)$. Then there is a simple algorithm to solve the 3SUM problem—Given a set of $n$ real numbers, do any three elements sum to zero?—in $P(n)+n\cdot Q(n)$ time. We pre-process all the numbers, then for each number $a_k$, we find the value of $a_i+a_j$ that is closest to $-a_k$; if it matches $-a_k$ exactly, we have found a solution to the 3SUM problem.

However, the fastest algorithm known for 3SUM runs in $O(n^2)$ time, and this algorithm is widely conjectured to be optimal. Moreover, there is a matching $\Omega(n^2)$ lower bound in a restricted but natural decision tree model of computation. For sets of integers, there are slightly subquadratic-time algorithms that "play games with bits", but even in the integer RAM model, 3SUM is conjectured to require $\Omega(n^2/\text{polylog}\,n)$ time.

So assuming that conjecture is correct, your problem either requires (near-)quadratic preprocessing time or (near-)linear query time.

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If you can use unlimited space and unlimited time during pre-processing, then the following solution meets your requirements:

  • During pre-processing, compute the set $\{a_i+a_j:1\le i\le j\le n\}$, and store this set in sorted order. This set can be generated and sorted in $O(n^2)$ time, and it takes $O(n^2)$ space to store it.

  • Now to answer a query (to find $a_i,a_j$ where $a_i+a_j$ is as close to $p$ as possible), simply do a binary search in this sorted list. That will take $O(\lg n)$ time.

If this solution is not acceptable, you need to think through your requirements more carefully and edit the question accordingly.

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  • $\begingroup$ Hi, and thank you! But your solution is the same as my solution #3, which is problematic due to O(n^2) pre-processing time. In my case, n is very large (e.g., 1m) and I must avoid O(n^2) operations. $\endgroup$ – Kevin Aug 17 '13 at 20:30

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