Editorial for ICPC NWERC 2014 F - Finding Lines

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Given $n$ $n$ points in $\mathbb R^2$ $R^{2}$ , is there a straight line passing through at least a $p$ $p$ fraction of them?

If line exists it is uniquely determined by any two of its points.
Probability that a random point lies on the line is $\ge p$ $\geq p$ .
Repeat $250$ $250$ times:
1. Pick two distinct points uniformly at random
2. Check if line defined by the points has enough points.
$Pr[\text{false negative}] \approx (1-p^2)^{250} \le (1-\frac{1}{25})^{250} < 4 \times 10^{-5}$ $P r [false negative] \approx (1 - p^{2})^{250} \leq (1 - \frac{1}{25})^{250} < 4 \times 10^{- 5}$ (cheated slightly – it's a bit worse)

Many other ways e.g. divide and conquer

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