deletions | additions       diff --git a/Smallest_Triangles_scan_L_to__.md b/Smallest_Triangles_scan_L_to__.md  index 3f09f82..ae67005 100644  --- a/Smallest_Triangles_scan_L_to__.md  +++ b/Smallest_Triangles_scan_L_to__.md 
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# Smallest Triangles  A triangle's area is determined by perpendicular distance to a point. As such, we can throw away the y-coordinates for both the segments and points.  - scan L to R  - we can reduce to a 1D problem  - x- coords and length of thing  if smallest and closest then match  for a and b *Handle the case where coordinates are equal*  if ## Algorithm  elementary intervals thing - \(L \leftarrow S \cup P: O(m+n)\)  - Sort \(L\) by ascending x-coordinate: \(O((m+n) \log (m + n) )\)  - T \(\leftarrow\) Empty List  - LeftPoint \(\leftarrow -\infty\)  - For each element in \(L\): O(m+n)   - If it's an interval, store it in \(T\)  - If it's a point denote it RightPoint:  - For each interval t in T: Amortised \(O(1)\) as we handle each interval exactly once across all loop iterations.  - Compare dist(t,LeftPoint) and dist(t, rightPoint). Emit a tuple of t and the argmin of these two distances: O(1).  - Remove t from the list: \(O(1)\)  ## Runtime  Running time is \(O((m+n) \log (m + n) )\) for the preprocessing and \(O(m+n)\) in the loop for a total of \(O((m+n) \log (m + n) )\).