algorithm - How to find minimal and maximal angle determined by three points selected from n points? -


given n points in 2-d plane, (0,0),(1,1), ... can select 3 points them construct angle. example, choose a(0, 0), b(1, 1), c(1, 0), angle abc = 45 degree, acb = 90 degree , cab = 45 degree.

my question how calculate max or min angle determined 3 points selected n points.

obviously, can use brute-force algorithm - calculate angels , find maximal , minimal value, using law of cosines calculate angles , pythagorean theorem calculate distances. efficient algorithm exist?

if i'm correct, brute-force runs in o(n^3): take every triplet, compute 3 angles, , store overall max.

you can improve o(n^2 * log(n)) it's trickier:

best_angle = 0 each point p1:     each point p2:         compute vector (p1, p2), , signed angle makes x-axis         store in  array     sort array    # o(n * log(n))     # traverse find best choice:     each alpha in a:          element beta in closest alpha+pi           # takes o(log n) because it's sorted. don't forget take account fact represents circle: both ends touch...         best_angle = max(best_angle, abs(beta - alpha))

the complexity o(n * (n + nlog(n) + n * log(n))) = o(n^2 * log(n))

of course can retrieve pt1, pt2 obtained best angle during loops.

there still better, feels doing work overall, if re-use previous computations of pt1 pt2, ..., ptn ...


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