Convex hull smallest convex set containing all the

Convex hull smallest convex set containing all the points

Convex hull smallest convex set containing all the points

Convex hull 3 smallest convex set containing all the points 2 4 1 start = 1 1. next = 2 = 3. prev 2. next = 3 = 4. prev 3. next = 4 = 1. prev 4. next = 1 = 2. prev representation = circular doubly-linked list of points on the boundary of the convex hull

Jarvis march (assume no 3 points colinear) s find the left-most point

Jarvis march (assume no 3 points colinear) s find the point that appears most to the right looking from s

Jarvis march (assume no 3 points colinear) s p find the point that appears most to the right looking from p

Jarvis march (assume no 3 points colinear)

Jarvis march (assume no 3 points colinear)

Jarvis march (assume no 3 points colinear) s point with smallest x-coord p s repeat PRINT(p) q point other than p for i from 1 to n do if i p and point i to the right of line (p, q) then q i p q until p = s

Jarvis march (assume no 3 points colinear) Running time = O(n. h)

Graham scan (assume no 3 points colinear) O(n log n) homework start with a simple polygon containing all the points fix it in time O(n)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear)

Graham scan (assume no 3 points colinear) A start B next(A) C next(B) repeat 2 n times if C is to the right of AB then A. next C; C. prev A B A A prev(A) else A B B C C next(C)

Closest pair of points

Closest pair of points

Closest pair of points 2 T(n/2) min(left, right)

Closest pair of points 2 T(n/2) min(left, right)

Closest pair of points 2 T(n/2) min(left, right)

Closest pair of points pre-processing X sort the points by x-coordinate Y sort the points by y-coordinate Closest-pair(S) if |S|=1 then return if |S|=2 then return the distance of the pair split S into S 1 and S 2 by the X-coord 1 Closest-pair(S 1), 2 Closest-pair(S 2) min( 1, 2) for points x in according to Y check 12 points around x, update if a closer pair found

Smallest enclosing disc

Smallest enclosing disc

Smallest enclosing disc Claim #1: The smallest enclosing disc is unique.

Smallest enclosing disc Claim #1: The smallest enclosing disc is unique.

Smallest enclosing disc SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x)

Smallest enclosing disc SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) SED-with-point(S, y) pick a random point x S (c, r) SED-with-point(S-{x}, y) if x Disc(c, r) then return (c, r) else return SED-with-2 -points(S, y, x)

Smallest enclosing disc SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) SED-with-point(S, y) pick a random point x S (c, r) SED-with-point(S-{x}, y) if x Disc(c, r) then return (c, r) else return SED-with-2 -points(S, y, x) SED-with-2 -point(S, y, z) pick a random point x S (c, r) SED-with-2 -points(S-{x}, y, z) if x Disc(c, r) then return (c, r) else return circle given by x, y, z

Running time ? SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) SED-with-point(S, y) pick a random point x S (c, r) SED-with-point(S-{x}, y) if x Disc(c, r) then return (c, r) else return SED-with-2 -points(S, y, x) SED-with-2 -point(S, y, z) pick a random point x S (c, r) SED-with-2 -points(S-{x}, y, z) if x Disc(c, r) then return (c, r) else return circle given by x, y, z

Running time ? SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) SED-with-point(S, y) pick a random point x S (c, r) SED-with-point(S-{x}, y) if x Disc(c, r) then return (c, r) else return SED-with-2 -points(S, y, x) SED-with-2 -point(S, y, z) pick a random point x S (c, r) SED-with-2 -points(S-{x}, y, z) if x Disc(c, r) then return (c, r) else return circle given by x, y, z O(n)

Running time ? SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) SED-with-point(S, y) pick a random point x S (c, r) SED-with-point(S-{x}, y) if x Disc(c, r) then return (c, r) else return SED-with-2 -points(S, y, x) T(n) = T(n-1) + T(n) = O(n) 2 n SED-with-2 -points O(n)

Running time ? SED(S) pick a random point x S (c, r) SED(S-{x}) if x Disc(c, r) then return (c, r) else return SED-with-point(S, x) T(n) = T(n-1) + T(n) = O(n) 2 n SED-with-point O(n)

Smallest enclosing disc md(I, B) = smallest enclosing disc with B on the boundary and I inside Claim #2: if x is inside md(I, B) then md(I {x}, B) = md(I, B)

Smallest enclosing disc md(I, B) = smallest enclosing disc with B on the boundary and I inside Claim #3: if x is outside of md(I, B) then md(I {x}, B) = md(I, B {x})

Smallest enclosing disc md(I, B) = smallest enclosing disc with B on the boundary and I inside Claim #3: if x is outside of md(I, B) then md(I {x}, B) = md(I, B {x}) x md(I, B) md(l {x}, B)

Smallest enclosing disc md(I, B) = smallest enclosing disc with B on the boundary and I inside Claim #3: if x is outside of md(I, B) then md(I {x}, B) = md(I, B {x}) Claim #2: if x is inside md(I, B) then md(I {x}, B) = md(I, B) Claim #1: md(I, B) is unique