Half%20Plane%20Intersections

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Half Plane Intersections

• Let H h1, h2, , hn be the set of linear
  constraints in two variables
• aix biy ? ci where ai, bi, and ci are
  constants such that at least one of ai bi is
  non-zero
• Geometrically each constraint is closed half
  plane bounded by line aix biy ci

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Half Plane Intersections

• Problem find all points (x,y) ? ?2 that satisfy
  constraints, that is, common intersection of n
  half planes
• What does this intersection look like?
• Each half plane is convex, and intersection
  convex sets is convex
• Boundary comes from lines bounding half planes ?
  segments

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Half Plane Intersections

• Examples of intersections
• bounded unbounded
  empty

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Algorithm for Intersect Half Plane (H)

• input a set of half planes H
• output convex polygonal region C ? h
• if H 1
• then Ch (input half plane)
• else 1. Split H into sets h1 h2 of size ?n/2?
  ?n/2?
• 2. C1 Intersect half planes (H1)
• 3. C2 Intersect half planes (H2)
• 4. C Intersect convex regions (C1, C2)
• end if
• end algorithm

h ? H
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Algorithm for Intersect Half Plane (H)

• Fact intersect convex regions (C1, C2) runs in
  time O(C1 C2)
• Running Time
• T(n) 2T(n/2) O(n) O(n log n)

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Algorithm for Intersect Half Plane (H)

• This is optimal time for computing intersection.
  To prove the lower bound we show that
• Sorting ?O(n) Common intersection of
  half-planes.
• Given n real numbers x1,..., xn
• Let Hi y ? 2xix xi2
• Once P H1?H2 ?...?Hn is formed, we may read
  off the xi's in
• sorted order by reading the slope of
  successive edges of P.
• But
• For many problems we dont need entire
  intersection, we just need one point in the
  intersection
• Is this any easier? We shall soon find out