Algorithms on grids

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Algorithms on grids

• Natasha Gelfand

Geometry Seminar Fall 2006
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Grids in computer science
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Grids in computer science

• Graphics and vision
• Collision detection, point location

• Simulation

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Grids for geometric proximity

• Isolate and localize interesting events
• Proximity is local
• Uniform grids
• Closest pair, k-Minimum enclosing disk
• Adaptive grids
• quadtrees

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Closest pair

• Given n points in the plane
• Return pair of pointsrealizing

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Grids for points

• Computing the grid takes linear time

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Sub-problem

• Given a set P and a distance r, verify in linear
  time if CP(P)ltr or CP(P) gt r
• Insert points sequentially
• If CP(P) lt r, p, q are in the same
  orneighboring cells
• Can we search a cellin constant time?

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Algorithm

• If some cell contains more than 9 points, then
  CP(P)ltr
• Algorithm
• Insert points into the grid
• If cell(p) contains more than 9 points, return
  CP(P) lt r

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Algorithm

• If some cell contains more than 9 points, then
  CP(P)ltr
• Algorithm
• Insert points into the grid
• If cell(p) contains more than 9 points, return
  CP(P) lt r
• Otherwise, compute
• Constant time per point,running time O(n)

r
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Closest pair

• Permute points Pltp1, p1, , pngt
• Let ri CP(p1, , pi)
• Can check if riltri-1 in linear time
• Good case ri ri-1
• Grid is already built, check in O(1) time
• Bad case ri lt ri-1
• Rebuild grid, O(i) time
• Trivial bound O(nk), when closest pair changes k
  times

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Analysis

• Let Xi 1 if ri ri-1, and 0 otherwise
• Running time

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Analysis

• Bound PrXi 1 Prri lt ri-1
• Likelihood that pi realizes CP(Pi)
• Expected running time

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k-Enclosing minimum disk

• Disk of minimum radius that contains k points
• Brute force O(nk)
• 2-Opt algorithm r(P,k) 2ropt(P,k)

k4
k3
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Non-uniform grid

• Partition P into horizontal strips with at most
  k/4 points in each strip
• Recursive median partitioning
• O(n/k) strips

Running time T(n) n 2T(n/2) Stop at n lt
k/4 O(nlog(n/k))
G
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Finite centers

• Claim Dopt(P,k) contains at least one
  intersection point of G
• Pf By contradiction

k/4 points
k/4 points
Dopt(P,k)
k points
At most k/2 points
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Algorithm

• For each grid intersection point g2G
• Compute smallest circle centered at p with k
  points
• k-th order statistic of p, g
• Expected time O(n)
• Return the best of (n/k)2 candidates
• Running time O(n(n/k)2)

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Correctness

• 2-Opt r(P,k) 2ropt(P,k)