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Algorithms on grids

Natasha GelfandGeometry 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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• Computing the grid takes linear time

Grids for points

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Sub-problem• Given a set P and a distance r, verify in

linear time if CP(P)<r or CP(P) > r

– Insert points sequentially

– If CP(P) < r, p, q are in the same orneighboring cells

– Can we search a cellin constant time?

r

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Algorithm• If some cell contains more than 9 points,

then CP(P)<r

• Algorithm:– Insert points into the grid– If cell(p) contains more than

9 points, return CP(P) < r

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Algorithm• If some cell contains more than 9 points,

then CP(P)<r

• Algorithm:– Insert points into the grid– If cell(p) contains more than

9 points, return CP(P) < r

– Otherwise, compute

• Constant time per point,running time O(n)

r

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Closest pair• Permute points P=<p1, p1, … , pn>

• Let ri = CP({p1, …, pi})– Can check if ri<ri-1 in linear time

• Good case: ri = ri-1

– Grid is already built, check in O(1) time

• Bad case: ri < 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 Pr[Xi = 1] = Pr[ri < 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)

k=3

k=4

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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

G

Running time:

T(n) = n + 2T(n/2)Stop at n < k/4

O(nlog(n/k))

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Finite centers• Claim: Dopt(P,k) contains at least one intersection

point of G• Pf: By contradiction:

At most k/2 points

Dopt(P,k)

k/4 points

k/4 pointsk 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)