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Computing Covers of Plane Forests Luis Barba Alexis Beingessner - - PowerPoint PPT Presentation
Computing Covers of Plane Forests Luis Barba Alexis Beingessner - - PowerPoint PPT Presentation
Computing Covers of Plane Forests Luis Barba Alexis Beingessner Prosenjit Bose Michiel Smid Previous Work Given a set T = { T 1 , T 2 , . . . , T m } of m pairwise non-crossing geometric trees with a total of n vertices in general position. The
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Previous Work
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Previous Work
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Previous Work
Beingessner and Smid 2012:
◮ Coverage can be computed in O(m2n2) time ◮ Worst case example with coverage of size Ω(n4). ◮ Problem is Θ(n4)
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Previous Work
Is slowness a consequence of bad inputs being “contrived”? Optimization to be had in structure of “real” inputs?
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Observations
Coverage of a single tree is it’s convex hull
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Observations
If two coverages overlap, their combined convex hull is covered
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The hull-cover
◮ Compute the convex hull, CH(Ti), of every tree Ti ∈ T ◮ If any two convex hulls overlap, replace them with their
convex hull
◮ Repeat until all convex hulls computed thusly are disjoint ◮ Resulting set of convex polygons is the hull-cover of T
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The hull-cover
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The hull-cover
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The hull-cover
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The hull-cover
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The hull-cover
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Approximation
Does this approximate the coverage?
◮ A terrible approximation (for already hard inputs) ◮ A great approximation (for natural inputs)
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Computing the hull-cover
Challenges:
◮ Finding pairwise intersection is fairly expensive ◮ Computing convex hulls is fairly expensive
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Weakly Disjoint Polygons
Let a weakly disjoint pair of convex polygons P, Q be a pair of convex polygons such that P \ Q and Q \ P are both connected sets of points, and P does not share a vertex with Q.
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Weakly Disjoint Polygons
A pair of polygons that are weakly disjoint, but not disjoint
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Weakly Disjoint Polygons
A pair of polygons that are not weakly weakly disjoint
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Weakly Disjoint Polygons
Lemma
If two convex polygons P, Q are weakly disjoint, then their boundaries intersect at at most two points.
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Weakly Disjoint Polygons
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Weakly Disjoint Polygons
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Weakly Disjoint Polygons
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Weakly Disjoint Polygons
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Weakly Disjoint Polygons
Lemma
If two convex polygons P, Q are weakly disjoint, but not disjoint, then one contains a vertex of the other.
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Weakly Disjoint Polygons
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Lemma
The convex hulls of two disjoint trees are weakly disjoint.
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Weakly Disjoint Polygons
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Blocked or Nested
Lemma
Assume R and S are two non-crossing trees whose convex hulls
- intersect. Then the convex hull of one is strictly inside the other,
- r there exists a pair of adjacent vertices on the convex hull of one
whose visibility is blocked by the other tree.
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Blocked or Nested
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Shoot and Insert
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Shoot and Insert
Ishaque et al. 2012: n pairwise disjoint polygonal obstacles can be preprocessed in O(n log n) time and space to support m permanent ray shootings in O((n + m) log2 n + m log m) time
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Algorithm
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Algorithm
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Algorithm
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Algorithm
1
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Algorithm
1 2
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Algorithm
1 2 3
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Algorithm
1 2 4 3
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Algorithm
1 2 4 3 5
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Algorithm
1 2 4 3 5 6
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Algorithm
1 2 4 3 5 6 7
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Algorithm
1 2 4 3 5 6 7 8
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Algorithm
1 2 4 3 5 6 7 8 9
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Algorithm
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Algorithm
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Analysis
O(n log2 n) time O(n log n) space
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