on quadtrees for point sets martin fürer penn state joint work with shiva kasiviswanathan martin...
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On Quadtrees for Point Sets
Martin FürerPenn State
Joint work with Shiva Kasiviswanathan
Martin FürerPenn State
Joint work with Shiva Kasiviswanathan
Dagstuhl 2006 On Quad Trees for Point Sets
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QuadtreeQuadtree
NW NE
SW SE
o
Dagstuhl 2006 On Quad Trees for Point Sets
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Picture vs. Point Set
Picture vs. Point Set
Picture: Stop when monochromatic Leaves labeled by color 0 or 4 children
Point Set: Stop when 1 point Leaves labeled by coordinates of point
0 to 4 children
Picture: Stop when monochromatic Leaves labeled by color 0 or 4 children
Point Set: Stop when 1 point Leaves labeled by coordinates of point
0 to 4 children
Dagstuhl 2006 On Quad Trees for Point Sets
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Compressed QuadtreesCompressed Quadtrees
Succinct representation for clustered point Sets
Replace each maximal path with degree 1 internal nodes by a single edge
n points --> 2n-1 node tree
Internal nodes labeled with their tile (size and position)
Succinct representation for clustered point Sets
Replace each maximal path with degree 1 internal nodes by a single edge
n points --> 2n-1 node tree
Internal nodes labeled with their tile (size and position)
Dagstuhl 2006 On Quad Trees for Point Sets
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SpannersSpanners
1+ - spanner: Subgraph of a weighted graph with distances increasing by at most a factor of 1+.
1+ - spanner: Subgraph of a weighted graph with distances increasing by at most a factor of 1+.
Dagstuhl 2006 On Quad Trees for Point Sets
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Our ApplicationOur Application
Find Spanners of (Unit) Disk Graphs
O(n/ε) edges vertex separators (hereditary)
Fast approximations of shortest paths
Find Spanners of (Unit) Disk Graphs
O(n/ε) edges vertex separators (hereditary)
Fast approximations of shortest paths
Dagstuhl 2006 On Quad Trees for Point Sets
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Construction of Quadtrees
Construction of Quadtrees
Time: Sorting-time + linear Operations: + - < and for given x, y with x<y, findlevel(x,y) = min k such that
x2k< y2k, also output x2k
Time: Sorting-time + linear Operations: + - < and for given x, y with x<y, findlevel(x,y) = min k such that
x2k< y2k, also output x2k
Dagstuhl 2006 On Quad Trees for Point Sets
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Simulated OperationSimulated Operation
Input (x1,x2), (y1,y2)
Decide whether x1|x2 < y1|y2
where for u = ∑iui2i and v = ∑ivi2i
u|v = ∑i(2ui+vi)4i is the shuffle
Input (x1,x2), (y1,y2)
Decide whether x1|x2 < y1|y2
where for u = ∑iui2i and v = ∑ivi2i
u|v = ∑i(2ui+vi)4i is the shuffle
Dagstuhl 2006 On Quad Trees for Point Sets
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AlgorithmAlgorithm
Sort points x by shuffle(x), (the shuffle of their coordinates)
For adjacent points x,y in the sorted order, compute
level(shuffle(x), shuffle(y)) Result: s1, l1, s2, l2, … sn-1, ln, sn
Each si is (the shuffle of) the root node of a (trivial) quadtree.
Sort points x by shuffle(x), (the shuffle of their coordinates)
For adjacent points x,y in the sorted order, compute
level(shuffle(x), shuffle(y)) Result: s1, l1, s2, l2, … sn-1, ln, sn
Each si is (the shuffle of) the root node of a (trivial) quadtree.
Dagstuhl 2006 On Quad Trees for Point Sets
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Continue AlgorithmContinue Algorithm
Recall: s1, l1, s2, l2, … sn-1, ln, sn
Push left part on a stack until li>li+1
Combine the trees with roots si and si+1 into one tree
Recall: s1, l1, s2, l2, … sn-1, ln, sn
Push left part on a stack until li>li+1
Combine the trees with roots si and si+1 into one tree
Dagstuhl 2006 On Quad Trees for Point Sets
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Open?Open?
Is Bichromatic Closest Pair easier, when the two sets of points are well separated? Dimensions 2 or d.
° ° °°°
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Is Bichromatic Closest Pair easier, when the two sets of points are well separated? Dimensions 2 or d.
° ° °°°
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