On Quadtrees for Point Sets Martin Fürer Penn State Joint work with Shiva Kasiviswanathan Martin Fürer Penn State Joint work with Shiva Kasiviswanathan

Dagstuhl 2006On Quad Trees for Point Sets2 Quadtree NW NE SWSE o

Dagstuhl 2006On Quad Trees for Point Sets3 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 2006On Quad Trees for Point Sets4 Compressed 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 2006On Quad Trees for Point Sets5 Spanners 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 2006On Quad Trees for Point Sets6 Our 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 2006On Quad Trees for Point Sets7 Construction of Quadtrees  Time: Sorting-time + linear  Operations: + - < and for given x, y with x<y, find level(x,y) = min k such that  x2 k  <  y2 k , also output  x2 k   Time: Sorting-time + linear  Operations: + - < and for given x, y with x<y, find level(x,y) = min k such that  x2 k  <  y2 k , also output  x2 k 

Dagstuhl 2006On Quad Trees for Point Sets8 Simulated Operation  Input (x 1,x 2 ), (y 1,y 2 )  Decide whether x 1 |x 2 < y 1 |y 2 where for u = ∑ i u i 2 i and v = ∑ i v i 2 i  u|v = ∑ i (2u i +v i )4 i is the shuffle  Input (x 1,x 2 ), (y 1,y 2 )  Decide whether x 1 |x 2 < y 1 |y 2 where for u = ∑ i u i 2 i and v = ∑ i v i 2 i  u|v = ∑ i (2u i +v i )4 i is the shuffle

Dagstuhl 2006On Quad Trees for Point Sets9 Algorithm  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: s 1, l 1, s 2, l 2, … s n-1, l n, s n  Each s i 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: s 1, l 1, s 2, l 2, … s n-1, l n, s n  Each s i is (the shuffle of) the root node of a (trivial) quadtree.

Dagstuhl 2006On Quad Trees for Point Sets10 Continue Algorithm  Recall: s 1, l 1, s 2, l 2, … s n-1, l n, s n  Push left part on a stack until l i >l i+1  Combine the trees with roots s i and s i+1 into one tree  Recall: s 1, l 1, s 2, l 2, … s n-1, l n, s n  Push left part on a stack until l i >l i+1  Combine the trees with roots s i and s i+1 into one tree

Dagstuhl 2006On Quad Trees for Point Sets11 Open?  Is Bichromatic Closest Pair easier, when the two sets of points are well separated? Dimensions 2 or d. ° ° °°° °  Is Bichromatic Closest Pair easier, when the two sets of points are well separated? Dimensions 2 or d. ° ° °°° °