1. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems Mugurel Ionut Andreica Politehnica University of Bucharest Computer Science Department

2. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 2 Summary Motivation Largest Empty Circle Largest Hyper-Rectangle with Fixed Aspect Ratio Circle Containment Hyper-Rectangle Containment Maximum Weight Subsequence Accepted by a NFA Conclusions

3. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 3 Motivation Extracting information from multidimensional data – applications in: –Business planning & integration –Statistics –Scientific research –Distributed systems Distributed, peer-to-peer message routing system –Peer identifiers: mapped into a metric space –Analysis of the distribution of identifiers in the metric space (empty zones, covering,...)

4. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 4 Largest Empty Circle (1/2) n points (x i,y i ) located inside a circle C find the largest circle C’, fully contained inside C, which contains none of the given points binary search the radius R of C’ => feasibility test –shrink the radius of C by R –inflate every point into a circle of radius R –decide if the union of the n circles covers the shrinked circle C => we need an algorithm for deciding if the union of n circles covers another circle

5. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 5 Largest Empty Circle (2/2) Compute a set S of all the x-coordinates of the intersection points of the circles, plus {x i, x i -R, x i +R} Maintain only the distinct x-coordinates in S (O(n 2 ) values) + sort them: xs 1 <xs 2 <...<xs M for each pair of consecutive coordinates (xs i, xs i+1 ) –xmid=(xs i +xs i+1 )/2 –each circle j intersects the vertical line x=xmid on an interval Ij –compute the union of the intervals (for each interval in the union, store the circle ids of its two endpoints) –compute the interval Ic (intersection of the shrinked circle C with x=xmid) –if union does not cover Ic => can increase R (in the binary search) –if all unions cover Ic => R is too large O(n 3 ·log(n)) (or O(n 2 ·(n+log(n)))

6. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 6 Largest Hyper-Rectangle with Fixed Aspect Ratio n d-dimensional points inside a d-dimensional hyper-rectangle R find the largest volume empty hyper-rectangle with fixed aspect ratio –length j =f j ·length 1 (j=1,...,d) (f 1 =1) binary search length 1 => feasibility test inflate every point to a hyper-rectangle with length 1 and fixed aspect ratio + shrink R by length i on dimension i (similar to the circle case) verify if the union of the hyper-rectangles covers R –clip every hyper-rectangle on the boundaries R –compute the volume V union of the union of the clipped hyper-rectangles – O(n d-1 ·log(n)) time –If V union =Volume(R) => R is covered => decrease length 1 ; otherwise, increase length 1

7. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 7 Circle Containment n circles (x i, y i, r i ) every two circles A and B: completely disjoint or A is included in B (or B in A) decide if each circle contains (is contained in) another circle consider left and right half-circles sweep a vertical line from left to right (considering only right half-circles) ; afterwards, do the same from right to left (with left half-circles only) maintain a balanced tree T of disjoint intervals (the intersections of the half-circles with the sweep line) when we reach the left half of a half-circle => we traverse all the intervals in T intersecting the vertical segment I=[y i -r i,y i +r i ] ; we adjust their length (it might be outdated) and check for inclusion –remove the intervals included in I (or I, if it is included in some interval from T) –insert (back) into T the adjusted, non-included intervals overall: O(n·log(n))

8. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 8 Hyper-Rectangle Containment n d-dimensional hyper-rectangles decide, for each hyper-rectangles, if it includes (is included in) another hyper-rectangle sweep line on the d-th coordinate associate a (2·d-2)-dimensional point p i to each rectangle i maintain a range tree R at left side of dimension d of rectangle i: insert p i into R with weight = the right side on dimension d-1 at right side of dimension d of rectangle i: –remove p i from R –range max query the range tree => find the hyper-rectangle Q with the largest (d-1)th dimension and s.t. the other dims are appropriate –we test if Q includes R

9. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 9 Maximum Weight Subsequence Accepted by a NFA sequence of n d-dimensional points with weights a non-deterministic finite automaton NFA –each directed edge (q(1),q(2)): d intervals [a(1),b(1)],..., [a(d),b(d)] –pass from q(1) to q(2) only if the difference between the coordinates of the next point (and the current one) on dim j is within [a(j),b(j)] find a non-contiguous subsequence of points of maximum total weight s.t. NFA reaches a final state W max (i,j)=the maximum weight of a subsequence ending at point i and reaching the state q(j) (dynamic programming) O(n·(m+|E|)·log d (n)) –m = #of states of NFA –|E|=# of (directed) edges of NFA –maintain a range tree for each state of the NFA very general model special cases: longest increasing subsequence, longest alternating (antimonotonic) subsequence

10. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 10 Conclusions Geometric data analysis problems –Largest Empty Circle –Largest Hyper-Rectangle with Fixed Aspect Ratio –Circle Containment –Hyper-Rectangle Containment –Maximum Weight Subsequence Accepted by a NFA Applications to many domains –my motivation: analysis of data from distributed systems with node identifiers mapped into a metric space

11. Efficient Algorithmic Techniques for Several Multidimensional Geometric Data Management and Analysis Problems 11 Thank You !