Aggregate-Max Nearest Neighbor Searching in the Plane Haitao Wang Utah State University CCCG 2013

Nearest neighbor searching Input: a set P of n red points Query: a query blue point q report the nearest red point to q

Aggregate/Group nearest neighbor searching Input: a set of n red points Query: a set Q of m query blue points report the “nearest” red point to Q

What is the “nearest neighbor” to Q? For any red point p in P, the aggregate distance from p to Q: the farthest distance from p to the points of Q that’s why it is called Aggregate-max The nearest neighbor of Q: the red point whose aggregate distance to Q is the minimum p

An application A group of people want to meet together given a set of candidate meeting locations find a location that minimizes the longest distance to all people

Previous work Heuristic algorithms given in the database area Minimum bounding method (Papadias et al, TODS 2005) R-trees (Li et al, TKDE 2011) Ω(n+m) time in the worst case (1+ε)-approximation result (Li et al, TKDE 2011) Aggregate-sum version 3-approximation (Li et al, SIGMOD 2011) (1+ε)-approximation (Agarwal et al, PODS 2012)

Our results Preprocessing: O(n log n) time and O(n log log n) space Query: O(mn1/2 logO(1) n) time Or, Preprocessing: O(n2+ε) time and space for any ϵ > 0 Query: O(m log n) time n: the number of input points m: the number of query points m<<n

Our results – the L1 distances Preprocessing: O(n log n) time and O(n) space Query: O(m+logn) time Top-k queries: Report the k nearest neighbors to Q Query: O(m+klogn) time vertical distance e horizontal distance The L1 distance of e = the horizontal distance + the vertical distance

A simple observation For any red point p, how to determine the farthest point q of Q to p? An observation: q is the point whose farthest Voronoi region contains p q p

A query algorithm For each blue point q that defines a farthest Voronoi region FVR(q) find the nearest neighbor of q in FVR(q) Return the shortest distance q Difficulty: How to find the nearest neighbor of q in FVR(q)? FVR(q)

A solution: maintain Voronoi diagrams Obtain the Voronoi diagram of the red points in FVR(q) Determine the Voronoi region that contains q q FVR(q)

The new problem: answering Voronoi diagram queries For a set of points in the plane, query: a triangle return (implicitly) the Voronoi diagram of the points in the triangle Our approach: Using the simplex range searching data structures (Matousek 92’, 93’, Chan 12’) For each canonical subset of points, maintain its Voronoi diagram explicitly

The L1 version For any red point p easy to determine the farthest point of Q to p one of the four extreme points The farthest Voronoi diagram of Q is determined by the four extreme points p

The key problem: determine the nearest neighbor of q in FVR(q)? The solution for L2: maintaining Voronoi diagrams A better solution for L1: using segment dragging queries FVR(q) q

A key observation on the shape of FVR(q): only three types Type-A bounded by two vertical half-lines and a segment of slope -1 in between On the right of the vertical line through q b is on the horizontal line through q assume q is the extreme point along the southwest direction FVR(q) a b q

The shape of FVR(q): three types Type-B bounded by a vertical half-line, a horizontal half-line, and a segment of slope -1 in between in the first quadrant with respect to q FVR(q) a b q

The shape of FVR(q): three types Type-C symmetric to the Type-A a q FVR(q) b

The shape of FVR(q): three types b q b b q q FVR(q) A B C

Determining nearest neighbor in type-A regions Partition FVR(q) into three subregions Find the nearest neighbor in each subregion a b q

The shape of FVR(q): three types b q b b q q A B C

Segment dragging queries Preprocessing: O(nlog n) time and O(n) space Query: O(log n) time out-of-corner queries: Mitchell 92’ parallel-track queries: Chazelle 88’ out-of-corner query parallel-track query

The query algorithm –a summary Given a query set Q: Compute the farthest Voronoi diagram O(m) time For each extreme point q, determine the nearest point of P in FVR(q) O(log n) time by segment dragging queries Total query time: O(m + log n)

Extended to the top-k queries After the nearest point is found in a subregion, keep searching the next nearest point by segment dragging queries a tempting approach: keep using another out-of-corner query does not work!! the interior of the purple triangle should not contain any point a q

Extended to the top-k queries Our approach: partition into three smaller regions: an out-of-corner query for the green region parallel-track queries for the blue regions Total query time: O(m+klog n) a q

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