Random Partition via Shifting

Random Partition via ShiftingTomer Margalit, 21/5/2012Table of ContentsIntroduction and fundamental properties of randomly shifted grids.Application minimal disk covering of points. We improve the running time of the trivial approach using randomly shifted grids.Shifting Quadtrees we will see how using random shifts when constructing Quadtrees can be beneficial.Approximation of an n-point Metric Space We will show how we approximate such a metric using a new data structure called an HST.Background

Example Points in square

Example Unshifted Partition

Example Shifted Partition

Example Shifted Points

Partition via Shifting 1Example Partition Function

Partition via Shifting 2Good Probability for a Nice DistributionIntuition 1

Intuition 2

ProofGeneralized Partition ShiftExample 2D Shifted Partition

Example 3D Partition

Good Probability for a Nice Distribution in Multidimensional spacesProof, ContinuedRemarks about Randomly Shifted GridsWe have shown that if we choose grids randomly, we get a good chance of separating two points if we have a coarse enough partition.Obviously, we will use this property to give randomized algorithms, and algorithms that have a high probability to succeed.Application Covering by DisksNow we show an application for these simple facts about random partitions.Given a set of n points in the plane, we would like to cover them by the minimal amount of unit disks.Apparently, the randomly shifted grids can improve the trivial algorithm by quite a bit.First we show the trivial algorithm, and then the improvement gained by using randomly shifted grids.

Disk Cover - Points

Disk Cover - Cover

Trivial CoverIntuition What Sets Two Covers Apart? (1)

Intuition What Sets Two Covers Apart? (2)

Intuition What Sets Two Covers Apart? (3)

Intuition What Sets Two Covers Apart? (4)

From Intuition to ProofThe intuition shows that two covers are equivalent if all the disks in both covers cover the same set of points.This intuition will direct us in providing the trivial algorithm.Instead of considering any cover, we consider only equivalence classes of covers.Very trivially, we can consider every partition of the points into (not necessarily disjoint) groups, and check if that cover exists for them.However, this approach gives us a lot of invalid covers.We can use some simple observations to cut that number.I. Every pair of input points (at distance under 2) defines two possible disks

II. Given a cover, we can move every disk to have at most 2 points on the edge

ProofThe first insight is obvious.The second insight is not that obvious though.To get the second claim, given a disk (that is part of a cover), we can move the disk downward until it hits a point.And then rotate it (around the point) until it hits another one.Move downward until a point is hit

Rotate until a second point is hit

Trivial Cover FeasibilityTrivial Cover Given the Minimal Cover SizeTrivial Cover Full AlgorithmTrivial Cover Final StatementMinimal Disk Cover - ImprovementMinimal Disk Cover Randomized AlgorithmAlgorithm

Algorithm 2Good Expectation of the Cover SizeProof 1Optimal Cover

Proof 2Proof 3Proof 4Proof 5Minimal Disk Cover Randomized Algorithm ApplicationsShifting QuadtreesWe now use the randomly shifted grids in combination with Quadtrees.Later on we will show applications for this approach.We start with one-dimensional Quadtrees, and later on extend the approach to higher dimensions.

Shifting Quadtrees One dimensional1 Dimensional Shifted Quadtree

Good Chance for the lca to be Close to the Level of the DistanceProof 1Proof 2CorollaryHigher Dimensional Quadtrees2D Randomly Shifted Quadtree

Hierarchical Representation of a Point SetWe now switch our discussion to metric spaces.From now on instead of speaking about euclidean spaces, we will talk about a metric space M with metric d.We will now show a new type of data structure (tree), that can be used to define a metric over a set of points.Hierarchically Well-Separated TreeHSTs 2HSTs 3t-ApproximationApproximating n-point Metrics by HSTsClaim: Given a weighted connected graph G=(V,E) on n vertices and m edges, it is possible to construct, in O(nlogn+m) time, a binary HST H that (n-1)-approximates the shortest path metric of G.To clarify, the shortest path metric of a graph G assigns to every two nodes the weight of the lightest path between them.We give the algorithm, and the proof of correctness afterwards.Algorithm 1Algorithm 2Example Run - MST

Example Run 1

Example Run 2

Example Run 3

Example Run 4

Example Run 5

Tightness of the ApproximationCorrectnessProofProof 2HST Approximating an n-point Metric The End.