Random Partition via Shifting

Random Partition via ShiftingSeminar on Geometric Approximation AlgorithmsSpeaker Dina BilchinskyTAUaOutline of the lectureDefinitions.Applications:Covering by Disks.Shifting Quadtrees.Hierarchical Representation of a Point Set:Low Quality Approximation by HST.Fast & Dirty HST in High Dimensions.Low Quality ANN Search.aOutline of the LectureIn this lecture we investigate a simple technique for partitioning a geometric domain.This idea can be extended to shifting multi resolution grid over .This yields some simple algorithms for Clustering and Nearest Neighbor search.

aShifted Partition of the Real Line>0 real number.b uniformly distributed number in [0, ].This induces a natural partition of the real line into intervals, by the function

Each interval has size .The origin is shifted to the right by an amount b.

aShifted Partition of the Real Line induce same partition , but not same function.b can be picked uniformly from any interval of length multiple of .Specifically from

aLemma 11.2For anyProof:

Assume , define

0 - r

aShifted Partition of Space - point in .b = , randomly and uniformly chosen from hypercube . - grid with origin b and side length .For a point the ID of the grid cell containing it

aLemma 11.3 - randomly shifted grid.B ball of radius or an axis parallel hypercube with side length .

aLemma 11.3 - Proof2r > the probability is 1.2r < : Project B into its coordinate.It becomes interval of length 2r. becomes one dimensional shifted grid . B is contained in a single cell of is contained in a single cell of

aApplications: Covering by DisksGiven a set P of n points in the plane we would like to cover them by a minimal number of unit disks.Canonical DisksType A The boundary circle contains two points of P.Type B The top point of this circle is a point of P.Any set of points can be covered by Canonical Disks only.

aNumber of Canonical DisksEvery pair of input points determines two possible disks.

If a pair of input points is at distance larger than 2, than the Canonical Disk they define is invalid.Therefore, there are such Canonical Disks.

We assume the cover uses only such disks.

aDisk Cover Disk Cover VerificationGiven k disks, we can verify the cover in

Lemma 11.4Given a set P of n points in the plane, we can compute in time, a cover of P by at most k unit disks, if such a cover exists.

For every point , check if it contained in one of the disks.

aLemma 11.4 - ProofUse the Verification Algorithm , trying all covers of size .The Algorithm returns the first cover found.

Running time Dominant by the last iteration.

There are different covers to consider.Each verification . Thus total running time .

aDisk CoverThe problem with this algorithm is that k might be quite large, say n/4.Fortunately, the shifting grid saves the day.

Theorem 11.5P set of n points in the plane. > 0 is a parameter.We can compute using randomized algorithm in time, a cover of P , by unit disks .

aTheorem 11.5 - ProofChoose and consider a randomly shifted grid .Compute used cells ,grid cells that contain points of P, by computing for each point its and sort it in hash table. - points of P falling into grid cell . Each can be covered by unit disks.

aTheorem 11.5 - proofFor each ,compute the minimum number of unit disks required to cover . ( Bounded by )By Lemma 11.4 we can compute it in .There are at most used cells.Thus the total running time is .

aTheorem 11.5 - proof

Overall Cover : merge together the covers of each grid cell .

aProof Bounding Expectation optimal solution.We will generate a feasible solution from is one of the possible solutions considered by the algorithm. - set of disks of the optimal solution that intersect .Consider the multi-set The algorithm returns for each grid cell minimal cover , that is of a size at most .

*

it returns smallest possible covera18Proof Bounding ExpectationThe cover returned by the algorithm is of a size at most .Disk of the optimal solution can appear in at most 4 times (can intersect at most 4 cells of the grid )Disk will appear in more than once it is not fully contained in a greed cell of .By Lemma 11.3 the probability for that is bounded by

aProof Bounding Expectation

The running time can be improved to

aShifting QuadtreesaShifting One Dimensional QuadtreeaBit IndexaShifting One Dimensional QuadtreeaExample Without Shifting

aLemma 11.7 aLemma 11.7 - proofaLemma 11.7 - proof

aCorollary 11.8

aHigher Dimensions QuadtreesaHigher Dimensions QuadtreesaLemma 11.9

aHierarchical Representation of a Point SetaMetric Space

aHierarchically Separated TreeaHSTaExample - HSTaMetric Space t-approximateaLemma 11.13a39Lemma 11.13 - Proofa Proof - Approx. factor

aSpannersaCorollary 11.14 aCorollary 11.15aCorollary 11.15 - Proof

aFast&Dirty HST in High DimensionThe above construction of HST has exponential dependency on the dimension.Next we will show how we can get an approximate HST of low quality , but in polynomial time in the dimension.aLemma 11.16aLemma 11.16- ProofaLemma 11.16 Proof(2)

aClaim 11.17aClaim 11.17 - Proofa Claim 11.17 Proof(2)

aClaim 11.17 Proof (3)

aTheorem 11.18aTheorem 11.18 - Proof

aDeterministic Construction of HSTaLemma 11.19aLemma 11.19 - ProofaLemma 11.19 Proof (2)aLemma 11.19 Proof(3)a60Lemma 11.19 Proof(4)aLemma 11.19 Proof(5)

aLow Quality ANN SearchaData Structure & Search ProcedureaAnalysis

aAnalysis(2)aLemma 11.20aLemma 11.20 Proof(1)aLemma 11.20 Proof(2)

aTheorem 11.21aThe Enda