Well-Separated Pair Decomposition

Seminar on Geometric Approximation Algorithms, Spring 2012 Eyal Altshuler Well-Separated Pair Decomposition

Topics To Be Covered • Motivation • WSPD – Basic Definitions • WSPD – The Construction Algorithm • Applications of WSPD • Approximating the diameter • Closest pair • Spanners • Approximated minimum spanning tree

Motivation • Given a set P of points in , We want to represent the distances between them efficiently • Suggestions: • explicitly listing the distances • listing arrays of coordinates • We will see a more compact representation

Motivation • Suppose that all we care about are approximate distances • Want to capture that: • s and q are close together as far as p is concerned • s and q have the “same distance” from p • A “spoiler” for today’s data structure: • () • We will see how to build it q p s

WSPD – Basic Definitions • [ All works in , we will concentrate on ] • Let P be a set of n points in

WSPD – Basic Definitions • Denote by • The set of all (unordered) pairs of points formed by two sets of points – A and B • Informally we will refer to as the pair of the sets A and B

WSPD – Basic Definitions • For a point set P, a pair decomposition of P is a set of pairs – • such that: • for every i • = for every i • =

WSPD – Basic Definitions • The pair Q and R is (-separated if where

WSPD – Basic Definitions Q R

WSPD – Basic Definitions • For a point set P, a well-separatedpair decomposition(WSPD) of P with parameter ( is a pair decomposition of P with a set of pairs- • such that, for any i, the sets and are ( separated

Example b a f c d e

Building a WSPD • How to represent a WSPD efficiently? • Construct a Tree T having the points of P as leaves. • Every Pair {} will be just a pair of nodes (, such that • denotes the points of P stored in the sub-tree of v

Reminder - Quadtrees b a f c d e

Reminder - Quadtrees b a f c d f e a d e c b

Reminder - Quadtrees • Denote by the cell that contains the points that are in the subtree of v • Working with compressed quadtrees – time construction

Building a WSPD • Build T as a compressed quadtree d f e a c b

Building a WSPD • Find the pairs of vertices that define a WSPD d f e a c b

Notes • The quadtree decomposes the points such that the diameter of a point set stored in a node drops quickly as we go down the tree • A given tree can be used to represent many possible WSPDs. We are looking for the WSPD that is “minimal”

Notes • Suppose we find, for a given , the pairs of nodes that represent -separated point sets • This is the WSPD compact representation we want to achieve • It will be convenient, for each point set in a pair, to choose an arbitrary representative. Calculating distances between pairs will be done using these representatives

The Construction Algorithm • Given – A set of n points P in • First, compute the quadtree T of P • Next, be greedy – by calling • algWSPD(root,root)

Some More Definitions • - the “diameter” of the cell associated with • for a leaf, otherwise

Examples b a f c v d f u a g e g d c e b ?

Examples b a f c v d f u a g e g d c e b

algWSPD(u,v) • If and then • return // Don’t pair a leaf with itself • Ifthen • Exchange and • Ifthen • return // - the children of • return

Analysis • Computing the com-pressed quadtree T of P can be done in time (see previous lecture) • We will prove that constructing an -WSPD takes time • running time

Analysis • Lemma 1: • Let be a cell of a grid G of with cell diameter . For , the number of cells in G at distance at most from is • Proof: y y y x y

Analysis • Lemma 2 (validity): • algWSPD terminates and computes a valid pair decomposition • Proof: • Termination – always stops if both and are leaves • Every pair of points is covered – by induction • Every pair is valid -

Analysis • Lemma 3: (-WSPD) • For the WSPD generated by algWSPD, we have that for any pair in the WSPD, and , for any ,

Analysis • Proof: ,

Analysis • We got that the algorithm always return a valid -WSPD. • Note that the running time of the algorithm is clearly linear in its output size. • So, what is the output size?

Analysis • For a pair computed by algWSPD, we have that- where denotes that parent of in T

Analysis • [ • Proof: • Trivial that- • Let us show that

Analysis • [ • Consider the set of recursive calls

Analysis • [ • Assume

Analysis • [ • Assume • Clearly,

Analysis • [ • Similarly consider the index t where- • Clearly,

Analysis • Lemma: (WSPD’s size) The number of pairs in the computed WSPD is • Corollary: For , one can construct an -WSPD of size , and the construction time is

Analysis • Proof: • Suppose in the output • Suppose was called by • “charge” for that • Show that can’t be charged too much

Analysis • If: • Then:

Analysis • In addition- • Because the pair is not in the output • Denote-

Analysis • Suppose the three possibilities • Each one can happen at most times • (show on board) • There are n possible s, thus the total number of charges is

Applications

remember from algWSPD • For an -WSPD , it holds, for any pair , that: • Will be very important in the application side

Applications – Approximating The Diameter • Given a set P of n points in , find a pair such that , where is the diameter of P • Approximation vs. time, space

Applications – Approximating The Diameter

Applications – Approximating The Diameter • The affect of choosing the correct epsilon • Compute a -WSPD W of P • From the result- • Return the representatives of that have maximal distance between them

Applications – Approximating The Diameter q s

Applications – Approximating The Diameter • Consider the pair realizing the diameter of P • Let be the pair that contain the points q s

Applications – Closest Pair • Let P be a set of points in . We would like to compute the closest pair. • (first lecture) • Let be an -WSPD. • Let us look at closest points p and q in P, and consider the pair that separates them

Well-Separated Pair Decomposition