seminar on geometric approximation algorithms speaker: alon horowitz

Well-separated pairs

decomposition

Seminar on Geometric Approximation AlgorithmsSpeaker: Alon Horowitz

• What is WSPD• How to construct and represent WSPD

efficiently• Applications of WSPD

Outline of this lecture

What is WSPD• Let P be a set of n points in Rd, and ¼ > ɛ > 0 a

parameter.• Say we want to represent all distances

between points of P.

Return all pairwise distances

Return all n points in P (of size dn)

pq

s

• We are interested in a representation that will capture the structure of the distances between the points.

≈ Close together as far as p is concerned

• As such, if we are interested in the closest pair among the three points, we will want to only check the distance between q and s.

• Example:{1,2,3} {1,2,3} = {{1,2},{1,3},{2,3}}

• Denote by the set of all the (unordered) pairs of points formed by the sets A and B.

• We will be informal and refer to as a pair of sets A and B.

Definitions

Definitions• For a point set P, a pair decomposition of P is a

set of pairs such that:1. Ai , Bi P for every i

2. Ai ∩ Bi = ɸ for every i

3.

• Translation: For any pair of distinct points p,q ϵ P, there is at least one pair {Ai ,Bi} ϵ such that p ϵ Ai and q ϵ Bi

Definitions• The pair Q and R is (1/ɛ)-separated if:

Max(diam(Q),diam(R)) ≤ ɛ∙d(Q,R)Where d(Q,R) = minqϵQ,sϵR

Max = } , {

1/ɛ

Min size ball Min size ball

1/ɛ

• Returning to the previous example:

p

q

s

The pairs {p} {q,s} and {q} {s} are 3-separated.

• We replaced the distance description made out of three pairs of points by distance between two pairs of sets.

Definitions• For a point set P, a well-separated pair

decomposition (WSPD) of P with parameter 1/ɛ is a pair decomposition of P with a set of pairs:

such that, for any i, the sets Ai and Bi are 1/ɛ-separated.

• In the last example we got: = { {{p},{q,s}} , {{q},{s}} }

How to construct and represent WSPD efficiently

• Representation• Construction algorithm• Analysis

Representation• Instead of maintaining such decomposition

explicitly, it is convenient to construct a tree T having the points of P as leaves.

• Now every pair (Ai , Bi) is just a pair of nodes (v , u) of T, such that Ai = Pv

and Bi = Pu , where Pv denotes the points of P stored in the subtree of v.

V

u

(Ai , Bi( = )P v , Pu{( = ),b c , }{e)}

P = {a,b,c,d,e,f}

• In our case, the tree we would use is a compressed quadtree of P.

Representation

• There are many possible WSPDs that can be represented using the tree.

• We will try to find a WSPD that is “minimal”.

The diameter of a point set

stored in a node drops quickly

Construction algorithmGiven a point set P in Rd:• Compute the compressed quadtree T of P.• Greedy – tries to put into the WSPD pairs of nodes

in the tree that are as high as possible:– Start from root (The pair {root,root}).– Check if current pair is well separated.– If not, replace the bigger node (diameter wise) with his

children.

Some definitions• □v := The quadtree cell associated with the node v

• Δ(v) := diam(□v)

• Δ(v) = 0 if Pv is either empty or a single point

• d(u,v) := d(□u,□v) = minpϵ□u,qϵ□v

V□

v

x

d(□v,□f) =d(v,f) =d(b,f)

0

Construction algorithm

Construction algorithm

2-WSPD

Analysis

• algWSPD terminates• algWSPD Computes a valid pair-decomposition• The computed pairs are (1/ɛ)-separated• The number of computed pairs is O(n/ɛd)• (1/ɛ)-WSPD construction time is O(nlogn + n/ɛd)

algWSPD terminates

• If u, v are leaves then Δ(u) = 0 and Δ(v) = 0• Δ(u) ≤ ɛ∙d(u,v) True• Always stops if both u and v are leafs• Always terminates

algWSPD Computes a valid pair-decomposition

• 1. For every i, Ai ,Bi corresponds to some Pu ,Pv and by definition Pu ,Pv P.

• 3. Every pair of points of P is covered by a pair of subsets {Pu ,Pv} output by the algWSPD algorithm By induction…

• Reminder: For a point set P, a pair decomposition of P is a set of pairs such that:1. Ai , Bi P for every i

2. Ai ∩ Bi = ɸ for every i

3.

• 2. Let {u,v} be an output pair• If Pu and Pv are single point, then Pu ϶ x ≠ y ϵ Pv

because of the first line in algWSPD Pu∩Pv =ɸ

• If Pu or Pv are not single point, then:

a:= max(diam(Pu),diam(Pv)) > 0

This implies that:d(Pu, Pv) ≥ d(□u,□v) = d(u,v) ≥ Δ(u)/ɛ ≥ a/ɛ > 0

Pu∩Pv =ɸ

algWSPD Computes a valid pair-decomposition

The computed pairs are (1/ɛ)-separated

• Proof: for every output pair {u,v}, we have by the design of the algorithm that:

Max(diam(Pu),diam(Pv)) ≤ max{Δ(u),Δ(v)} ≤ ɛ∙d(u,v)

Also, for any qϵPu and sϵPv we have:

d(u,v) = d(□u,□v) ≤ d(Pu,Pv) ≤ d(q,s)

since Pu □u and Pv □v

• Reminder: The pair Q and R is (1/ɛ)-separated if:Max(diam(Q),diam(R)) ≤ ɛ∙d(Q,R)

Where d(Q,R) = minqϵQ,sϵR

The number of computed pairs is O(n/ɛd)• First, a short Lemma:• Let □ be a cell of a grid G of Rd with cell diameter x.

For y ≥ x, the number of cells in G at distance at most y from □ is O((y/x)d).

• Proof: by figure:

x

y+1y+1

d = 2

O(([2(y+1)+x]/x)2) = O((y/x)2) In d dimensions: O((y/x)d)

• Proof: Let {u,v} be a pair appearing in the output• Let's consider the sequence of recursive calls that

led to this output. In particular, Let's assume that the last recursive call to algWSPD(u,v) was issued by algWSPD(u,v’), where v’ = p(v) (the parent of v in T)

• This implies that Δ(u) ≤ Δ(v’)• Furthermore, the fact that algWSPD(u,v’) was

invoked implies that algWSPD(p(u),a(v’)) has been considered and then p(u) was split.

• This implies that Δ(a(v’)) ≤ Δ(p(u))• To summarize: Δ(u) ≤ Δ(v’) ≤ Δ(a(v’)) ≤ Δ(p(u))

The number of computed pairs is O(n/ɛd)

• Let us prove that each node v’ is charged at most O(1/ɛd) times

• Since the pair {u,v’} was not output by algWSPD (despite being considered), we conclude:

Δ(v’) > ɛ∙d(u,v) d(u,v) < Δ(v’)/ɛ := r• Because we proved: Δ(u) ≤ Δ(v’) ≤ Δ(p(u)) then

there are 3 possibilities:– Δ(u) = Δ(v’)– Δ(v’) = Δ(p(u))– Δ(u) < Δ(v’) < Δ(p(u))


Possible?

• Δ(u) = Δ(v’): by the lemma we proved, the number of u nodes that: Δ(u) = Δ(v’) and d(u,v’) < Δ(v’)/ɛ := r is at most O((r/ Δ(v’))d) = O(1/ɛd).


x

y+1y+1

Since v’ has at most 2d

children, this type of charge can happen at most O(2d (1/ɛ∙ d))

• Δ(v’) = Δ(p(u)): by the same argument, the number of p(u) nodes that: d(p(u),v’) ≤ d(u,v’) < r is at most O(1/ɛd). Since also p(u) has at most 2d children at most O(2d 2∙ d (1/ɛ∙ d))

• Δ(u) < Δ(v’) < Δ(p(u)): Let □’ be the cell in G containing □u. Observe that □u □’ □p(u). In addition:

d(□’,□v’) ≤ d(□u,□v’) = d(u,v’) < r.


□p(

u)

□u

□v’

□ ’

As before, it follows that there are at most O(1/ɛd) cells like □’, and as before, total number of charges is at most O(2d (1/ɛ∙ d)).

• In conclusion, v’ can be charged at most O(2d 2∙ d (1/ɛ∙ d)) = O(1/ɛd) times.

• Since there are O(n) nodes in T, the total number of pairs generated by algWSPD is O(n/ɛd).


• Every point of P is present in O(1/ɛd) pairs.• Since running time of algWSPD is linear in the output

size, and quadtree construction time is O(nlogn) we conclude:

(1/ɛ)-WSPD construction time is O(nlogn + n/ɛd)

Applications of WSPD

• Closest pair• All nearest neighbors• Spanners• Approximating the Minimum Spanning Tree

Closest pair• Let P be a set of points in Rd, we would like to

compute the closest pair.

• Lemma: let W be a (1/ɛ)-WSPD of P, for ɛ ≤ ½. There exists a pair {u,v}ϵW, such that:– |Pu| = |Pv| = 1– is the distance of the closest pair.

• Proof: Let p,q be the closest pair and let {u,v}ϵW be the pair such that pϵPu and qϵPv

• Let assume by contradiction that there is an additional point sϵ Pu

Closest pair

Contradiction to p,q being the closest pair

Algorithm:• Compute 2-WSPD of P• Scan all pairs of W• Compute distance between pairs {u,v} which are

singletons• Return the closest pair encountered

Closest pair

• Given a set P of points in Rd, we would like to compute for each point qϵP its nearest neighbor in P.

• Is nearest neighbor a symmetrical relationship?

All nearest neighbors

q is the nearest neighbor to p, but s is the nearest neighbor to q

All nearest neighborsAlgorithm:• Compute 4-WSPD of P• Scan all pairs of W• Compute distance between pairs {u,v} such that

Pu or Pv is a singleton

• For each Pu = {p}, record for p the closest point to it in Pv

• Return the recorded nearest point for every point p in P

• Lemma: Let p be a point in P and let q be the nearest neighbor to p in P\{p}, then there exists a pair {u,v}ϵW such that Pu={p} and qϵPv

• Proof: Consider {u,v}ϵW such that pϵPu and qϵPv


• If Pu contained any other point except p then contradiction to q being the nearest neighbor to p

Diam(Pu) ≤ ɛd(Pu, Pv) ≤ ɛ||p-q|| ≤ ||p-q||/4

• Let P be a set of n points in the plane, then one can solve the all nearest neighbor problem, in O(n(logn + logɸ(P)) time, where ɸ is the spread of P


• Difficulties: according to the algorithm: Compute distance between pairs {u,v} such that

Pu or Pv is a singleton For each Pu = {p}, record for p the closest point to

it in Pv

• What if Pv is very big???

SpannersDefinitions:• dG(q,s) := distance of the shortest path between

vertices q,s in weighted graph G.• A t-spanner of a set of points P in Rd is a weighted

graph G whose vertices are the points of P, and for any q,sϵP, we have:

dG is a metric (complies with the triangle inequality)

• Let P be a set of n points in Rd and let 1 ≥ ɛ > 0 be a parameter

• we would like to compute a (1+ɛ)-spanner of P with O(n/ɛd) edges in O(nlogn + n/ɛd) time

Spanners

• (1+ɛ)-spanners approximate the complete graph with a relative error ɛ

Spanners

Algorithm:• Set δ = ɛ/c , where c ≥ 16• Compute a (1/δ)-WSPD of P• For every pair {u,v}ϵW, add an edge between

{repu, repv} with weight

Spanners

• Return resulting graph G

Analysis:• We will show that for any pair x,yϵP:– ||x-y|| ≤ dG(x,y)

– dG(x,y) ≤ (1+ɛ)||x-y||

Proof:• ||x-y|| ≤ dG(x,y) is trivial… why?

Triangle inequality

Spanners

||x-y ||

dG(x,y)

• dG(x,y) ≤ t||x-y|| by induction on the distance of the pairs:• Let's assume that for any pair z,wϵP:

||z-w||<||x-y|| dG(z,w) ≤ (1+ɛ)||z-w||

• The pair x,y must appear in some pair {u,v}ϵW, where xϵPu and yϵPv , thus: (*)

Spanners

• Also: (**)

• We conclude:dG(x,y) ≤ dG(x,repu) + dG(repu,repv) + dG(repv,y)

≤ (1+ɛ) ||rep∙ u-x|| + dG(repu,repv) + (1+ɛ) ||rep∙ v-y||

= (1+ɛ) ||rep∙ u-x|| + ||repu-repv|| + (1+ɛ) ||rep∙ v-y||

≤ 2 (1+ɛ)δ∙||repu-repv|| + ||repu-repv||

≤ (1+2δ+2ɛδ) ||rep∙ u-repv||

≤ (1+2δ+2ɛδ)(1+2δ) ||x-y||∙≤ (1+ɛ) ||x-y||∙

Spanners

)*(

δ = ɛ/c and c ≥ 16

)**((repu,repv)

Is an edge

Induction hypothesis

Approximating the Minimum Spanning Tree• Given a set P of n points in Rd, we would like to

compute a spanning tree T of P such that:w(T) ≤ (1+ɛ)w(M)

where M is the minimum spanning tree of P, and w(T) is the total weight of the edges of T.

Approximating the Minimum Spanning Tree

Algorithm:• Compute a (1+ɛ)-spanner G of P• Compute the minimum spanning tree T of G• Return T as the approximate minimum spanning

tree

Running Time:• Computing a minimum spanning tree of a graph,

with n vertices and m edges takes O(nlogn + m) time

Computing T takes O(nlogn + n/ɛd) time

• We need to prove that T is the required approximation

Proof:• π(q,s) : = shortest path between q and s in G• M:= the minimum spanning tree of P

• Since G is a (1+ɛ)-spanner, for any q,sϵP: w(π(q,s)) ≤ (1+ɛ)||q-s||

• Let’s look at G’ = (P,E) which is a connected subgraph of G, where E =


• Since G is a (1+ɛ)-spanner:

• Since G’ is a connected spanning subgraph of G:w(T) ≤ w(G’) ≤ (1+ɛ)w(M)


That’s All…

100 pointsOn a Circle

seminar on geometric approximation algorithms speaker: alon horowitz

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