introduction outline the problem domain network design spanning trees steiner trees triangulation...

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

The Problem Domain

General network design problem:

Given a set S of n points in Rd, how to construct a good network that connects these points ?

Property 1:Any network connecting a set of n points must have at least n-1 edges.

Sparse network – the number of edges is linear to the number of points. (i.e. O(n) edges)

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Network Design

Size

Network quality measures:

Stretch factor

Weight

Degree Diameter Connectivity - Fault Tolerance

Network quality measures (cont.)

Load Factor

Number of Steiner points

e

(To be covered)

Bottle Neck

Motivation by examples

Scandinavian rail network (prior 2000)

Copenhagen

Malmo

Motivation by examples

Stretch FactorImagine set of highways connecting cities.We need to upgrade the highway system in the cost-effective manner.

Should we upgrade ALL existing highways ?Upgrade only carefully selected subset of existing highways !

Significance of Network Analysis

What is the size, weight, stretch factor, diameter, degree or connectivity of the network ?

Queries examples:

What is the farthest pair of cites in the network ?For which pair of the cites the stretch factor is the largest ?Which edge (or K edges) should be added to the network to achieve the greatest decrease in the stretch factor and/or load factor ?

What is the total length of the edges to be added to the network to achieve a desired fault-tolerance and stretch factor without destroying the planarity ?

Network Design Tradeoffs

Bounded degree vs. the small diameter

Small stretch factor vs. the small size/weight

Thus, network quality measures can be thought of as multicriteria optimization problems

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Spanning Trees

Spanning Tree: given set S of n points,

ST(S) is an acyclic connected graph on these points.

Spanning tree is good in sense that it has minimum number of edges

Sylvester(1857),Cayley(1889) formula:A set of n points has exactly nn-2 spanning trees

Spanning Trees Let T be a spanning tree of the set S.

Weight ωt(T) is defined to be the sum of the length of its edges

length of an edge {p,q} is the Euclidian distance |pq| between p and q

Minimum Spanning Tree - MST(S) of set S is a spanning tree of minimum weight.

MST Properties

A MST(S) is a shortest network connecting the points of S

In MST(S), each point has degree at most 6.

Let S be a set of n points in the plane, then:

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Steiner Trees

We allow additional points (Steiner points) to be added to the graph, in order to reduce the total length

We define: Steiner Minimum Tree - SMT(S) be a graph

with minimum total length(weight)

Interconnect set S of points by a network (graph) of shortest length, where the length is the sum of the lengths of all edges. MST…?

Steiner Trees Ex.1

C

A B

XMST(S) =SMT(S) =

Steiner Trees

Ex.2

C

A B

X1

MST(S) =SMT(S) =

D

X2

Steiner Trees Lemma (1.1.4)

Let S be a finite set of points in Rd, then:

ωt(MST(S)) ≤ 2 ∘ ωt(SMT(S)) Proof :

Let : SMT(S)=T

we will construct a spanning tree T* of S having at most twice the weight of T

Steiner Trees (cont.)

Proof (construction):

C

A B

X

SMT(S)=TW=

Euler’s tour: W’

ωt(W’ ) = ωt(W ) = 2∘ ωt(T )

Steiner Trees (cont.)

As result we have:T* that visits each point of S exactly onceωt(T*) ≤ 2∘ωt(T)

remove all Steiner points

For each point p of S, remove all visits to p except the first one

We will construct T* of points of S from W’ (Euler’s tour) applying the short-cuts technique:

Steiner Trees (cont.)

Recall:

C

A B

X

Euler tour: W’

?

Steiner Trees (cont.) Is factor 2 is the best possible…?

23

Gilbert & Pollack (1968) conjectured that: is the best possible

That is, for any finite set S of points of R2 :

ωt(MST(S)) ≤ ∘ ωt(SMT(S))2

3

After 20 years, Du&Hwang settled the conjecture

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Triangulation Define:

Convex Hull of set of points S in a real vector space V is the minimal Convex Set containing S

Y

XX

Y

In Euclidian Space, an object is convex if :

for every pair of points within the object, every point on the straight line segment that joins them is also within the object

A Convex set A non-Convex set

Triangulation Convex Hull

S =

Triangulation

Let S be a set of points in the planeA triangulation is a partition of the convex hull

into triangles , such that vertices of these triangles are exactly points of S

General:Subdivision of geometric object into simplices

In particular: In the plane it is a subdivision into triangles,

hence the name.

Triangulation Ex.3

S =

Delaunay Triangulation

Boris Delaunay (1934)

Delaunay triangulation for a set S of points in the plane is a triangulation DT(S) that: no point of S, is inside the circumcircle of any

triangle in DT(S)

Property:DT(S) contains MST(S)

Delaunay Triangulation

Example

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Spanners Spanner Definition.

Let S be a set of n points in Rd and let t ≥ 1 be a real number. A t-spanner for S is an undirected graph G with vertex set S, such that for any two points p and q of S, there is a path in G between p and q, whose length is less than or equal to t|pq|. Any path satisfying this condition is called a t-spanner path between p and q.

Spanners

1-spanner 1.5-spanner 3-spanner

Spanners

Stretch Factor Definition.

Let S be a set of n points in Rd and let G be a Euclidean graph with vertex set S. The stretch factor of G is the smallest real number t such that G is a t-spanner of S.

Spanners

Spanner Property.Let S be a set of n points in Rd and let t ≥ 1 be a real number. Any t-spanner of S has:

1. Number of edges - at least n – 1

2. Weight - at least ωt(MST(S))

Spanners

Basic Spanner Problem

Let S be a set of n points in Rd and let t ≥ 1 be a real number.

Does there exist a t-spanner for S having at most ctdn edges, where ctd is a real number that depends only on t and d?

If so, how much time does it take to compute such a t-spanner?

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Spanners Application Approximating Minimum Spanner Trees

Let S be a set of n points in the plane.

Since any MST(S) of S is contained in Delanay triangulation DT(S) of S, we can compute MST(S) in the following way:

1. Compute DT(S) in O(n log n) time.

2. Compute MST of DT(S), which is in fact an Euclidean MST of the set S.

Since DT(S) contains only linear number of edges, entire algorithm runs O(n log n)

Theorem

Let S be a set of n points in Rd, let t ≥ 1 be a real number and let G be an arbitrary t-spanner for S. A minimum spanning tree of G is t-approximate minimum spanning tree of S, that is the weight of any MST of G is at most

t∘ωt(MST(S))

Proof

Let T be a minimum spanning tree of S, and number its edges arbitrarily as e1, e2,…en-1. Consider edge ei. Since G is a t-spanner for S, there exists a t-spanner path Pi in G between the endpoints of ei. Thus the length wt(Pi) of Pi is at most t times the length of ei.

It follows that:

1

1

1

1

))((*)(*)(n

i

n

iii SMSTwttewttPwt

Proof (cont.) Let G’ be the subgraph of G, whose edge set is

the union of the edge sets of the paths Pi, 1 ≤ i ≤ n-1.

Then G’ is a connected graph with vertex set S and its weight is at most t*wt(MST(S)).

Since the weight of the minimum spanning tree of G is less than of equal to the weight of G’, the proof is complete.

Introduction Outline

The Problem Domain Network Design Spanning Trees Steiner Trees Triangulation Technique Spanners Spanners Application Simple Greedy Algorithm

Greedy Spanner AlgorithmAlgorithm PathGreedy (S, t)

Input: Set S of n points in Rd , t > 1

Output: G=(S, E) – a t-spanner for S

Sort the ( ) pairs of distinct points in nondecreasing order of their distances and store them in list L

E {} G (S, E) for each pair (u,v) є L (considering pairs in sorted order)

let dG(u,v) be the length of the shortest path in G between u and v

if dG(u,v) > t |u,v| add (u,v) to E

Output G

2

n

Greedy Spanner Algorithm (example 1)

A

CD

B Consider:

S = {A, B, C, D}, t = 2

AD = BC = 3

AB = DC = 4

AC = BD = 5

Sorting Pairs of S:

L = {AD, BC, AB, DC, AC, BD}

Deciding (D, C): (dG(D,C) = 10 > 2 * 4)? True! DC is added to G

Deciding (A, C): (dG(A,C) = 7 > 2 * 5)? False! AC is not added to G

Deciding (B, D): (dG(B,D) = 7 > 2 * 5)? False! BD is not added to G

G

Greedy Spanner Algorithm (example 2)

A

CD

B Consider:

S = {A, B, C, D}, t = 1.3

AD = BC = 3

AB = DC = 4

AC = BD = 5

Sorting Pairs of S:

L = {AD, BC, AB, DC, AC, BD}

Deciding (D, C): (dG(D,C) = 10 > 1.3 * 4)? True! DC is added to G

Deciding (A, C): (dG(A,C) = 7 > 1.3 * 5)? True! AC is added to G

Deciding (B, D): (dG(B,D) = 7 > 1.3 * 5)? True! BD is added to G

G