GRAPH SPANNERS

by

S.Nithya

Spanner Definition- Informal

A geometric spanner network for a set of points is a graph G in which each pair of vertices is connected by a “short” path.

Spanner Definition-Formal(1)

Let S be a finite set of points in the plane and let t>1 be a real number.

Let G=(S,E) be a (directed or undirected) graph with vertex set S in which edges are drawn as straight-line segments joining two vertices.

Let d_G(p,q) = Euclidean length Let |pq| = Euclidean distance

Spanner Definition-Formal(2)

We say that G is a t-spanner for S, if d_G(p,q) < t |pq| for any two vertices p and q of S. The smallest value of t for which G is a t-spanner for S is called the stretch factor of G .

SPARSENESS

Let Weight (G) denote the sum of all edge weights of a n-vertex graph GLet Size (G) denote the number of edges in G.Then,

1. A graph is sparse in size if it has a few edges.2. A graph is sparse in weight if its total edge

weight is small.

OBJECTIVE

To keep stretch factors constant.

Algorithm to Construct Spanners(1)

Input : A weighted graph G,

A positive parameter r.

The weights need not be unique.

Output : A sub graph G’.

Algorithm to Construct Spanners(2)

Step 1: Sort E by non-decreasing weight.

Step 2: Set G’ = { }.

Step 3: For every edge e = [u,v] in E, compute P(u,v), the shortest path from u to v in the current

G’.

Step 4: If, r.Weight(e) < Weight(P(u,v)),

then, add e to G’,

else, reject e.

Algorithm to Construct Spanners(3)

Step 5:Repeat the algorithm for the next edge in E, and so on.

This algorithm is very simple and easy to implement

and has many interesting properties.

PROPERTIES(1)

1.G’ is a r-spanner of G.

2.Let C be any simple cycle in G’,then

size(C) > r+1.This proves the size

sparseness.

3.Let C be any simple cycle in G’ and let e be

any edge in C, then Weight(C – {e}) > r.Weight(e)

This proves the weight sparseness.

PROOF(1)

4. MST(G) is contained in G’.

Proof:

Let the sequence { } = G0’,G1’,…,Gsize’ = G’.

Let the sequence { } = M0,M1,…,Msize(E) = MST(G).

In both the graphs ,at any stage the sub-graph will be a collection of connected components, which will finally become one component.

PROOF(2)

The only difference is that there will be no cycle in a Kruskal’s algorithm.

To prove by induction, for all i, the number of connected components of Mi is the same as that of Gi’, and each component of Mi is contained in a corresponding component of Gi’.

To prove this, let us assume that both algorithms are being run simultaneously.

PROOF(3)

Assume that the hypothesis is true for some i. Let the i + 1th edge to be considered be e = [u,v].Case 1: u and v belong to the same component of Mi .Then e forms a cycle with Mi, and hence it is not included in Mi+1.Since they belong to the same component in Mi, by the hypothesis they also belong to the same corresponding component of Gi’.Now whether e is included or not in Gi+1’, the hypothesis remains true.

PROOF(4)

Case 2: u and v belong to different components

of Mi.Then e does not form a cycle with Mi, and hence it is included in Mi+1.Here,2 components of Mi merge to form 1 component in Mi+1.By the hypothesis u and v belong to different corresponding components of Gi’.Thus,the distance from u to v in Gi’ is infinite. Thus e will be added in Gi+1’,the hypothesis remains true.

PROOF(5)

Since Msize(E) = MST(G), and Gsize’(E) = G’,

the property, MST(G) is contained in G’, is proved.

PLANAR GRAPHS

Definition: A planar graph is one that can be drawn on a plane in such a way that there are no "edge crossings," i.e. edges intersect only at their common vertices.

PROPERTY

If all the faces of an n-vertex connected planar graph G have sizes >= r, then

Size(G) <= (n-2)(1 + (2/r-2))

Proof:

If we traverse the boundary of each face and mark the edges encountered, every edge in the graph will be marked twice. Euler's formula for planar graphs states that n-m+f = 2,where

PROOF(1)

N number of vertices.

M size of the graph.

f number of faces.

Since the size of each face >= r, we have f.r<=2m.Thus,

(m+2-n)r<= 2m,

m(r-2)<=(n-2)r,

m<=(n-2)(1+(2/r-2))

FACTS

This proof exists only when r <= 2n-2 because this is the maximum possible face size of a connected planar graph.

APPLICATIONS

Unit edged spanners appear in distributed systems, communication network design and genetics.

Spanners are used to design routing tables in a communication network.

Designs synchronizers which is a distributed scheme that simulates synchrony on an asynchronous distributed system.

CONCLUSIONS

The concept of Spanners were generalized. Spanner algorithm was introduced. Some lemma about arbitrary edge weighted graphs and

planar graphs were proved.

MOTIVATION

I am the only one doing this topic. I read the papers and acquired most of the interest. A very interesting concept on the whole.

THANK YOU!!! (for putting up with me)