1. GRAPH SPANNERS

2. 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.

3. 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

4. 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 .

5. SPARSENESS
Let Weight (G) denote the sum of all edge weights of a n-vertex graph G
Let Size (G) denote the number of edges
in G.
Then,
A graph is sparse in size if it has a few edges.
A graph is sparse in weight if its total edge weight is small.

6. OBJECTIVE
To keep stretch factors constant.

7. 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’.

8. 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.

9. 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.

10. 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.

11. PROPERTIES(1)
1.G’ is a r-spanner of G.

12. PROPERTIES(1) 2.Let C be any simple cycle in G’,then
size(C) > r+1.This proves the size sparseness.

13. PROPERTIES(1) 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.

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

15. 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.

16. 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.

17. 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.

18. PROOF(5)
Since Msize(E) = MST(G), and G’size(E) = G’, the property, MST(G) is contained in G’, is proved.

19. 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.

20. 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

21. 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))

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

23. 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.