Lower Bound for Sparse Euclidean Spanners Presented by- Deepak Kumar Gupta(Y6154), Nandan Kumar Dubey(Y6279), Vishal Agrawal(Y6541)

Road Map Introduction To Spanners Algorithm to Construct Spanners  Outlier Algorithm  Properties of Spanners from this Algorithm Lower Bound for Sparse Euclidean Spanners  Introduction  Related Work  Terminologies  Proof(as presented in Paper) References

Introduction A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. The number k is the dilation or stretch factor. Given a set of points V = (v 1,v 2,v 3,…..,v n ) and a graph G(V,E). Define the weight of an edge e = (v i,v j ) as w(e).The weight of a graph G’(V’,E’) is defined as, w(G’) = Σ w(e i ) where e i Є E’. The shortest path between nodes v i and v j denoted by P G (v i,v j ) is the smallest weight path that connects v i and v j in G. The minimum link path, denoted by Π(v i,v j ) is the one with the smallest number of edges. Define the diameter of the graph as Δ(G) = max |Π(v i,v j )| where 1≤i,j ≤n A subgraph G’(V,E’) of G is a t-spanner of G if for any v i,v j Є V, W(P G’ (v i,v j )) / W(P G (v i,v j )) ≤ t

SPANNERS 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, 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 Input : A weighted graph G, A positive parameter r. The weights need not be unique. Output : A sub graph G’. ALGORITHM SPANNER(G(V,E),r) begin sort E by non-decreasing weight; Set G’ = (V,{ }). For every edge e = [u,v] in E do begin computer P(u,v), the shortest path from u to v in the current G’; If( r.Weight(e) < Weight(P(u,v))) then, add e to G’; end; output G’; end;

PROPERTIES The following Lemmas describe the properties of the output graph G’ G’ is a r-spanner of G. Let C be any simple cycle in G’,then size(C) > r+1. Let C be any simple cycle in G’ and let e be any edge in C, then Weight(C – {e}) > r.Weight(e) MST(G) is contained in G’.

G’ is a r-spanner of G. PROOF: PROPERTIES

Let C be any simple cycle in G’,then size(C) > r+1. PROOF: PROPERTIES

Let C be any simple cycle in G’ and let e be any edge in C, then Weight(C – {e}) > r.Weight(e) PROOF: PROPERTIES

MST(G) is contained in G’. PROOF: PROPERTIES

Related Work

Points V = (v 1,v 2,v 3,…..,v n ) is a set of n ordered points in R 1, such that any two consecutive points are unit distance apart. A block of nodes [i:j] is defined as = (v i, i+2,…..,v j ), and v i and v j are referred to as endpoints of the block. Let X(v k ) be the covering of a node v k, defined as the number of edges that span over v k, i.e., the number of edges (v i,v j ) such that i<k<j. Set X(G) = max X(v) where vЄV. Two edges (v i,v j ) and (v k,v l ) intersect if i<k<j<l. A graph is called a stack if it only contains non-intersecting edges. A cluster in a stack graph G is a maximal subgraph G’ = (V’,E’) induced by V’ = [i:j] such that edge (v i,v j ) Є E’, and no edge in E/E’ spans over any point in V’. Terminologies and Notation

PROOF(1) Lemma1: If X(n,) ≥ g(n), where g(n) is a concave function, then w(n, δ) = Ω(ng(n)). Proof:

PROOF(2) Lemma2: For any graph G = (V,E), there is a stack graph S = (V, ε) such that X(S) ≤ X(G) and Δ (S) ≤ (X(G)+1) Δ (G). Proof:

PROOF(3) We now present the relation between the diameter and covering of stack graphs.

PROOF(4) Putting everything together.

RESULT Theorem: Corollary:

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.

REFERENCES On Sparse Spanners of Weighted Graphs : By Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph and José Soares. Lower Bound for Sparse Euclidean Spanners : By Pankaj K. Agarwal, Yusu Wang, Peng Yin.