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 = (v1,v2,v3,…..,vn) and a graph G(V,E).

Define the weight of an edge e = (vi,vj) as w(e).The weight of a graph G’(V’,E’) is defined as, w(G’) = Σ w(ei) where ei Є E’ .

The shortest path between nodes vi and vj denoted by PG(vi,vj) is the smallest weight path that connects vi and vj in G. The minimum link path, denoted by Π(vi,vj) is the one with the smallest number of edges.

Define the diameter of the graph as Δ(G) = max |Π(vi,vj)| where 1≤i,j ≤n

 A subgraph G’(V,E’) of G is a t-spanner of G if for any vi,vj Є V, 

W(PG’(vi,vj)) / W(PG(vi,vj)) ≤ t

SPANNERS

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,

• 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.

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’.

Lemma1: G’ is a r-spanner of G.

PROOF: On board

PROPERTIES

Lemma2: Let C be any simple cycle in G’, then size(C) > r+1.

PROOF: On board

PROPERTIES

Lemma3: Let C be any simple cycle in G’ and let e be any edge in C,

then Weight(C – {e}) > r.Weight(e).

PROOF: On board

PROPERTIES

Lemma4: MST(G) is contained in G’.

PROPERTIES

Proof continued…

Abstract

Given a one-dimensional graph G such that any two consecutive nodes are unit distance away, and such that the minimum number of links between any two nodes (the diameter of G) is O(log n), we prove an Ω(n log n / log log n) lower bound on the sum of lengths of all the edges (i.e., the weight of G).

Lower Bound for Sparse Euclidean Spanners

This one-dimensional graph problem is related to the partial sum problem, where given an array of numbers A[1],….,A[n], one would like to construct a data structure of small size so that a partial sum like S(i,j) = ΣA[k] where i≤ k ≤ j can be computed efficiently. Roughly speaking, the query time there corresponds to the diameter in our case, while the canonical sets usually constructed for the data structures there correspond to the edge set in our graphs.

Related Work

Points V = (v1,v2,v3,…..,vn) is a set of n ordered points in R1 , such that any two consecutive points are unit distance apart. A block of nodes [i:j] is defined as = (vi,i+2,…..,vj) , and vi and vj are referred to as endpoints of the block.

Let X(vk) be the covering of a node vk, defined as the number of edges that span over vk, i.e., the number of edges (vi,vj) such that i<k<j.

Set X(G) = max X(v) where vЄV.

Two edges (vi,vj) and (vk,vl) 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 (vi,vj) Є E’, and no edge in E/E’ spans over any point in V’.

Terminologies and Notation

PROOF(1)

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

Proof: Let G = (V,E) be a with diameter δ, covering X(n, δ), and weight w(n, δ). Heavy node: X(v) ) ≥ g(n/6)•Claim: |Vh| ≥ n/2 (This implies the lemma)•Proof by contradiction(on board)

•Use g(n/k) ≥ g(n)/k for k>1 for concave function.

PROOF(2)The next lemma allows us to focus only on the covering and diameter of a stack graph.

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).

Intuition: In order to obtain a stack graph, we wish to split an edge e if it intersects other edges. However, we have to do it in a way such that we do not introduce new intersections while removing an old one.

Proof: In particular, given a graph G’ = (V’,E’), where V’ = [i:j] is a block of nodes from V, let E’ = E’\{(vi,vj)} if edge (vi,vj) exists.

Remaining proof on board

The above process produces a stack graph without increasing the covering for any vertex v Є V. Furthermore, a split on an edge e Є E happens only if e intersects some edge that covers its left endpoint, and each split removes at least one such intersection. This implies that Δ (S) ≤ (X(G)+1) Δ (G).

PROOF(3)

We now present the relation between the diameter and covering of stack graphs.Claim: n = |V| ≤ (2.Δ(S))X(S)+2 --(2.1)

PROOF(4)

Putting everything together. By (2.1) and Lemma 2, for any graph G with n nodes, there is a corresponding stack graph S also with n nodes:

RESULT

Theorem: Given any 1-dimensional graph G with unit distance between consecutive nodes and Δ(G) = O(log n), then w(G) = Ω(n log n / log log n).

Corollary: There is a graph that any of its t-spanners with diameter O(log n) has Ω(w(T) log n / log log n) weight.

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.