Lower Bound for Sparse
Euclidean Spanners
Presented by- Deepak Kumar Gupta(Y6154),
Nandan Kumar Dubey(Y6279),
Vishal Agrawal(Y6541)
Road Map
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Introduction To Spanners
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Algorithm to Construct Spanners
 Outlier Algorithm
 Properties of Spanners from this Algorithm
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Lower Bound for Sparse Euclidean Spanners
 Introduction
 Related Work
 Terminologies
 Proof(as presented in Paper)
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References
Introduction
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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 nvertex graph G
Let Size (G) denote the number of edges in G.
Then,
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A graph is sparse in size if it has a few edges.
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A graph is sparse in weight if its total edge weight is small.
OBJECTIVE: To keep stretch factors constant.
Algorithm to Construct Spanners
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Input : A weighted graph G, A positive parameter r.
The weights need not be unique.
Output : A sub graph G’.
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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’
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G’ is a r-spanner of G.
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Let C be any simple cycle in G’,then size(C) > r+1.
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Let C be any simple cycle in G’ and let e be any edge in C,
then Weight(C – {e}) > r.Weight(e)
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MST(G) is contained in G’.
PROPERTIES
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Lemma1: G’ is a r-spanner of G.
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PROOF: On board
PROPERTIES
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Lemma2: Let C be any simple cycle in G’, then size(C) > r+1.
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PROOF: On board
PROPERTIES
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Lemma3: Let C be any simple cycle in G’ and let e be any edge in C,
then Weight(C – {e}) > r.Weight(e).
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PROOF: On board
PROPERTIES
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Lemma4: MST(G) is contained in G’.
Proof continued…
Lower Bound for Sparse Euclidean Spanners
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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).
Related Work
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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.
Terminologies and Notation
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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.
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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.
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Set X(G) = max X(v) where vЄV.
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Two edges (vi,vj) and (vk,vl) intersect if i<k<j<l.
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A graph is called a stack if it only contains non-intersecting edges.
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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’.
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.
--(2.1)
Claim: n = |V| ≤ (2.Δ(S))X(S)+2
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
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Unit edged spanners appear in distributed systems, communication
network design and genetics.
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Spanners are used to design routing tables in a communication
network.
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Designs synchronizers which is a distributed scheme that simulates
synchrony on an asynchronous distributed system.
REFERENCES
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On Sparse Spanners of Weighted Graphs : By Ingo Althöfer,
Gautam Das, David Dobkin, Deborah Joseph and José Soares.
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Lower Bound for Sparse Euclidean Spanners : By Pankaj K.
Agarwal, Yusu Wang, Peng Yin.