1. Spanners With Low Average Stretch
2/25/2019
Spanners With Low Average Stretch
Presented By: James Cassidy
April 10, 2003
Chapter 18

2. 18.1 Basic Properties of Average Stretch Spanners
Definition [Average stretch]:
Let G = (V,E,) be a weighted multigraph and H a spanning subraph of G. Assume that the edge weights (e) are normalized so that the lightest edge has weight 1. For every subset of edges E’, denote
Cost(E’,H,G) =
Av_Stretch(G,H) =
Corollary :
For ever weighted n-vertex graph G=(V,E,), there exists a spanning subgraph H=(V,E’) such that |E’| = O(n) and Av_Stretch(H) = O(log n).
Corollary :
For every weighted n-vertex graph G=(V,E,) and for ever k > 1, there exists a (polynomial-time constructible) spanning subgraph H=(V,E’) such that |E’|<nn1/k and Av_Stretch(H)2k+1.
Lemma :
For every k1 and constant 0<c<1, there exist a constant d>0 and (infinitely many) n-vertex graphs G = (V,E) such that for every spanning subgraph H of G, if Av_Stretch(H) < ck, then H has at least dn1+1/k edges.
April 10, 2003
18.1 Basic Properties of Average Stretch Spanners

3. 18.1 Basic Properties of Average Stretch Spanners
Proof:
By the second claim of Lemma , there exist (infinitely man) n-vertex graphs G = (V,E) with girth k or higher and at least ¼n1+1/k edges. Fix d = (1 - c)/4. Consider such a graph G, and suppose that H is a spanning subgraph of G with fewer than dn1+1/k edges. Then the number of edges of G that were not taken into H is at least (c/n)n1+1/k.The high girth of G implies that for each of these edges, the shortest path connecting its endpoints in H is of length k or more. Therefore each of these edges contributes at least k to the summation in the expression for Av_Stretch(H). The edges of H contribute 1 each. Hence
Av_Stretch(H)  = (1 - c) + ck  ck.
Lemma :
For every integer r  3 and n-vertex, m-edge graph G = (V,E) with Girth(G)  r, m  n1+2/(r-2)+n.
For every integer r  3, there exist (infinitely many) n-vertex, m-edge graphs G = (V,E) with Girth(G)  r and m ¼n1+1/r.
Corollary :
For ever constant 0 < c < 1, there exist a constant d > 0 and n-vertex graphs G=(V,E) such that for every spanning subgraph H of G, if H has fewer than dn edges, then Av_Stretch(H)  clog(n).
April 10, 2003
18.1 Basic Properties of Average Stretch Spanners

4. Low Average Stretch Trees
April 10, 2003

5. 18.2 Low Average Stretch Trees
Definition [Minimum average stretch]:
Letting T range over all spanning trees of G, define
Wheel Graph
Spanning Tree of Wheel Graph
Example 1: Rings with diagonals.
Given n even, let Wn be the wheel graph consisting of an n-vertex ring Cn together with the chords joining antipodal points on the ring. Consider the following spanning trees for Wn. Let T1 be a tree consisting of all the edges of the ring save one. Then
A better choice would be the tree T2 consisting of a path of n/2 edges on Cn and the (n/2-1) diagonals having one endpoint in the path. Here we have
Hence Sopt(Wn)8/3-14/(3n).
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18.2 Low Average Stretch Trees

6. 18.2 Low Average Stretch Trees
Example 2: Two-dimensional grids.
Lower bounding the value of Sopt(Gn) for the 2-dimensional grid Gn is difficult, and we only illustrate the construction by example: the tree T illustrated in Figure18.2 has Av_Stretch(T) = O(log(n)). Note that other choices of a spanning tree for the grid, such as the tree of Figure 18.3, might be far worse.
Figure 18.2
Figure 18.3
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18.2 Low Average Stretch Trees

7. Av_Stretch(T)=Cost(E,T,G)/|E|½(aln(n)-1). Thus, Sopt(G)½(aln(n)-1).
Theorem :
There exists a positive constant c such that, for n sufficiently large, there exists an unweighted n-vertex graph G such that Sopt(G)cln(n).
Proof:
The second claim of Lemma implies, by an appropriate choice of parameters, that there exists a positive constant a such that, for n sufficiently large, there exists an n-vertex graph G with 2n edges such that every cycle in G is of length at least aln(n). (In particular, there are 4-regular graphs of girth aln(n) constructed by growing a tree to depth approximately log4n and appropriately linking the leaves.) Let T be any spanning tree in G. Then distT(u,v)/(e)  aln(n)-1 for any nontree edge e=(u,v). Since more than half the edges are nontree edges, it follows that, for ever T,
Av_Stretch(T)=Cost(E,T,G)/|E|½(aln(n)-1).
Thus, Sopt(G)½(aln(n)-1).
It is also known that the n-vertex grid necessitates average stretch (log n), and this result can also be generalized to multidimensional grids.
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Lower Bound on Average Stretch Trees

8. Constructing Average Stretch Trees on Unweighted Graphs
April 10, 2003

9. 18.3 Constructing Average Stretch Trees on Unweighted Graphs
Lemma :
For every n-vertex weighted multigraph G=(V,E,), there exists a multigraph G’=(V,E’,’) with at most n(n+1) edges such that Sopt(G)2•Sopt(G’)
Overview of the construction:
April 10, 2003
18.3 Constructing Average Stretch Trees on Unweighted Graphs

10. 18.3 Constructing Average Stretch Trees on Unweighted Graphs
Set j  1 and Gj  G.
Set x  x(n) as defined above.
While Ej  do:
Invoke S  AV_PARTe(Gj) to partition Gj into clusters.
Construct an SPT TC for each cluster C  S.
For each edge e of each of the constructed trees, add the corresponding edge of the original graph G to the output tree T.
Construct the next multigraph Gj+1 by contracting each clust of S into a single vertex, discarding covered edges (connecting endpoints in the same cluster) from Ej and replacing each intercluster edge by a new edge connecting the corresponding contracted vertices.
Set j = j + 1.
AV_STR_TREE
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18.3 Constructing Average Stretch Trees on Unweighted Graphs

11. 18.3 Constructing Average Stretch Trees on Unweighted Graphs
Theorem [Average stretch-unweighted]:
There exists a constant c such that, for n sufficiently large, every n-vertex unweighted multigraph G satisfies Sopt(G)  exp(c ).
Lemma :
|Ej|  |E|/xj for every j  1.
Corollary :
Algorithm AV_STR_TREE terminates after at most (3lnn)/(lnx) iterations.
Lemma :
In iteration j, the radius of each cluster is bounded above by yj+1.
Lemma :
Cost(E,T,G)  4x(9ln n) (3ln n)/(ln x) |E|.
Lemma :
Av_Stretch(G,T) = exp(O( )).
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18.3 Constructing Average Stretch Trees on Unweighted Graphs

12. Constructing Average Stretch Trees on Weighted Graphs
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13. 18.4 Constructing Average Stretch Trees on Weighted Graphs
Overview
Merge increases radius increases by weight of heaviest merged edge instead of just by one
Growing cluster cannot be controlled
Problems
Modifications
E broken into classes Ei, i1, according to weights
Ei contains weights in range [yi-1,yi)
y chosen parameter y = y(n,x)
x same as in unweighted case
Intuitively, we handle light edges first.
Each class Ei first used in iteration i
Each class Ei used for O(ln n/ln x) iterations after first use
y controls two functions
Clusters in graph Gj in jth iteration will have radius at most y/3
Governs weight ranges of classes (see second point)
Radius of j-cluster is bounded by rj  yj+1
Each merged layer increases radius by: yj + 2rj-1  3yj.
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18.4 Constructing Average Stretch Trees on Weighted Graphs

14. Construction Algorithm
Parameters
θ = 3ln n/ln x,
ψ = 9θln n,
y = xψ.
Ei = {e|ω(e) [yi-1,yi)}.
Eij = set of edges from Ei still uncovered at beginning of iteration j
V(p)=Γpun(v) \ Γp-1un(v)
set of vertices at unweighted distance exactly p from u in Ĝ.
Êji(p)= set of edges of Êji that join a vertex in V(p) with a vertex in V(p)V(p-1)
Set S  .
While Ĝ  do:
Choose arbitrarily a center vertex u in Ĝ.
Let p* be the least p such that for all 1  I  j,
If no such p exists, then C  V; Else C 
Set S  S ∪C and remove the vertices of C from V
AV_PART(Ĝ) ^ ^
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18.4 Constructing Average Stretch Trees on Weighted Graphs

15. 18.4 Constructing Average Stretch Trees on Weighted Graphs
Analysis
Lemma : |Eji|  |Ei|/xj-1 for every 1  i  j.
Lemma : In iteration j, the radius of each cluster is bounded above by yj+1.
Lemma : Cost(E,T,G)  4x2ψθ+1|E|.
Lemma [Average stretch]:
There exists a constant c such that for n sufficiently large, every n-vertex weighted multigraph G satisfies Sopt(G) 
April 10, 2003
18.4 Constructing Average Stretch Trees on Weighted Graphs

16. Light Trees With Low Average Stretch
Construct tree T with simultaneously low Av_Stretch(T) and low ω(T).
Construct G’ = (V,E’) for G as in previous section, using k=log n.
Stretch(G’) = O(log n)
|E(G’)| = O(nΛ·log n)
ω(G’) = O(Λ ·log n · ω(MST))
Compute shorted path Ў(e) in G’ for every edge e E
Supp(e’) = {e E | e’  Ў(e)}
Define g(e’) =
Construct a weighted multigraph G’’(V,E’’, ω’’) by taking each edge e’ of G’ with multiplicity g(e’).
Construct a tree T for G’’ as in Section 18.4
April 10, 2003
18.5 Light Trees With Low Average Stretch

17. 18.5 Light Trees With Low Average Stretch
Analysis
Lemma :
T is no heavier than G’, whose weight is guaranteed to be  O(Λ·log n) times heavier than ω(MST).
So, T  O(Λ·log n) ·ω(MST)
Lemma : The average stretch of the tree T satisfies
Av_Stretch(G,T)  c’log n·exp(c·sqrt(log n·log log n))
for some constants c,c’
Theorem :
There exist constants c,c’,c’’ such that, given an n-vertex weighted graph G=(V,E,ω), there is a polynomially constructible spanning tree T such that
Av_Stretch(T)  c’log n·exp(c·sqrt(log n·log log n)) and ω(T)  c’’Λ·log n· ω(MST)
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18.5 Light Trees With Low Average Stretch

18. 18.5 Light Trees With Low Average Stretch
Proof for Lemma
Proof:
For an edge e = (u,w) in a graph H, denote distH(e) = distH(u,w).
First observe that
Since G’ guarantees a stretch factor of Av_Stretch(G’) Stretch(G’)=O(log n), it follows
that there exists a constant c such that |E’’|  c|E|·log n.
Next, observe that for every edge e E,
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18.5 Light Trees With Low Average Stretch

19. Proof for Lemma 18.5.1 (continued)
Therefore
Combining the last two equations yields
April 10, 2003
Chapter 18 (Basic Properties of Average Stretch Spanners)