157
Pairwise spanners T. Kavitha ( School of Tech. & Computer Science, TIFR Mumbai) [January 9, 2015; IISc Bangalore] . – p.1/43

T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

  • Upload
    others

  • View
    3

  • Download
    0

Embed Size (px)

Citation preview

Page 1: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

T. Kavitha

( School of Tech. & Computer Science, TIFR Mumbai)

[January 9, 2015; IISc Bangalore]

. – p.1/43

Page 2: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The problem

Input: an undirected graph G = (V,E) on n vertices.

. – p.2/43

Page 3: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The problem

Input: an undirected graph G = (V,E) on n vertices.

We want a sparse subgraph that well-approximatesevery u-v distance, where (u, v) ∈ V × V .

. – p.2/43

Page 4: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The problem

Input: an undirected graph G = (V,E) on n vertices.

We want a sparse subgraph that well-approximatesevery u-v distance, where (u, v) ∈ V × V .

such a subgraph is called a spanner of G [PS89].

. – p.2/43

Page 5: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An example

���

���

��������

���

���

����

��������

��������

���

���

���

���

����

����

������������������������������������������������������������������

������������������������������������������������������������������

��������������������������

���������������������������������������������������������������������������������������������������������������������������������������

���������������������������������������������������������������������������������������������������������������������������������������

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������

����������������������������������������������������������������

����������������������������������������������������������������

On the left is G with(

n2

)

edges while the subgraph H

on the right has only n− 1 edges.

for all (u, v): dH(u, v) ≤ dG(u, v) + 1.. – p.3/43

Page 6: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Stretch of a spanner H

For all pairs of vertices (u, v):

. – p.4/43

Page 7: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Stretch of a spanner H

For all pairs of vertices (u, v):

dH(u, v) ≤ α · dG(u, v) + β =⇒ H is an(α, β)-spanner

. – p.4/43

Page 8: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Stretch of a spanner H

For all pairs of vertices (u, v):

dH(u, v) ≤ α · dG(u, v) + β =⇒ H is an(α, β)-spanner

for every integer k ≥ 1, G admits a(2k − 1, 0)-spanner of size O(n1+1/k) [ADDJS93].

. – p.4/43

Page 9: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Partition the vertex set into disjoint clusters:

����

. – p.5/43

Page 10: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Partition the vertex set into disjoint clusters:

Let C1, . . . , Ci−1 be the clusters constructed so far.

����

. – p.5/43

Page 11: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Partition the vertex set into disjoint clusters:

Let C1, . . . , Ci−1 be the clusters constructed so far.

start with any vertex u outside C1 ∪ · · · ∪ Ci−1:Ci = {u}

����

. – p.5/43

Page 12: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Partition the vertex set into disjoint clusters:

Let C1, . . . , Ci−1 be the clusters constructed so far.

start with any vertex u outside C1 ∪ · · · ∪ Ci−1:Ci = {u}

����u

. – p.5/43

Page 13: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Initially Ci = {u}.

����

����

����

����

����

����

. – p.6/43

Page 14: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Initially Ci = {u}.

Nbr(Ci) = unclustered neighbors of vertices in Ci

����

����

����

����

����

����

. – p.6/43

Page 15: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Initially Ci = {u}.

Nbr(Ci) = unclustered neighbors of vertices in Ci

while |Nbr(Ci)| > n1/k · |Ci| do: Ci = Ci ∪Nbr(Ci).

����

����

����

����

����

����

. – p.6/43

Page 16: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

Initially Ci = {u}.

Nbr(Ci) = unclustered neighbors of vertices in Ci

while |Nbr(Ci)| > n1/k · |Ci| do: Ci = Ci ∪Nbr(Ci).

����

����

����

����

����

����

u

. – p.6/43

Page 17: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

When |Nbr(Ci)| ≤ n1/k · |Ci|, the cluster Ci stopsgrowing.

����

����

����

����

����

����

����

����

����

����

����

����

����

. – p.7/43

Page 18: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

When |Nbr(Ci)| ≤ n1/k · |Ci|, the cluster Ci stopsgrowing.

����

����

����

����

����

����

����

����

����

����

����

����

����

u

. – p.7/43

Page 19: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A simple proof [HZ97]

When |Nbr(Ci)| ≤ n1/k · |Ci|, the cluster Ci stopsgrowing.

����

����

����

����

����

����

����

����

����

����

����

����

����

u

radius(Ci) ≤ k − 1 since the scaling factor is > n1/k.. – p.7/43

Page 20: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The spanner H

Finally V is partitioned into C1 ∪ C2 ∪ · · · ∪ Ct.

�����

���

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

����

. – p.8/43

Page 21: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The spanner H

Finally V is partitioned into C1 ∪ C2 ∪ · · · ∪ Ct.

�����

���

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

����

. – p.8/43

Page 22: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The spanner H

Finally V is partitioned into C1 ∪ C2 ∪ · · · ∪ Ct.

�����

���

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

����

Initialize H = forest defined by the clusters.

. – p.8/43

Page 23: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The final H

For any vertex v and cluster C s.t. v /∈ C and v has aneighbor in C:

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

. – p.9/43

Page 24: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The final H

For any vertex v and cluster C s.t. v /∈ C and v has aneighbor in C:

add to H exactly one edge between v and C

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

. – p.9/43

Page 25: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The final H

For any vertex v and cluster C s.t. v /∈ C and v has aneighbor in C:

add to H exactly one edge between v and C

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

. – p.9/43

Page 26: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The final H

For any vertex v and cluster C s.t. v /∈ C and v has aneighbor in C:

add to H exactly one edge between v and C

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

total number of red edges ≤ n1/k(∑

i |Ci|) = n1+1/k.. – p.9/43

Page 27: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The stretch in H

Consider any edge (a, b) missing in H

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

. – p.10/43

Page 28: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The stretch in H

Consider any edge (a, b) missing in H

∃ edge (b, x) in H where x is in a’s cluster

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

����

. – p.10/43

Page 29: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The stretch in H

Consider any edge (a, b) missing in H

∃ edge (b, x) in H where x is in a’s cluster

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

���� a

bx

. – p.10/43

Page 30: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The stretch in H

Consider any edge (a, b) missing in H

∃ edge (b, x) in H where x is in a’s cluster

����

����

����

����

����

����

����

�����

���

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

����

����

����

����

���� a

bx

diameter of each cluster is ≤ 2k − 2, soδH(a, b) ≤ 2k − 1.

. – p.10/43

Page 31: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Spanners with additive stretch

For every integer k ≥ 1, G has a subgraph of sizeO(n1+1/k) such that ∀(u, v) ∈ V × V :

δH(u, v) ≤ (2k − 1) · δG(u, v)

. – p.11/43

Page 32: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Spanners with additive stretch

For every integer k ≥ 1, G has a subgraph of sizeO(n1+1/k) such that ∀(u, v) ∈ V × V :

δH(u, v) ≤ (2k − 1) · δG(u, v)

H can also be computed efficiently.

. – p.11/43

Page 33: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Spanners with additive stretch

For every integer k ≥ 1, G has a subgraph of sizeO(n1+1/k) such that ∀(u, v) ∈ V × V :

δH(u, v) ≤ (2k − 1) · δG(u, v)

H can also be computed efficiently.

Are there sparse subgraphs H s.t. ∀(u, v):δH(u, v) ≤ δG(u, v) +O(1)?

. – p.11/43

Page 34: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Spanners with additive stretch

For every integer k ≥ 1, G has a subgraph of sizeO(n1+1/k) such that ∀(u, v) ∈ V × V :

δH(u, v) ≤ (2k − 1) · δG(u, v)

H can also be computed efficiently.

Are there sparse subgraphs H s.t. ∀(u, v):δH(u, v) ≤ δG(u, v) +O(1)?

Such an H is called a purely additive spanner.

. – p.11/43

Page 35: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Multiplicative vs Additive spanners

To bound multiplicative stretch, we do it edge by edge.

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

������������������ ������������������������������������������������������������������ ����

. – p.12/43

Page 36: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Multiplicative vs Additive spanners

To bound multiplicative stretch, we do it edge by edge.

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���������������� �������������������������������������������������������������������� ����

≤ δG(u, v) +O(1)

u

u

v

v

. – p.12/43

Page 37: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Purely additive spanners

A (1, 2) spanner of size O(n1.5) [ACIM99].

. – p.13/43

Page 38: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Purely additive spanners

A (1, 2) spanner of size O(n1.5) [ACIM99].

A (1, 4) spanner of size O(n1.4) [C13].

. – p.13/43

Page 39: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Purely additive spanners

A (1, 2) spanner of size O(n1.5) [ACIM99].

A (1, 4) spanner of size O(n1.4) [C13].

A (1, 6) spanner of size O(n1.33) [BKMP05].

. – p.13/43

Page 40: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Purely additive spanners

A (1, 2) spanner of size O(n1.5) [ACIM99].

A (1, 4) spanner of size O(n1.4) [C13].

A (1, 6) spanner of size O(n1.33) [BKMP05].

no other results for purely additive spanners areknown.

. – p.13/43

Page 41: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Purely additive spanners

A (1, 2) spanner of size O(n1.5) [ACIM99].

A (1, 4) spanner of size O(n1.4) [C13].

A (1, 6) spanner of size O(n1.33) [BKMP05].

no other results for purely additive spanners areknown.

size O(n1+δ) vs additive stretch O(n1−3δ

2 ) [C13]

. – p.13/43

Page 42: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A (1, 2) spanner of size O(n3/2)

Low degree vertex: one with degree at most h(h =

√n))

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.14/43

Page 43: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A (1, 2) spanner of size O(n3/2)

Low degree vertex: one with degree at most h(h =

√n))

Special high degree vertex: each such vertex shouldclaim ≥ h unclaimed high degree neighbors

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.14/43

Page 44: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A (1, 2) spanner of size O(n3/2)

Low degree vertex: one with degree at most h(h =

√n))

Special high degree vertex: each such vertex shouldclaim ≥ h unclaimed high degree neighbors

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.14/43

Page 45: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A (1, 2) spanner of size O(n3/2)

Low degree vertex: one with degree at most h(h =

√n))

Special high degree vertex: each such vertex shouldclaim ≥ h unclaimed high degree neighbors

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

there will be at most n/h special (high degree) vertices. – p.14/43

Page 46: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The edge set of H

All edges incident on low-degree vertices andunclaimed high degree vertices are in H.

. – p.15/43

Page 47: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The edge set of H

All edges incident on low-degree vertices andunclaimed high degree vertices are in H.

H contains a BFS tree rooted at each special vertex.

. – p.15/43

Page 48: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The edge set of H

All edges incident on low-degree vertices andunclaimed high degree vertices are in H.

H contains a BFS tree rooted at each special vertex.

The size of H is O(nh+ n2/h).

. – p.15/43

Page 49: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The edge set of H

All edges incident on low-degree vertices andunclaimed high degree vertices are in H.

H contains a BFS tree rooted at each special vertex.

The size of H is O(nh+ n2/h).

Since h =√n, the size of H is O(n3/2).

. – p.15/43

Page 50: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Bounding the stretch in H

Consider any pair of vertices (a, b).

����

����

����

����

����

����

����������

����������

. – p.16/43

Page 51: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Bounding the stretch in H

Consider any pair of vertices (a, b).

If no special vertex is adjacent to any vertex inSP (a, b), then δH(a, b) = δG(a, b).

����

����

����

����

����

����

����������

����������

. – p.16/43

Page 52: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Bounding the stretch in H

Consider any pair of vertices (a, b).

If no special vertex is adjacent to any vertex inSP (a, b), then δH(a, b) = δG(a, b).

Else there is a vertex on SP (a, b) with a specialvertex x as its neighbor

����

����

����

����

����

����

����������

����������

. – p.16/43

Page 53: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Bounding the stretch in H

Consider any pair of vertices (a, b).

If no special vertex is adjacent to any vertex inSP (a, b), then δH(a, b) = δG(a, b).

Else there is a vertex on SP (a, b) with a specialvertex x as its neighbor

����

����

����

����

����

����

�������������������������������������������������������������������������� ������������������������

����������

ℓ1 ℓ2

a cb

x

. – p.16/43

Page 54: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Bounding the stretch in H

Consider any pair of vertices (a, b).

If no special vertex is adjacent to any vertex inSP (a, b), then δH(a, b) = δG(a, b).

Else there is a vertex on SP (a, b) with a specialvertex x as its neighbor

����

����

����

����

����

����

�������������������������������������������������������������������������� ������������������������

����������

ℓ1 ℓ2

a cb

x

δH(a, b) ≤ δH(a, x) + δH(x, b) ≤ ℓ1 + 1 + ℓ2 + 1.. – p.16/43

Page 55: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A generalization: an (S × V )-spanner [KV13]

We are given a set of sources S ⊆ V .

. – p.17/43

Page 56: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A generalization: an (S × V )-spanner [KV13]

We are given a set of sources S ⊆ V .

Compute a sparse subgraph H such thatδH(s, v) ≤ δG(s, v) + 2 for all s ∈ S and v ∈ V .

. – p.17/43

Page 57: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A generalization: an (S × V )-spanner [KV13]

We are given a set of sources S ⊆ V .

Compute a sparse subgraph H such thatδH(s, v) ≤ δG(s, v) + 2 for all s ∈ S and v ∈ V .

As before, we have low degree vertices (those withdegree ≤ h ≈ (n|S|)1/4) and high degree vertices.

. – p.17/43

Page 58: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Each special vertex x should be able to claim at least hunclaimed high degree neighbors

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.18/43

Page 59: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Each special vertex x should be able to claim at least hunclaimed high degree neighbors

these high degree neighbors claimed by x form acluster centered at x

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.18/43

Page 60: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Each special vertex x should be able to claim at least hunclaimed high degree neighbors

these high degree neighbors claimed by x form acluster centered at x

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

. – p.18/43

Page 61: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Each special vertex x should be able to claim at least hunclaimed high degree neighbors

these high degree neighbors claimed by x form acluster centered at x

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

����

���� �

���

H is initialized to the forest defined by clusters alongwith all edges incident on unclustered vertices.

. – p.18/43

Page 62: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Any shortest path p is one of two types:

. – p.19/43

Page 63: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Any shortest path p is one of two types:

p has < τ clustered vertices (where τ ≈√

n/|S|)

. – p.19/43

Page 64: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Any shortest path p is one of two types:

p has < τ clustered vertices (where τ ≈√

n/|S|)

p has ≥ τ clustered vertices

. – p.19/43

Page 65: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Any shortest path p is one of two types:

p has < τ clustered vertices (where τ ≈√

n/|S|)

p has ≥ τ clustered vertices

If p has ≥ τ clustered vertices, then there are ≥ τ/3

special vertices adjacent to p.

. – p.19/43

Page 66: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Any shortest path p is one of two types:

p has < τ clustered vertices (where τ ≈√

n/|S|)

p has ≥ τ clustered vertices

If p has ≥ τ clustered vertices, then there are ≥ τ/3

special vertices adjacent to p.

Sample special high degree vertices w.p. ≈ 3/τ to getvery special vertices.

. – p.19/43

Page 67: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Add to H a BFS tree rooted at each very specialvertex.

����

����

����

����

����

����

����

����

. – p.20/43

Page 68: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Add to H a BFS tree rooted at each very specialvertex.

when SP (s, u) has ≥ τ clustered vertices, we haveδH(s, u) ≤ δG(s, u) + 2

����

����

����

����

����

����

����

����

. – p.20/43

Page 69: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner

Add to H a BFS tree rooted at each very specialvertex.

when SP (s, u) has ≥ τ clustered vertices, we haveδH(s, u) ≤ δG(s, u) + 2

����

����

����

����

����

����

����

����

s u

. – p.20/43

Page 70: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner of size O(n5/4|S|1/4)

For those clustered v s.t. SP (s, v) has ≤ τ clusteredvertices:

����

����

����

����

����

����

����

����

����

. – p.21/43

Page 71: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner of size O(n5/4|S|1/4)

For those clustered v s.t. SP (s, v) has ≤ τ clusteredvertices:

����

����

����

����

����

����

����

����

����s

w

uvℓ

≤ ℓ

. – p.21/43

Page 72: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner of size O(n5/4|S|1/4)

For those clustered v s.t. SP (s, v) has ≤ τ clusteredvertices:

����

����

����

����

����

����

����

����

����s

w

uvℓ

≤ ℓ

Add to H the missing edges of one such pathSP (s, w), where w ∈ C is closest to s (in G).

. – p.21/43

Page 73: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × V )-spanner of size O(n5/4|S|1/4)

For those clustered v s.t. SP (s, v) has ≤ τ clusteredvertices:

����

����

����

����

����

����

����

����

����s

w

uvℓ

≤ ℓ

Add to H the missing edges of one such pathSP (s, w), where w ∈ C is closest to s (in G).

so we have δH(s, v) ≤ δG(s, v) + 2 here. – p.21/43

Page 74: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 4) spanner of size O(n1.4)

Compute a clustering with h = n0.4 log0.2 n, letS = {special vertices}.

����

����

����

����

����

������

������

. – p.22/43

Page 75: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 4) spanner of size O(n1.4)

Compute a clustering with h = n0.4 log0.2 n, letS = {special vertices}.

add the (S × V )-spanner with additive stretch 2

����

����

����

����

����

������

������

. – p.22/43

Page 76: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 4) spanner of size O(n1.4)

Compute a clustering with h = n0.4 log0.2 n, letS = {special vertices}.

add the (S × V )-spanner with additive stretch 2

����

����

����

����

����

������������������������

������

≤ ℓ+ 3

a cb

s

. – p.22/43

Page 77: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 4) spanner of size O(n1.4)

Compute a clustering with h = n0.4 log0.2 n, letS = {special vertices}.

add the (S × V )-spanner with additive stretch 2

����

����

����

����

����

������������������������

������

≤ ℓ+ 3

a cb

s

so δH(a, b) ≤ δH(a, s) + δH(s, b) ≤ δG(a, c) + 1 + ℓ+ 3

= δG(a, b) + 4.. – p.22/43

Page 78: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × S) spanner

We are again given a subset S ⊆ V .

. – p.23/43

Page 79: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × S) spanner

We are again given a subset S ⊆ V .

Compute a sparse subgraph H such thatδH(s1, s2) ≤ δG(s1, s2) + 2 for all s ∈ S.

. – p.23/43

Page 80: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × S) spanner

We are again given a subset S ⊆ V .

Compute a sparse subgraph H such thatδH(s1, s2) ≤ δG(s1, s2) + 2 for all s ∈ S.

Low degree vertices have degree ≤ h =√

|S|

. – p.23/43

Page 81: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An (S × S) spanner

We are again given a subset S ⊆ V .

Compute a sparse subgraph H such thatδH(s1, s2) ≤ δG(s1, s2) + 2 for all s ∈ S.

Low degree vertices have degree ≤ h =√

|S|

As before, some high degree vertices are unclusteredand the rest are clustered around special vertices.

. – p.23/43

Page 82: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Let p be the shortest path between s1 and s2.

����

����

����

����

����

����

����

����

. – p.24/43

Page 83: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Let p be the shortest path between s1 and s2.

����

����

����

����

����

����

����

����

s1 s2p = SP (s1, s2)

. – p.24/43

Page 84: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Let p be the shortest path between s1 and s2.

����

����

����

����

����

����

����

����

s1 s2p = SP (s1, s2)

p is one of two types:

. – p.24/43

Page 85: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Let p be the shortest path between s1 and s2.

����

����

����

����

����

����

����

����

s1 s2p = SP (s1, s2)

p is one of two types:

there is no cluster C incident on p such thatδH(C, s1) ≤ δp(C, s1) and δH(C, s2) ≤ δp(C, s2)

. – p.24/43

Page 86: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Or ∃C on p such that

δH(C, s1) ≤ δp(C, s1) and δH(C, s2) ≤ δp(C, s2)

����

����

����

����

����

����

����

����

. – p.25/43

Page 87: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Or ∃C on p such that

δH(C, s1) ≤ δp(C, s1) and δH(C, s2) ≤ δp(C, s2)

����

����

����

����

����

����

����

����

s1 s2

C

≤ ℓ1 ≤ ℓ2

ℓ1 ℓ2

. – p.25/43

Page 88: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

Or ∃C on p such that

δH(C, s1) ≤ δp(C, s1) and δH(C, s2) ≤ δp(C, s2)

����

����

����

����

����

����

����

����

s1 s2

C

≤ ℓ1 ≤ ℓ2

ℓ1 ℓ2

so we already have δH(s1, s2) ≤ δG(s1, s2) + 2 in thiscase

. – p.25/43

Page 89: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

If there is no such cluster then we add all the missingedges of p to H.

����

����

����

����

����

����

����

����

. – p.26/43

Page 90: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

If there is no such cluster then we add all the missingedges of p to H.

Now every cluster incident on p reduces its distance toeither s1, s2.

����

����

����

����

����

����

����

����

. – p.26/43

Page 91: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

If there is no such cluster then we add all the missingedges of p to H.

Now every cluster incident on p reduces its distance toeither s1, s2.

����

����

����

����

����

����

����

����

s1 s2p = SP (s1, s2)

. – p.26/43

Page 92: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Path-buying

If there is no such cluster then we add all the missingedges of p to H.

Now every cluster incident on p reduces its distance toeither s1, s2.

����

����

����

����

����

����

����

����

s1 s2p = SP (s1, s2)

So δH(s1, C) or δH(s2, C) has strictly decreased and wecharge such (s, C) pairs to pay for the edges added.

. – p.26/43

Page 93: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The size of H

This charging mechanism bounds the number of suchedges added to H by O(|S|·(number of clusters)).

. – p.27/43

Page 94: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The size of H

This charging mechanism bounds the number of suchedges added to H by O(|S|·(number of clusters)).

The number of clusters ≤ n/h = n/√

|S|.

. – p.27/43

Page 95: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The size of H

This charging mechanism bounds the number of suchedges added to H by O(|S|·(number of clusters)).

The number of clusters ≤ n/h = n/√

|S|.

So the number of edges added by path-buying isO(n

|S|).

. – p.27/43

Page 96: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

The size of H

This charging mechanism bounds the number of suchedges added to H by O(|S|·(number of clusters)).

The number of clusters ≤ n/h = n/√

|S|.

So the number of edges added by path-buying isO(n

|S|).

The number of edges incident on unclustered verticesis O(nh), which is again O(n

|S|).

. – p.27/43

Page 97: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 6) spanner of size O(n4/3)

Compute a clustering with h = n1/3 and letS = {special vertices}

����

����

����

����

����

����

������������������������������������������������������������������ �����������������������

�������

������

������

����

����

. – p.28/43

Page 98: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 6) spanner of size O(n4/3)

Compute a clustering with h = n1/3 and letS = {special vertices}

add the (S × S)-spanner with additive stretch 2

����

����

����

����

����

����

������������������������������������������������������������������ �����������������������

�������

������

������

����

����

. – p.28/43

Page 99: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 6) spanner of size O(n4/3)

Compute a clustering with h = n1/3 and letS = {special vertices}

add the (S × S)-spanner with additive stretch 2

����

����

����

����

����

�����

�����

������

������

������

����

����

≤ ℓ+ 4

a b

s1 s2

. – p.28/43

Page 100: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

An all-pairs (1, 6) spanner of size O(n4/3)

Compute a clustering with h = n1/3 and letS = {special vertices}

add the (S × S)-spanner with additive stretch 2

����

����

����

����

����

�����

�����

������

������

������

����

����

≤ ℓ+ 4

a b

s1 s2

so δH(a, b) ≤ δH(a, s1) + δH(s1, s2) + δH(s2, b)

≤ δG(a, b) + 6.. – p.28/43

Page 101: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

D-Preservers [BCE03]

Input: G = (V,E) and D ∈ Z+

. – p.29/43

Page 102: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

D-Preservers [BCE03]

Input: G = (V,E) and D ∈ Z+

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) s.t. dG(u, v) ≥ D

. – p.29/43

Page 103: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

D-Preservers [BCE03]

Input: G = (V,E) and D ∈ Z+

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) s.t. dG(u, v) ≥ D

such an H is called a D-preserver

. – p.29/43

Page 104: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

D-Preservers [BCE03]

Input: G = (V,E) and D ∈ Z+

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) s.t. dG(u, v) ≥ D

such an H is called a D-preserver

a D-preserver of size O(n2/D) can be computed inpolynomial time.

. – p.29/43

Page 105: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise Preservers [CE05]

Input: G = (V,E) along with P ⊆ V × V

. – p.30/43

Page 106: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise Preservers [CE05]

Input: G = (V,E) along with P ⊆ V × V

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) ∈ P

. – p.30/43

Page 107: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise Preservers [CE05]

Input: G = (V,E) along with P ⊆ V × V

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) ∈ P

such an H is called a P-preserver

. – p.30/43

Page 108: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise Preservers [CE05]

Input: G = (V,E) along with P ⊆ V × V

find a sparse subgraph H where

dH(u, v) = dG(u, v) ∀(u, v) ∈ P

such an H is called a P-preserver

a P-preserver of size O(min(n√

|P|, n+ |P|√n))

can be computed in polynomial time.

. – p.30/43

Page 109: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

Find a sparse subgraph such that

δH(u, v) ≤ δG(u, v) + β ∀(u, v) ∈ P

. – p.31/43

Page 110: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

Find a sparse subgraph such that

δH(u, v) ≤ δG(u, v) + β ∀(u, v) ∈ P

such an H is a P-spanner with additive stretch β

. – p.31/43

Page 111: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

Find a sparse subgraph such that

δH(u, v) ≤ δG(u, v) + β ∀(u, v) ∈ P

such an H is a P-spanner with additive stretch β

size O(n · |P|1/3) and additive stretch 2 [KV13]

. – p.31/43

Page 112: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

Find a sparse subgraph such that

δH(u, v) ≤ δG(u, v) + β ∀(u, v) ∈ P

such an H is a P-spanner with additive stretch β

size O(n · |P|1/3) and additive stretch 2 [KV13]

size O(n · |P|2/7) and additive stretch 4 [K15]

. – p.31/43

Page 113: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Pairwise spanners

Find a sparse subgraph such that

δH(u, v) ≤ δG(u, v) + β ∀(u, v) ∈ P

such an H is a P-spanner with additive stretch β

size O(n · |P|1/3) and additive stretch 2 [KV13]

size O(n · |P|2/7) and additive stretch 4 [K15]

size O((n · |P|1/4) and additive stretch 6 [K15]

. – p.31/43

Page 114: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner of size O(n|P|1/3)

We have to ensure SP (u, v) for each (u, v) ∈ P iswell-approximated in the final H.

����

����

����

����

����

����

����

����

. – p.32/43

Page 115: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner of size O(n|P|1/3)

We have to ensure SP (u, v) for each (u, v) ∈ P iswell-approximated in the final H.

Run the clustering step with parameter h ≈ |P|1/3 andwe get the starting subgraph H.

����

����

����

����

����

����

����

����

. – p.32/43

Page 116: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner of size O(n|P|1/3)

We have to ensure SP (u, v) for each (u, v) ∈ P iswell-approximated in the final H.

Run the clustering step with parameter h ≈ |P|1/3 andwe get the starting subgraph H.

����

����

����

����

����

����

����

����

u vp = SP (u, v)

. – p.32/43

Page 117: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner of size O(n|P|1/3)

We have to ensure SP (u, v) for each (u, v) ∈ P iswell-approximated in the final H.

Run the clustering step with parameter h ≈ |P|1/3 andwe get the starting subgraph H.

����

����

����

����

����

����

����

����

u vp = SP (u, v)

Sample each special vertex with probability ≈ 3/τ

(τ = n/|P|2/3) to get very special vertices. – p.32/43

Page 118: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 2

Add to H a BFS tree rooted at each very specialvertex.

. – p.33/43

Page 119: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 2

Add to H a BFS tree rooted at each very specialvertex.

So if SP (u, v) has ≥ τ clustered vertices, thenδH(u, v) ≤ δ(u, v) + 2.

. – p.33/43

Page 120: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 2

Add to H a BFS tree rooted at each very specialvertex.

So if SP (u, v) has ≥ τ clustered vertices, thenδH(u, v) ≤ δ(u, v) + 2.

For (u, v) ∈ P:

. – p.33/43

Page 121: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 2

Add to H a BFS tree rooted at each very specialvertex.

So if SP (u, v) has ≥ τ clustered vertices, thenδH(u, v) ≤ δ(u, v) + 2.

For (u, v) ∈ P:

if SP (u, v) has < τ clustered vertices then add to H

all missing edges of SP (u, v).

. – p.33/43

Page 122: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 2

Add to H a BFS tree rooted at each very specialvertex.

So if SP (u, v) has ≥ τ clustered vertices, thenδH(u, v) ≤ δ(u, v) + 2.

For (u, v) ∈ P:

if SP (u, v) has < τ clustered vertices then add to H

all missing edges of SP (u, v).

so δH(u, v) = δ(u, v) in this case.. – p.33/43

Page 123: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Run the clustering step with parameter h ≈ |P|2/7 andwe get the starting subgraph H.

. – p.34/43

Page 124: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Run the clustering step with parameter h ≈ |P|2/7 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

. – p.34/43

Page 125: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Run the clustering step with parameter h ≈ |P|2/7 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

those with many clustered vertices

. – p.34/43

Page 126: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Run the clustering step with parameter h ≈ |P|2/7 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

those with many clustered vertices

those with only a few clustered vertices

. – p.34/43

Page 127: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Run the clustering step with parameter h ≈ |P|2/7 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

those with many clustered vertices

those with only a few clustered vertices

those with an intermediate number of clusteredvertices

. – p.34/43

Page 128: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Shortest paths with either many clustered vertices oronly a few clustered vertices are easy to handle.

����

����

������������������������������������������������������������������������������������������������������

. – p.35/43

Page 129: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Shortest paths with either many clustered vertices oronly a few clustered vertices are easy to handle.

To deal with paths of the third type:

����

����

������������������������������������������������������������������������������������������������������

. – p.35/43

Page 130: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Shortest paths with either many clustered vertices oronly a few clustered vertices are easy to handle.

To deal with paths of the third type:

����

��������������������������������������������������������������������������������������������������������u v

q r

. – p.35/43

Page 131: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

Shortest paths with either many clustered vertices oronly a few clustered vertices are easy to handle.

To deal with paths of the third type:

����

��������������������������������������������������������������������������������������������������������u v

q r

Add to H all missing edges in the prefix q and suffix r

of each such path.. – p.35/43

Page 132: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We select some special vertices so that

����

����

������������������������������������������������������������������������������������������������������

. – p.36/43

Page 133: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We select some special vertices so that

each prefix has an adjacent selected special vertex

����

����

������������������������������������������������������������������������������������������������������

. – p.36/43

Page 134: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We select some special vertices so that

each prefix has an adjacent selected special vertex

each suffix has an adjacent selected special vertex

����

����

������������������������������������������������������������������������������������������������������

. – p.36/43

Page 135: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We select some special vertices so that

each prefix has an adjacent selected special vertex

each suffix has an adjacent selected special vertex

����

����

����������������������������������������������������������������������������������������������������u v

q r

. – p.36/43

Page 136: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We buy at most one SP (x, y) for x ∈ C1 and y ∈ C2,where C1 and C2 are clusters centered at selectedspecial vertices.

����

����

. – p.37/43

Page 137: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We buy at most one SP (x, y) for x ∈ C1 and y ∈ C2,where C1 and C2 are clusters centered at selectedspecial vertices.

����

��������������������������������������������������������������������������������������������������������u v

q r

C1 C2

. – p.37/43

Page 138: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 4

We buy at most one SP (x, y) for x ∈ C1 and y ∈ C2,where C1 and C2 are clusters centered at selectedspecial vertices.

����

��������������������������������������������������������������������������������������������������������u v

q r

C1 C2

This ensures δH(u, v) ≤ δG(u, v) + 4 in this case.. – p.37/43

Page 139: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

Run the clustering step with parameter h ≈ |P|1/4 andwe get the starting subgraph H.

. – p.38/43

Page 140: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

Run the clustering step with parameter h ≈ |P|1/4 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

. – p.38/43

Page 141: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

Run the clustering step with parameter h ≈ |P|1/4 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

those with only a few clustered vertices

. – p.38/43

Page 142: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

Run the clustering step with parameter h ≈ |P|1/4 andwe get the starting subgraph H.

We classify SP (u, v) into one of three types:

those with only a few clustered vertices

in this case we add to H all missing edges in thispath.

. – p.38/43

Page 143: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the second type, let p = SP (u, v).

����

����

������������������������������������������������������������������������������������������������������

. – p.39/43

Page 144: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the second type, let p = SP (u, v).

����

��������������������������������������������������������������������������������������������������������u v

q r

. – p.39/43

Page 145: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the second type, let p = SP (u, v).

����

��������������������������������������������������������������������������������������������������������u v

q r

each of the middle clusters improves its distance toeither all clusters in the prefix q or all clusters in thesuffix r.

. – p.39/43

Page 146: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the second type, let p = SP (u, v).

����

��������������������������������������������������������������������������������������������������������u v

q r

each of the middle clusters improves its distance toeither all clusters in the prefix q or all clusters in thesuffix r.

in this case we add to H all missing edges in p.. – p.39/43

Page 147: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the third type, let p = SP (u, v).

����

����

����������������������������������������������������������������������������������������������������

. – p.40/43

Page 148: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the third type, let p = SP (u, v).

There is a middle cluster C that satisfiesδH(C,A) ≤ δp(C,A) and δH(C,B) ≤ δp(C,B).

����

����

����������������������������������������������������������������������������������������������������

. – p.40/43

Page 149: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

To define the third type, let p = SP (u, v).

There is a middle cluster C that satisfiesδH(C,A) ≤ δp(C,A) and δH(C,B) ≤ δp(C,B).

����

����

������������������������������������������������������������������������������������������������������u v

q

A BC

r

. – p.40/43

Page 150: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

In this case we buy all edges in the prefix q and suffix r.

���������������������������������������������������������������������������������������������������������

���

. – p.41/43

Page 151: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

In this case we buy all edges in the prefix q and suffix r.

�����������������������������������������������������������������������������������������������������������

���u v

q r

. – p.41/43

Page 152: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

A P-spanner with additive stretch 6

In this case we buy all edges in the prefix q and suffix r.

�����������������������������������������������������������������������������������������������������������

���u v

q r

So we have δH(u, v) ≤ δG(u, v) + 6 here.

. – p.41/43

Page 153: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Open problems

+2 P-spanner of size O(n|P|1/4)?

. – p.42/43

Page 154: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Open problems

+2 P-spanner of size O(n|P|1/4)?

then all these cases: P = V × V , P = S × S,P = S × V would be corollaries of this result

. – p.42/43

Page 155: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Open problems

+2 P-spanner of size O(n|P|1/4)?

then all these cases: P = V × V , P = S × S,P = S × V would be corollaries of this result

A pairwise spanner of size o(n|P|1/4) and O(1) additivestretch?

. – p.42/43

Page 156: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Open problems

+2 P-spanner of size O(n|P|1/4)?

then all these cases: P = V × V , P = S × S,P = S × V would be corollaries of this result

A pairwise spanner of size o(n|P|1/4) and O(1) additivestretch?

this would imply an all-pairs of size o(n4/3) and O(1)

additive stretch. – p.42/43

Page 157: T. Kavitha ( School of Tech. & Computer Science, TIFR ...math.iisc.ac.in/~nmi/Kavitha_Telikepalli.pdf · The problem Input: an undirected graph G =(V,E)on n vertices. We want a sparse

Thank you!

. – p.43/43