T. Kavitha ( School of Tech. & Computer Science, TIFR...

Pairwise spanners

T. Kavitha

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

[January 9, 2015; IISc Bangalore]

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The problem

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

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The problem

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

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The problem

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].

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An example

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On the left is G with(

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

Stretch of a spanner H

For all pairs of vertices (u, v):

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

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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].

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A simple proof [HZ97]

Partition the vertex set into disjoint clusters:

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

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

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

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Initially Ci = {u}.

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Initially Ci = {u}.

Nbr(Ci) = unclustered neighbors of vertices in Ci

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Initially Ci = {u}.

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

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Initially Ci = {u}.

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

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

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

The spanner H

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

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The spanner H

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The spanner H

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Initialize H = forest defined by the clusters.

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The final H

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

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The final H

add to H exactly one edge between v and C

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The final H

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The final H

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total number of red edges ≤ n1/k(∑

i |Ci|) = n1+1/k.. – p.9/43

The stretch in H

Consider any edge (a, b) missing in H

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The stretch in H

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

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The stretch in H

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The stretch in H

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

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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)

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δH(u, v) ≤ (2k − 1) · δG(u, v)

H can also be computed efficiently.

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δH(u, v) ≤ (2k − 1) · δG(u, v)

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

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δH(u, v) ≤ (2k − 1) · δG(u, v)

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.

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Multiplicative vs Additive spanners

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

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Multiplicative vs Additive spanners

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

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≤ δG(u, v) +O(1)

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Purely additive spanners

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

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

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

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no other results for purely additive spanners areknown.

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no other results for purely additive spanners areknown.

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

2 ) [C13]

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

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

√n))

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√n))

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

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√n))

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√n))

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

The edge set of H

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

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The edge set of H

H contains a BFS tree rooted at each special vertex.

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The edge set of H

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

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The edge set of H

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

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

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Bounding the stretch in H

Consider any pair of vertices (a, b).

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

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

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ℓ1 ℓ2

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ℓ1 ℓ2

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

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

We are given a set of sources S ⊆ V .

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

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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.

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An (S × V )-spanner

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

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

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

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Any shortest path p is one of two types:

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p has < τ clustered vertices (where τ ≈√

n/|S|)

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n/|S|)

p has ≥ τ clustered vertices

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n/|S|)

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

special vertices adjacent to p.

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n/|S|)

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.

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

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

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

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

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

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≤ ℓ

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≤ ℓ

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

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≤ 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

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

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

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

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≤ ℓ+ 3

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≤ ℓ+ 3

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

= δG(a, b) + 4.. – p.22/43

An (S × S) spanner

We are again given a subset S ⊆ V .

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An (S × S) spanner

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

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An (S × S) spanner

Low degree vertices have degree ≤ h =√

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An (S × S) spanner

Low degree vertices have degree ≤ h =√

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

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Path-buying

Let p be the shortest path between s1 and s2.

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Path-buying

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s1 s2p = SP (s1, s2)

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Path-buying

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s1 s2p = SP (s1, s2)

p is one of two types:

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Path-buying

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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)

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Path-buying

Or ∃C on p such that

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

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Path-buying

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≤ ℓ1 ≤ ℓ2

ℓ1 ℓ2

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Path-buying

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≤ ℓ1 ≤ ℓ2

ℓ1 ℓ2

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

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Path-buying

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

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Path-buying

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

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Path-buying

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s1 s2p = SP (s1, s2)

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Path-buying

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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.

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The size of H

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

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The size of H

The number of clusters ≤ n/h = n/√

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The size of H

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

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The size of H

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

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

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

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

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

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≤ ℓ+ 4

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≤ ℓ+ 4

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

≤ δG(a, b) + 6.. – p.28/43

D-Preservers [BCE03]

Input: G = (V,E) and D ∈ Z+

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find a sparse subgraph H where

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

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such an H is called a D-preserver

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such an H is called a D-preserver

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

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Pairwise Preservers [CE05]

Input: G = (V,E) along with P ⊆ V × V

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dH(u, v) = dG(u, v) ∀(u, v) ∈ P

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dH(u, v) = dG(u, v) ∀(u, v) ∈ P

such an H is called a P-preserver

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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.

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Pairwise spanners

Find a sparse subgraph such that