Near Optimal Streaming algorithms for Graph Spanners

Surender Baswana

IIT Kanpur

Graph spanner :

a subgraph which is sparse and still preserves all-pairs approximate distances.

t-spanner

G=(V,E) : an undirected graph, |V|=n, |E|=m, t > 1

δ(u,v) : distance between u and v in G.

A subgraph GS= (V,ES), where ES is a subset of E such that

for all u,v ε V,

δ(u,v) ≤ δS(u,v) ≤ t δ(u,v)

t : stretch of the spanner.

Sparseness versus stretch

• Consider a graph modeling some network

• Edges correspond to possible links.

• Each edge has certain cost.

Aim : to select as few edges as possible without

increasing the pair wise distance too much.

t-spanner

• Computing a t-spanner of smallest possible size is NP-complete.

• For a graph on n vertices, how large can a t-spanner be ?

vu

t-spanner

• Computing a t-spanner of smallest possible size is NP-complete.

• For a graph on n vertices, how large can a t-spanner be ?

u v

t-spanner

• Computing a t-spanner of smallest possible size is NP-complete.

• For a graph on n vertices, how large can a t-spanner be ?

u v 2-spanner may require Ω(n2) edges

t-spanner

• [Erdös 1963, Bollobas, Bondy & Simonovits]

“There are graphs on n vertices for which every 2k-spanner or

a (2k-1)- spanner has Ω(n1+1/k) edges.”

G=(V,E) ALGORITHM

GS=(V,ES), |ES|=O(n1+1/k) GS is (2k-1)-spanner

Algorithms for t-spanner(RAM model)

Stretch Size Running time

Das et al., 1991

2k-1 O(n1+1/k) O(mn1+1/k)

Deterministic

Roditty et al.

2004

2k-1 O(n1+1/k) O(n2+1/k)

Deterministic

B & Sen, 2003

2k-1 O(kn1+1/k) O(km)

Randomized

Roditty et al., 2005

2k-1 O(kn1+1/k) O(km)

Deterministic

Algorithms for t-spanner(RAM model)

Stretch Size Running time

Das et al., 1991

2k-1 O(n1+1/k) O(mn1+1/k)

Deterministic

Roditty et al.

2004

2k-1 O(n1+1/k) O(n2+1/k)

Deterministic

B & Sen, 2003

2k-1 O(kn1+1/k) O(km)

Randomized

Roditty et al., 2005

2k-1 O(kn1+1/k) O(km)

Deterministic

• avoids distance computation altogether.• near optimal algorithms in parallel, external-memory, distributed environment

Computing a t-spanner in streaming environment

Input : n, m, k, and a stream of edges of an unweighted graph

Aim : to compute a (2k-1)-spanner

Efficiency measures :

1. number of passes

2. space (memory) required

3. time to process the entire stream

Computing a t-spanner in streaming environment

Input : n, m, k, and a stream of edges of an unweighted graph

Aim : to compute a (2k-1)-spanner

Algo 1 : Streaming model

Efficiency measures :

1. number of passes 1

2. space (memory) required O(kn1+1/k)

3. time to process the entire stream O(m)

Computing a t-spanner in streaming environment

Input : n, m, k, and a stream of edges of an unweighted graph

Aim : to compute a (2k-1)-spanner

[Feigenbaum et al., SODA 2005]

Efficiency measures :

1. number of passes 1

2. space (memory) required O(kn1+1/k) for (2k+1)-spanner

3. time to process the entire stream O(mn1/k)

Computing a t-spanner in streaming environment

Input : n, m, k, and a stream of edges of a weighted graph

Aim : to compute a (2k-1)-spanner

Algo 2 : StreamSort model

Efficiency measures :

1. number of passes O(k)

2. working memory required O(log n) bits

3. time spent in one stream pass O(m)

Relation to previous results

B. & Sen, 2003 Feigenbaum et al.,

2005 Algo 1

Algo 2

• slightly different hierarchy • simple buffering technique

Algorithm 1

Intuition

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Intuition

Spanner edge

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Intuition

Spanner edge

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Cluster

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C(x) : center of cluster containing x

Radius : maximum distance from center to a vertex in the cluster

Clustering : a set of disjoint clusters

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K-1

Preprocessing : Clustering for the initial (empty) graph

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Sampling probability = n-1/k

Preprocessing : Clustering for the initial (empty) graph

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Sampling probability = n-1/k

Preprocessing : Clustering for the initial (empty) graph

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Sampling probability = n-1/k

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n1-1/k

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n1/k

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Preprocessing : Clustering for the initial (empty) graph

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n1-1/k

n1-2/k

n1/k

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Sampling probability = n-1/k

Preprocessing : Clustering for the initial (empty) graph

Processing the stream of edges

• Each vertex u at level i<k-1 wishes to move to higher levels.

Condition for upward movement :

“an edge (u,v) such that Ci(v) is a sampled cluster”

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From perspective of a vertex u …

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From perspective of a vertex u …

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From perspective of a vertex u …

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From perspective of a vertex u …

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From perspective of a vertex u …

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From perspective of a vertex u …

Processing an edge (u,v)

If Ci(v) is a sampled cluster : Ci+1(u) Ci+1(v);

add (u,v) to spanner;

u moves to level i+1 (or even higher)

Else if Ci(v) was not adjacent to u earlier :

add edge (u,v) to spanner;

Else Discard (u,v)

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n1-1/k

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n1/k

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Size and stretch of spanner

• Expected number of spanner edges contributed by a vertex

= O(k n1/k).

• Radius of a cluster at level i is at most i.

For each edge discarded, there is a path in spanner

of length (2i+1)

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i

Size and stretch of spanner

• Expected number of spanner edges contributed by a vertex

= O(k n1/k).

• Radius of a cluster at level i is at most i.

A single pass streaming algorithm

A (2k-1)-spanner of expected size O(kn1+1/k)

Running time of the algorithm

u

i

If Ci(v) is a sampled cluster : Ci+1(u) Ci+1(v); add (u,v) to spanner; u moves to level i+1 (or even higher)Else if Ci(v) was not adjacent to u earlier θ(n1/k) time add edge (u,v) to spanner; Else Discard (u,v)

v

Slight modification

• Each vertex u keeps two buffers for storing edges incident from clusters at its present level.

1. Temp(u)

2. Es(u)

• Whenever u moves to higher level, move all the edges of

Temp(u) and Es(u) to the spanner.

Modified algorithm

i

If Ci(v) is a sampled cluster : Ci+1(u) Ci+1(v); add (u,v) to spanner; u moves to level i+1 (or even higher)

Else add (u,v) to Temp(u) and Prune(u) if Temp(u) ≥ ES(u)

u v

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Adding edges to Temp(u)

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Adding edges to Temp(u)

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Prune(u)

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Time complexity analysis

• Prune(u) can be executed in O(|Temp(u)| + |Es(u)|) time using an

an auxiliary O(n) space.

• when is Prune(u) executed ?

Time complexity analysis

• Prune(u) can be executed in O(|Temp(u)| + |Es(u)|) time using an

an auxiliary O(n) space.

• Prune(u) is executed only when |Temp(u)| ≥ |Es(u)|

Time complexity analysis

• Prune(u) can be executed in O(|Temp(u)| + |Es(u)|) time using an

an auxiliary O(n) space.

• Prune(u) is executed only when |Temp(u)| ≥ |Es(u)|

• We can charge O(1) cost to each edge in Temp(u).

Time complexity analysis

• Prune(u) can be executed in O(|Temp(u)| + |Es(u)|) time using an

an auxiliary O(n) space.

• Prune(u) is executed only when |Temp(u)| ≥ |Es(u)|

• We can charge O(1) cost to each edge in Temp(u).

• An edge is processed in Temp(u) at most once.

Time complexity analysis

• Prune(u) can be executed in O(|Temp(u)| + |Es(u)|) time using an

an auxiliary O(n) space.

• Prune(u) is executed only when |Temp(u)| ≥ |Es(u)|

• We can charge O(1) cost to each edge in Temp(u).

• An edge is processed in Temp(u) at most once.

Total time spent in processing the stream = O(m)

Size of (2k-1)-spanner

• Expected size of |Es(u)| = O(n1/k)

• Temp(u) never exceeds |Es(u)| +1.

Expected size of (2k-1)-spanner is O(k n1+1/k)

Conclusion

THEOREM 1 :

Given any k ε N, a (2k-1)-spanner of expected size O(kn1+1/k) for any unweighted graph can be computed in one Stream pass with O(m)

time to process the entire stream of edges.

THEOREM 2 :

Given any k ε N, a (2k-1)-spanner of expected size O(kn1+1/k) for any weighted graph can be computed in O(k) StreamSort passes with O(log n) bits of working memory.