Near Optimal Streaming algorithms for Graph Spanners

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Near Optimal Streaming algorithms for Graph
Spanners

• Surender Baswana
• IIT Kanpur

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

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t-spanner

• G(V,E) an undirected graph, Vn, Em, t gt
  1
• d(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 e V,
• d(u,v) dS(u,v) t
  d(u,v)
• t stretch of the spanner.

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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.

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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 ?

v
u
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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 ?

v
u
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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 ?

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2-spanner may require O(n2) edges
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t-spanner

• Erdös 1963, Bollobas, Bondy Simonovits
• There are graphs on n vertices
  for which every 2k-spanner or
• a (2k-1)- spanner has O(n11/k)
  edges.

G(V,E)
GS(V,ES), ESO(n11/k) GS is
(2k-1)-spanner
ALGORITHM
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Algorithms for t-spanner(RAM model)
Stretch Size Running time
Das et al., 1991 2k-1 O(n11/k) O(mn11/k) Deterministic
Roditty et al. 2004 2k-1 O(n11/k) O(n21/k) Deterministic
B Sen, 2003 2k-1 O(kn11/k) O(km) Randomized
Roditty et al., 2005 2k-1 O(kn11/k) O(km) Deterministic
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Algorithms for t-spanner(RAM model)
Stretch Size Running time
Das et al., 1991 2k-1 O(n11/k) O(mn11/k) Deterministic
Roditty et al. 2004 2k-1 O(n11/k) O(n21/k) Deterministic
B Sen, 2003 2k-1 O(kn11/k) O(km) Randomized
Roditty et al., 2005 2k-1 O(kn11/k) O(km) Deterministic

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

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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

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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(kn11/k)
• 3. time to process the entire stream O(m)

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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(kn11/k) for
  (2k1)-spanner
• 3. time to process the entire stream O(mn1/k)

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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)

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Relation to previous results

• slightly different hierarchy
• simple buffering technique

B. Sen, 2003
Feigenbaum et al., 2005
Algo 1
Algo 2
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• Algorithm 1

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Intuition
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Intuition
Spanner edge
u
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Intuition
Spanner edge
u
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Cluster
v
o
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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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Preprocessing Clustering for the initial
(empty) graph
K
K-1
2
1
0
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Preprocessing Clustering for the initial
(empty) graph
K
K-1
2
1
0
Sampling probability n-1/k
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Preprocessing Clustering for the initial
(empty) graph
K
K-1
2
1
0
Sampling probability n-1/k
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Preprocessing Clustering for the initial
(empty) graph
K
0
n1/k
K-1
n1-2/k
2
n1-1/k
1
n
0
Sampling probability n-1/k
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Preprocessing Clustering for the initial
(empty) graph
K
0
n1/k
K-1
n1-2/k
2
n1-1/k
1
n
0
Sampling probability n-1/k
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Processing the stream of edges

• Each vertex u at level iltk-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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K
K-1
2
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0
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K
K-1
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K
K-1
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K
K-1
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K
K-1
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K
K-1
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K
K-1
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K
K-1
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K
K-1
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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
i1
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Processing an edge (u,v)
x
i1
y

• If Ci(v) is a sampled cluster Ci1(u) ?
  Ci1(v)
• 
  add (u,v) to spanner
• 
  u moves to level i1 (or even higher)
• Else if Ci(v) was not adjacent to u earlier
• add edge (u,v) to
  spanner
• Else Discard (u,v)

u
x
i
y
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K
0
n1/k
K-1
n1-2/k
2
n1-1/k
1
n
0
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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 (2i1)

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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.
• A single pass streaming
  algorithm
• A (2k-1)-spanner of expected size
  O(kn11/k)

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Running time of the algorithm
i
u
v
If Ci(v) is a sampled cluster Ci1(u) ?
Ci1(v)
add (u,v) to spanner
u moves to
level i1 (or even higher) Else if Ci(v) was not
adjacent to u earlier ?(n1/k) time
add edge (u,v) to
spanner Else Discard (u,v)
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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.

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Modified algorithm
i
If Ci(v) is a sampled cluster Ci1(u) ?
Ci1(v)
add (u,v) to spanner
u moves to
level i1 (or even higher) Else add (u,v) to
Temp(u) and Prune(u) if Temp(u) ES(u)
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Adding edges to Temp(u)
u
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Adding edges to Temp(u)
u
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Prune(u)
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 ?

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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.
• Prune(u) is executed only when Temp(u)
  Es(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.
• Prune(u) is executed only when Temp(u)
  Es(u)
• We can charge O(1) cost to each edge in Temp(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.
• 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.

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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.
• 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)
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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
  n11/k)

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Conclusion

• THEOREM 1
• Given any k e N, a (2k-1)-spanner of
  expected size O(kn11/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 e N, a (2k-1)-spanner of
  expected size O(kn11/k) for any weighted graph
  can be computed in O(k) StreamSort passes with
  O(log n) bits of working memory.