Decomposing Vertex Connectivity and the Cost of Multiple Broadcasts - - PowerPoint PPT Presentation

STRUCO Meeting, November 2013 Fabian Kuhn

Decomposing Vertex Connectivity

and the

Cost of Multiple Broadcasts

Fabian Kuhn University of Freiburg, Germany

Based on joint work with Mohsen Ghaffari (MIT) and Keren Censor-Hillel (Technion)

STRUCO Meeting, November 2013 Fabian Kuhn 2

Multi-Message Broadcast

STRUCO Meeting, November 2013 Fabian Kuhn

• a.k.a. gossip, token dissemination, …

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Multi-Message Broadcast

STRUCO Meeting, November 2013 Fabian Kuhn

Communication Assumptions

• For simplicity: synchronous model
• In each round:

Each node can send a message to each neighbor

• Message size: 𝑷(𝐦𝐩𝐡 𝒐) bits, 𝑷(𝟐) broadcast messages

– a.k.a. CONGEST model [Peleg 2000]

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Multi-Message Broadcast

STRUCO Meeting, November 2013 Fabian Kuhn

Goal: (Globally) broadcast 𝑂 messages Which message should be forwarded to neighbors?

• It doesn’t matter…
• 𝐸: diameter
• Optimal pipelining on a path of length 𝑒 gives 𝑃(𝑒 + 𝑂)

– 𝑬 + 𝑶 is asymptotically optimal in general

• What about networks with better connectivity?

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Broadcasting Multiple Messages

Strategy: In each round, each node forwards an “unforwarded” message to its neighbors Total time for 𝑶 broadcasts ≤ 𝑬 + 𝑶

[Topkis ‘85]

STRUCO Meeting, November 2013 Fabian Kuhn

Two natural variants… Edge-Capacitated Model

• Message size 𝑃(log 𝑜)
• Nodes can send different messages to different neighbors
• Classic CONGEST model

Node-Capacitated Model

• Message size 𝑃 log 𝑜
• Have to send the same message to all neighbors
• Communication by local broadcasts

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Communication Model

STRUCO Meeting, November 2013 Fabian Kuhn

Basic assumption:

• store-and-forward algorithms

Each message 𝑵:

• Edges on which 𝑁 is forwarded induce a spanning tree!

Throughput (𝑶 messages):

• 𝑂 spanning trees, one for each message
• Optimize throughput:

– try to use each edge as few times as possible

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Multi-Broadcast with Edge Capacities

STRUCO Meeting, November 2013 Fabian Kuhn

Spanning Tree Packing: set of edge-disjoint spanning trees

• sp. tree packing of size 𝑡 ⟺ 𝑡 edge-disjoint sp. trees

Proof sketch:

• Each spanning tree gets ≈ 𝑂 𝑡

messages

• Spanning trees don’t interfere with each other
• Use pipelining on each spanning tree

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Packing Spanning Trees

Spanning tree packing of size 𝑡 ⟹ throughput Ω(𝑡)

STRUCO Meeting, November 2013 Fabian Kuhn

𝑯 has edge connectivity 𝝁:

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Edge Connectivity

• min. cut

Thm: 𝐻 has ≤ 𝜇 edge-disjoint spanning trees. Thm: 𝐻 has ≥ 𝜇 2 edge-disjoint spanning trees. [Tutte ’61, Nash-Williams ‘61]

𝝁 edges

STRUCO Meeting, November 2013 Fabian Kuhn

𝑯 has edge connectivity 𝝁:

• This is tight:

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Edge Connectivity

• min. cut

Thm: 𝐻 has ≤ 𝜇 edge-disjoint spanning trees. Thm: 𝐻 has ≥ 𝜇 2 edge-disjoint spanning trees. [Tutte ’61, Nash-Williams ‘61]

𝝁 edges

STRUCO Meeting, November 2013 Fabian Kuhn

Nodes 𝑻𝑵 forward message 𝑵 Every other node needs to get the message:

• 𝑻𝑵 is a dominating set

One source ⟹ nodes in 𝑇𝑁 are connected to each other

• 𝑻𝑵 is a connected dominating set (CDS)

One CDS for each message 𝑁

• Use each node in as few CDSs as possible

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Vertex-Capacitated Networks?

STRUCO Meeting, November 2013 Fabian Kuhn

CDS packing of size 𝒅

• 𝑑 vertex-disjoint connected dominating sets

Fractional CDS packing of size 𝒅

• CDSs 𝑇1, … , 𝑇𝑢 and weights 𝜇1, … , 𝜇𝑢 such that

𝜇𝑗

𝑢 𝑗=1

= 𝑑, ∀𝑤 ∈ 𝑊 𝐻 : 𝜇𝑗

𝑗:𝑤∈𝑇𝑗

≤ 1

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Packing Connected Dominating Sets

STRUCO Meeting, November 2013 Fabian Kuhn

Proof sketch:

• Distribute msg. among CDSs (according to weight)

– Time-share between CDSs according to weight

• Use pipelining on each CDS (optimal throughput)
• Throughput Ω(𝑑):

– Tracking routes gives CDS for each message – Each nodes used at most 𝑃(𝑂 𝑑 ) times – CDS 𝑇 used by ℓ messages ⟹ weight of 𝑇 is Θ(ℓ𝑑 𝑂 )

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CDS Packings and Throughput

Fractional CDS packing of size 𝒅 ⟺ throughput 𝛁(𝒅) Some Intuition

STRUCO Meeting, November 2013 Fabian Kuhn

𝑯 has vertex connectivity 𝒍:

• Vertex cut 𝐷 ⊆ 𝑊 𝐻

– Each msg. needs to be forwarded by some node in 𝐷 throughput ≤ 𝑙

• Can we find a (fractional) CDS packing of size Ω(𝑙)?

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Vertex Connectivity

𝑙 = 5 𝑂 messages

Thm: Size of largest fractional CDS packing ≤ 𝑙

STRUCO Meeting, November 2013 Fabian Kuhn

• Joint work with Mohsen Ghaffari and Keren Censor-Hillel

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CDS Packing Results

Thm: There is a family of graphs with vertex connectivity 𝑙 and maximum fractional CDS packing size 𝑃(𝑙 log 𝑜 ). Thm: Every graph with vertex connectivity 𝑙 ≥ 1 has a fractional CDS packing of size Ω(1 + 𝑙 log 𝑜 ). Thm: Every graph with vertex connectivity 𝑙 ≥ 1 has a CDS packing of size Ω(1 + 𝑙 log5 𝑜 ).

STRUCO Meeting, November 2013 Fabian Kuhn

CDS results/techniques lead to other interesting results

• Proof idea: Fractional CDS packing construction can also be

applied to sampled sub-graph.

• Tight up to factor 𝑃

log 𝑜 .

• No non-trivial results of this kind where known before!

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Vertex Sampling Results

Thm: If each node of a 𝑙-vertex connected graph is indep. sampled with probability 𝑞, the vertex connectivity of the induced sub-graph is Ω(𝑙𝑞2 log3 𝑜 ). Thm: When sampling with prob. 𝑞 = 𝛿 ⋅ log (𝑜) 𝑙 , the induced sub-graph is connected w.h.p.

STRUCO Meeting, November 2013 Fabian Kuhn

Graph 𝑯 is 𝝁-edge connected:

• 𝑰: sub-graph induced when independently sampling each

edge with probability 𝑞 = Ω log 𝑜 𝜇 .

• 𝐼 is a connected graph, w.h.p. [Lomonosov and Poleskii ‘71]
• The edge connectivity of 𝐼 is Ω(𝜇𝑞), w.h.p. [Karger ‘94]
• Both results are tight

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Edge Sampling

STRUCO Meeting, November 2013 Fabian Kuhn

• A cut with value 𝛽𝜇 in 𝐻 has expected value 𝑞 ⋅ 𝛽𝜇 in 𝐼
• Chernoff bound: the probability that the value is far by a

factor ≥ (1 + 𝜁) is 𝑓−Θ(𝜁2𝑞𝛽𝜇)

• Union bound: values of all cuts are close to expectation
• Main tool: number of edge cuts of size ≤ 𝛽𝜇 is 𝑃 𝑜2𝛽

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Cuts After Sampling

𝑞 𝑞 𝑞 𝑞

STRUCO Meeting, November 2013 Fabian Kuhn

𝑯 is 𝝁-edge connected:

• Number of edges cuts of size ≤ 𝛽𝜇 is 𝑃 𝑜2𝛽 [Karger ‘94]
• Number of min. edge cuts is 𝑃 𝑜2

𝑯 is 𝒍-vertex connected:

• Number of min. vertex cuts can be Θ 2𝑙 𝑜 𝑙

2 .

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Number of Cuts

Our results: Tools for analyzing vertex connectivity

STRUCO Meeting, November 2013 Fabian Kuhn

Proof Sketch:

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Vertex Sampling Proof

𝑰 ⊆ 𝑯: Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p.

STRUCO Meeting, November 2013 Fabian Kuhn

Proof Sketch: Virtual graph 𝑯′ with 𝑴 = 𝚰(𝐦𝐩𝐡 𝒐) layers Edge between copies of same node or of neigboring nodes

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Vertex Sampling Proof

𝑰 ⊆ 𝑯: Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p.

STRUCO Meeting, November 2013 Fabian Kuhn

Proof Sketch: Virtual graph 𝑯′ with 𝑴 = 𝚰(𝐦𝐩𝐡 𝒐) layers Edge between copies of same node or of neigboring nodes

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Vertex Sampling Proof

𝑰 ⊆ 𝑯: Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p.

STRUCO Meeting, November 2013 Fabian Kuhn

Node set 𝑿′ ⊆ 𝑾′ is projected to 𝑿 ⊆ 𝑾: 𝑥 ∈ 𝑋 ⟺ 𝑋′ contains a copy of 𝑥

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Virtual Graph

𝑿′ connected 𝑿′ dominating 𝑿 connected 𝑿 dominating

⟺ ⟺

STRUCO Meeting, November 2013 Fabian Kuhn

• Sample virtual nodes with probability

𝒓 = 𝟐 − 𝟐 − 𝒒 𝟐 𝑴

≈ 𝒒

𝑴

• Sample real node 𝑤 iff 𝑤 is in the projection of the sampled

virtual nodes (at least one copy of 𝑤 sampled in 𝐻′)

• Happens with probability 𝟐 − 𝟐 − 𝒓 𝑴 = 𝒒
• Show that sampling in 𝑯′ gives a CDS

Idea: sample layer by layer and study progress

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Coupling Argument

STRUCO Meeting, November 2013 Fabian Kuhn

Claim: After sampling 𝑀 2 layers, the sampled nodes form dominating set. Proof Sketch:

• Sampling probability in 𝐻 after 𝑀 2

= Θ(log 𝑜) layers is Θ log 𝑜 𝑙

• Domination follows directly because every node in 𝐻 has

degree ≥ 𝑙

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Domination

STRUCO Meeting, November 2013 Fabian Kuhn

Recall Menger’s theorem:

• In a 𝑙-vertex connected graph 𝐻 = (𝑊, 𝐹), any two nodes

are connected by 𝑙 internally vertex-disjoint paths Assume: 𝐻 is 𝑙-vertex connected, 𝑇 ⊆ 𝑊 is a dominating set Components of 𝑯[𝑻]:

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Connectivity

𝒗 𝒘

STRUCO Meeting, November 2013 Fabian Kuhn

Assume: 𝐻 = (𝑊, 𝐹), 𝑇 ⊆ 𝑊 a dominating set Definition: For a component 𝐷 of 𝐻[𝑇], a connector path is a path with ≤ 2 internal nodes connecting 𝐷 to another component 𝐷′ of 𝐻[𝑇]. Menger & Domination of 𝑻:

• 𝐻 𝑙-vertex connected ⟹ there are ≥ 𝑙 such paths!

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Connector Paths

𝑫′ 𝑫′′′ 𝑫 𝑫′′

STRUCO Meeting, November 2013 Fabian Kuhn

Consider a layer ℓ > 𝑴 𝟑

• Nodes 𝑇<ℓ sampled by layers < ℓ form a dominating set

Sampling of layer ℓ:

• Virtual nodes sampled with probability

𝑟 ≈ 𝑞 𝑀 = 1 𝑀 ⋅ 𝛾 log 𝑜 𝑙 = Θ 1 𝑙 Component of 𝑯 𝑻<ℓ :

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Connectivity

𝑫

• ≥ 𝑙 connector paths
• each of them sampled with

prob Ω(1 𝑙 ) in layer ℓ

• 𝐷 connected to another

component with const. prob.

STRUCO Meeting, November 2013 Fabian Kuhn

Fast Merging:

• On each layer ℓ > 𝑀 2

, each component is connected to at least one other component with at least constant prob.

• With at least constant probability, the number of

connected components of the induced sub-graph is reduced by a constant factor

• After 𝑃(log 𝑜) layers, we have connectivity, w.h.p.

– Initially, #components is 𝑃(𝑜)

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Connectivity

STRUCO Meeting, November 2013 Fabian Kuhn

• Sampling directly gives CDS packing of size Ω

𝑙 log 𝑜 From size 𝛁 𝒍 to 𝛁 𝒍 …

• also based on virtual graph and layering
• construct all the CDSs at the same time
• carefully choose / assign connector paths

– make progress for all CDSs (reduce overall # of components)

• Gives fractional CDS packing

– different virtual copies of the same node in 𝐻 can go to different CDSs – Each node is in at most 𝑃 log 𝑜 CDSs

• CDS packing:

Use random layers of real nodes instead of virtual nodes

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(Fractional) CDS Packing

STRUCO Meeting, November 2013 Fabian Kuhn

• Algorithm works in the CONGEST model with

– Messages of size 𝑃(log 𝑜) – Capacities at nodes (comm. by local broadcast)

Lower bound

• If 𝑙 is not known, Ω

𝐸 + 𝑜 𝑙 rounds needed

• Proof based on techniques from [Das Sarma et al. ‘12]

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Distributed Construction

(Fractional) CDS packings of the same quality can be computed in a distributed way in time 𝑃 (𝐸 + 𝑜).

STRUCO Meeting, November 2013 Fabian Kuhn

Classic Locality Decompositions

• Decompose graph into clusters of small diameter
• Preserve locality, cluster graph sparse, low chrom. #, …

– e.g., [Awerbuch,Goldberg,Luby,Plotkin ’89], [Awerbuch,Peleg ‘90], [Linial,Saks ’93] – leads to efficient algorithms in the LOCAL model

(Fractional) CDS and Spanning Tree Packings

• Decompose nodes / edges of a graph 𝐻 into components
• Components are connected and they “span” 𝐻
• Also useful as a generic tool to build distributed alg.?

– if we want to exploit the inherent parallelism in networks – for CONGEST algorithms…

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Network/Graph Decompositions

STRUCO Meeting, November 2013 Fabian Kuhn

• Time to construct fractional CDS packing: 𝑃

(𝑛)

• Fastest known non-trivial approximation of vertex conn.
• Best known algorithms:

– compute 𝑙 exactly [Gabow ‘00]: 𝑃 𝑜2𝑙 + min 𝑜𝑙3.5, 𝑜1.75𝑙2 – 2-approximation [Henzinger ’97]: 𝑃 min 𝑜2.5, 𝑜2𝑙

• Distributed algorithm: 𝑃

min

𝑜 𝑙 , 𝐸 +

𝑜

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Approximating Vertex Connectivity

Fractional CDS packing construction gives 𝑷(𝐦𝐩𝐡 𝒐)- approximation of the vertex connectivity 𝑙 of 𝐻.

STRUCO Meeting, November 2013 Fabian Kuhn

• Close the log / polylog gaps!
• CDS packings as a useful primitive (e.g., for distr. alg.)?
• Other applications of the techniques in distr. algorithms?
• Other uses of the layering / virtual graph idea?

– In particular, when dealing with graph connectivity… – Idea also appears in the context of edge sampling in [Alon ‘95]

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Open Problems

STRUCO Meeting, November 2013 Fabian Kuhn 35

Thanks for your attention!

Questions, Comments?