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) Fabian Kuhn STRUCO Meeting, November 2013
Multi-Message Broadcast Fabian Kuhn STRUCO Meeting, November 2013 2
Multi-Message Broadcast • a.k.a. gossip, token dissemination, … Fabian Kuhn STRUCO Meeting, November 2013 3
Multi-Message Broadcast 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] Fabian Kuhn STRUCO Meeting, November 2013 4
Broadcasting Multiple Messages Goal: (Globally) broadcast 𝑂 messages Strategy: In each round, each node forwards an “ unforwarded ” message to its neighbors Which message should be forwarded to neighbors? • It doesn’t matter… Total time for 𝑶 broadcasts ≤ 𝑬 + 𝑶 [Topkis ‘85] • 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? Fabian Kuhn STRUCO Meeting, November 2013 5
Communication Model 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 Fabian Kuhn STRUCO Meeting, November 2013 6
Multi-Broadcast with Edge Capacities 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 Fabian Kuhn STRUCO Meeting, November 2013 7
Packing Spanning Trees Spanning Tree Packing: set of edge-disjoint spanning trees Spanning tree packing of size 𝑡 ⟹ throughput Ω(𝑡) • sp. tree packing of size 𝑡 ⟺ 𝑡 edge-disjoint sp. trees Proof sketch: messages • Each spanning tree gets ≈ 𝑂 𝑡 • Spanning trees don’t interfere with each other • Use pipelining on each spanning tree Fabian Kuhn STRUCO Meeting, November 2013 8
Edge Connectivity 𝑯 has edge connectivity 𝝁 : min. cut 𝝁 edges Thm: 𝐻 has ≤ 𝜇 edge-disjoint spanning trees. edge-disjoint spanning trees. Thm: 𝐻 has ≥ 𝜇 2 [Tutte ’61, Nash - Williams ‘61] Fabian Kuhn STRUCO Meeting, November 2013 9
Edge Connectivity 𝑯 has edge connectivity 𝝁 : min. cut 𝝁 edges Thm: 𝐻 has ≤ 𝜇 edge-disjoint spanning trees. edge-disjoint spanning trees. Thm: 𝐻 has ≥ 𝜇 2 [Tutte ’61, Nash - Williams ‘61] • This is tight: Fabian Kuhn STRUCO Meeting, November 2013 10
Vertex-Capacitated Networks? Nodes 𝑻 𝑵 forward message 𝑵 Every other node needs to get the message: • 𝑻 𝑵 is a dominating set One source ⟹ nodes in 𝑇 𝑁 are connected to each other • 𝑻 𝑵 is a connected dominating set (CDS) One CDS for each message 𝑁 • Use each node in as few CDSs as possible Fabian Kuhn STRUCO Meeting, November 2013 11
Packing Connected Dominating Sets CDS packing of size 𝒅 • 𝑑 vertex-disjoint connected dominating sets Fractional CDS packing of size 𝒅 • CDSs 𝑇 1 , … , 𝑇 𝑢 and weights 𝜇 1 , … , 𝜇 𝑢 such that 𝑢 𝜇 𝑗 = 𝑑, ∀𝑤 ∈ 𝑊 𝐻 : 𝜇 𝑗 ≤ 1 𝑗=1 𝑗:𝑤∈𝑇 𝑗 Fabian Kuhn STRUCO Meeting, November 2013 12
CDS Packings and Throughput Fractional CDS packing of size 𝒅 ⟺ throughput 𝛁(𝒅) Proof sketch: Some Intuition • 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 ) times – Each nodes used at most 𝑃(𝑂 𝑑 ) – CDS 𝑇 used by ℓ messages ⟹ weight of 𝑇 is Θ(ℓ𝑑 𝑂 Fabian Kuhn STRUCO Meeting, November 2013 13
Vertex Connectivity 𝑯 has vertex connectivity 𝒍 : 𝑙 = 5 𝑂 messages • Vertex cut 𝐷 ⊆ 𝑊 𝐻 – Each msg. needs to be forwarded by some node in 𝐷 throughput ≤ 𝑙 Thm: Size of largest fractional CDS packing ≤ 𝑙 • Can we find a (fractional) CDS packing of size Ω(𝑙) ? Fabian Kuhn STRUCO Meeting, November 2013 14
CDS Packing Results • Joint work with Mohsen Ghaffari and Keren Censor-Hillel 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 log 5 𝑜 ) . Fabian Kuhn STRUCO Meeting, November 2013 15
Vertex Sampling Results CDS results/techniques lead to other interesting 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(𝑙𝑞 2 log 3 𝑜 ) . • Proof idea: Fractional CDS packing construction can also be applied to sampled sub-graph. Thm: When sampling with prob. 𝑞 = 𝛿 ⋅ log (𝑜) 𝑙 , the induced sub-graph is connected w.h.p. • Tight up to factor 𝑃 log 𝑜 . • No non-trivial results of this kind where known before! Fabian Kuhn STRUCO Meeting, November 2013 16
Edge Sampling 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 Fabian Kuhn STRUCO Meeting, November 2013 17
Cuts After Sampling • A cut with value 𝛽𝜇 in 𝐻 has expected value 𝑞 ⋅ 𝛽𝜇 in 𝐼 𝑞 𝑞 𝑞 𝑞 • Chernoff bound: the probability that the value is far by a factor ≥ (1 + 𝜁) is 𝑓 −Θ(𝜁 2 𝑞𝛽𝜇) • Union bound: values of all cuts are close to expectation • Main tool: number of edge cuts of size ≤ 𝛽𝜇 is 𝑃 𝑜 2𝛽 Fabian Kuhn STRUCO Meeting, November 2013 18
Number of Cuts 𝑯 is 𝝁 -edge connected: • Number of edges cuts of size ≤ 𝛽𝜇 is 𝑃 𝑜 2𝛽 [Karger ‘94] • Number of min. edge cuts is 𝑃 𝑜 2 𝑯 is 𝒍 -vertex connected: • Number of min. vertex cuts can be Θ 2 𝑙 𝑜 𝑙 2 . Our results: Tools for analyzing vertex connectivity Fabian Kuhn STRUCO Meeting, November 2013 19
Vertex Sampling Proof 𝑰 ⊆ 𝑯 : Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p. Proof Sketch: Fabian Kuhn STRUCO Meeting, November 2013 20
Vertex Sampling Proof 𝑰 ⊆ 𝑯 : Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p. Proof Sketch: Virtual graph 𝑯′ with 𝑴 = 𝚰(𝐦𝐩𝐡 𝒐) layers Edge between copies of same node or of neigboring nodes Fabian Kuhn STRUCO Meeting, November 2013 21
Vertex Sampling Proof 𝑰 ⊆ 𝑯 : Sub-graph with nodes sampled independently with probability 𝒒 ≥ 𝜸 𝐦𝐩𝐡 (𝒐) 𝒍 ⟹ 𝑰 connected, w.h.p. Proof Sketch: Virtual graph 𝑯′ with 𝑴 = 𝚰(𝐦𝐩𝐡 𝒐) layers Edge between copies of same node or of neigboring nodes Fabian Kuhn STRUCO Meeting, November 2013 22
Virtual Graph Node set 𝑿 ′ ⊆ 𝑾′ is projected to 𝑿 ⊆ 𝑾 : 𝑥 ∈ 𝑋 ⟺ 𝑋 ′ contains a copy of 𝑥 ⟺ 𝑿′ connected 𝑿 connected 𝑿′ dominating ⟺ 𝑿 dominating Fabian Kuhn STRUCO Meeting, November 2013 23
Coupling Argument • Sample virtual nodes with probability ≈ 𝒒 𝒓 = 𝟐 − 𝟐 − 𝒒 𝟐 𝑴 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 Fabian Kuhn STRUCO Meeting, November 2013 24
Domination layers, the sampled nodes form Claim: After sampling 𝑀 2 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 Fabian Kuhn STRUCO Meeting, November 2013 25
Connectivity 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] : 𝒘 𝒗 Fabian Kuhn STRUCO Meeting, November 2013 26
Connector Paths 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] . 𝑫′′ 𝑫′ 𝑫 𝑫′′′ Menger & Domination of 𝑻 : • 𝐻 𝑙 -vertex connected ⟹ there are ≥ 𝑙 such paths! Fabian Kuhn STRUCO Meeting, November 2013 27
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