"Randomized Gossip Methods" by Dahlia Malkhi

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1938: first successful xerographic image 1979: first PC, Alto

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1982, CACM 1987, PODC 2000, FOCS

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Motivation 5

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Enters: ”Random Phone Call” Framework • Framework definition: – synchronous rounds – each node initiates one connection to any number of requests! – full network, accurate membership, choose partners at random • Protocols in this talk: – rumor mongering – failure detection – network discovery a ha! allow nodes to learn addresses and use them

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Demonstration of Randomized Gossip informed

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Demonstration of Randomized Gossip informed

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Demonstration of Randomized Gossip 2 ⁄ 2 ⁄

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Demonstration of Randomized Gossip 4 ⁄ 4 ⁄ 4 ⁄ 4 ⁄

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Demonstration of Randomized Gossip 8 ⁄ 8 ⁄ 8 ⁄ 8 ⁄ 8 ⁄ 8 ⁄ 8 ⁄ 8 ⁄

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Demonstration of Randomized Gossip

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Complexity of Gossip Processes in Full Networks • n nodes • rounds to completion – lower bound – upper bound • PUSH message complexity • PULL message complexity • PUSH-PULL message complexity

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Complexity of Gossip Processes in Full Networks – in a round, informed has informed number of nodes – in a round, every node interacts with expected 1 node – in a round where number of informed > log n • with high probability, informed nodes interact with < 2x informed nodes • informed grows by at most factor 3 – rounds = Ω(log n) Round complexity lower bound:

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Complexity of Gossip Processes in Full Networks – in a round where fraction of informed nodes < ½ : • probability of successful PUSH of an informed node > ½ • expected number of successful PUSH’s > informed / 2 • with probability > 1-1/n , number of successful PUSH’s > informed / 3 – in O(log n) rounds, informed grows to n/2 Round complexity PUSH upper bound, first phase:

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Complexity of Gossip Processes in Full Networks – in a round where fraction of informed nodes ≥ ½ : • probability of successful PUSH to an uninformed node ≥ 1- 1 − ) * */, ≅ 1− 0)/, • expected number of successful PUSH’s to uninformed nodes ≥ uninformed / 2 • with probability > 1-1/n , number of successful PUSH’s > uninformed / 3 – in O(log n) rounds, uninformed shrinks to zero Round complexity PUSH upper bound, second phase (coupon collector):

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Complexity of Gossip Processes in Full Networks – in a round where informed is a small constant: • non-negligible probability of no successful PULL from informed node 1 − 1 *0) 1*234567 ≅ 01*234567 – in a round where number of informed ≥ log n : • with high probability informed grows by constant factor – in O(log n) rounds informed grows to n/2 • but slow start! Round complexity PULL upper bound, first phase:

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Complexity of Gossip Processes in Full Networks – in a round where uninformed = cn : • probability of unsuccessful PULL by an uninformed node = c • expected number of unsuccessful PULLs by uninformed nodes = nc2 – .9nc2 with high probability • in R rounds, uninformed shrinks to ,; – in O(loglog n) rounds uninformed shrinks to zero Round complexity PULL upper bound, second phase:

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Complexity of Gossip Processes in Full Networks • rounds to completion [Demers et al. PODC 1987] – lower bound – upper bound • message complexity – connections – transmissions • PUSH message complexity • PULL connection complexity • PUSH-PULL message complexity [Karp et al. FOCS 2000]

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Scalability of Randomized Gossip • fault tolerant • simple • round complexity • rounds sufficient and necessary for completion – synchronous rounds • message complexity – number of interactions – number of transmissions – size of transmissions • full network • precise membership knowledge

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Failure Detection via Randomized Gossip Methods

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22 IFIP Middleware 1998

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Gossip-Style FD • Scale heartbeats: – Use gossip instead of multicast – Each node generates new heartbeat counter in every round 23

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Demonstration of Gossip-style FD 1 1

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2 1 1 2 Demonstration of Gossip-style FD

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Demonstration of Gossip-style FD 3 2 2 3 1 1 1 1

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Demonstration of Gossip-style FD 4 2 2 3 1 1 1 4 1 3 2 2

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Gossip-Style FD • each node generates new heartbeat counter in every round • heartbeat j expected to arrive everywhere by round j + Tfail – from each node, heartbeat j+1 to follow heartbeat j in • expected one round • worst case Tfail rounds • stopped heartbeat at round j expected be noticed everywhere by round j + Tfail – from failed node, stopped heartbeat at round j noticed in • gap of up to Tfail rounds ;; keep tombstone for 2x Tfail rounds 28

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Scalability of Gossip-Style FD • Periodic multicast is hard to scale – Every node sends heartbeats by everyone – Much of the gossip is redundant or stale – Dividing gossip into packets leads to slow failure detection and error-prone convergence 29

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30 DSN 2002

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Scalable Failure Detection 31 Gossip-Style FD gossips a lot of redundant and stale information Instead, gossip only alerts!

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SWIM • Failure Detector: – In a round, every node probes another node at random – a failed probe reinforced by peers • Weak Membership Service: – Spread alerts via gossip • Every faulty node detected (completeness) • Constant connection and message overhead per node per round • Separate failure detection from alert dissemination 32

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Demonstration of SWIM A B A is faulty! A is faulty! co-probe A probe A

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Demonstration of SWIM A B A is faulty! A is faulty! A is faulty! co-probe A probe A

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Demonstration of SWIM A B A is faulty! A is faulty! A is faulty! A is faulty! co-probe A probe A

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Demonstration of SWIM A C B A is faulty! A is faulty! A is faulty! co-probe A probe A

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Demonstration of SWIM A C B A is faulty! D co-probe A probe A

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Detection Completeness in SWIM • A failure is detected in expected one SWIM round – with non-negligible probability, only in log n ! • A failure detection alert is disseminated in O(log n) rounds – half of the system will learn only after in O(log n) rounds • The same failure will cause repeated co-probing and detection • Is randomized probing better than – Steady heartbeat (e.g., along a ring) ? – Stable connection (e.g., TCP/IP) ? – Leases ? 38

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Demonstration of SWIM A B A A is faulty! co-probe A probe A A is faulty! A is faulty!

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Demonstration of SWIM A C B A is faulty! D A is faulty! A is faulty! A is faulty! co-probe A probe A

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Demonstration of SWIM A E A is faulty! co-probe A probe A A is alive!!!

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False Failure Detection in SWIM • Suspicion – Alive -- Confirmation • Competing gossips may arrive in arbitrary order • Node incarnation – incremented by suspicion – suspicion overridden by alive overridden by confirmation 42

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Demonstration of SWIM A C B A is faulty! A is faulty! A is faulty! co-probe A probe A A is alive!!

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Demonstration of SWIM A D C B A is faulty! E co-probe A probe A A is alive!!!

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SWIM Patches • Suspicion – Alive -- Confirmation • Incarnations • Deterministic probing • Deterministic gossiping • Weak membership service 45

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Gossip in Arbitrary Graphs With and Without Network Discovery

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Gossip in Arbitrary Graphs • clearly not always log • might even be super-linear • O(log Φ) ⁄ rounds to completion [Giakkoupis STACS 2011]

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Gossip Process in Arbitrary Graphs • ≝ set of informed vertices at beginning of round • ≝ uninformed vertices • using PULL, each uniformed vertex in U – has b () informed neighbors – has () neighbors – learns with probability (b ()/deg ) – increases volume of edges in by expected deg ∗ (b ()/deg ) • total increase in edge-‐‑volume of I equals |E(I, U)|, the volume of edge-‐‑cut: j b() k 1* l ?? U I

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Gossip Process in Arbitrary Graphs by definition, edge-‐‑cut is larger than graph’s conductance Φ ≝ min n⊆p q(n , p0n ) k3r p this means vol st) ≥ vol s + s,s ≥ vol s ∗ (1+Φ) finish in O(log Φ) ⁄ ?? s

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Network Discovery 50 1999, PODC

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Definition of Network Discovery Problem • send information about neighbors • they become new neighbors of recipient

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• send information about neighbors • they become new neighbors of recipient v Definition of Network Discovery Problem Γ(v)

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Complexity of Network Discovery • Flood: send information about new neighbors to all initial neighbors – completes in diameter rounds – sends diameter x degree x n messages • Swamp: send information about all neighbors to all neighbors – completes in O(log n) rounds – sends O(n x n) large messages • Pointer jump: PULL from one random neighbor information about all its neighbors – may take O(n) rounds • Name Dropper: PUSH to one random neighbor information about all neighbors – completes in O(log2 n) rounds 53

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v Demonstration of Network Discovery Γ(v)

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v Demonstration of Network Discovery Γ(v) Γ(v) new neighbors PUSH neighbors

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v Demonstration of Network Discovery Γ(v) new neighbors PUSH neighbors Γ(v) Γ(v)

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Demonstration of Network Discovery (2)

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Demonstration of Network Discovery (2)

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Demonstration of Network Discovery (2)

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Demonstration of Network Discovery (2)

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Demonstration of Network Discovery (3)

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Demonstration of Network Discovery (3)

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Demonstration of Network Discovery (3)

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• for every path u – w – v – X: the set of neighbors of v, u – with constant probability • Either X grows by constant factor – if half of X has 2|X| neighbors • or some node in X contacts u w u v Analyzing Name Dropper X

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Complexity of Network Discovery • Name-Dropper [Harchol-Balter, Leighton, Lewin, PODC 1999]: – O(log n log D) rounds – Arbitrary directed graphs • Improvement [Kutten, Peleg and Vishkin, TCS 2003] – O(log n) rounds

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Gossip by Network Discovery? teaser: O(log n) + O(log n) = ?

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Breaking the Logarithmic Bound • synchronous rounds • each node initiates one connection number of informed nodes at most doubles each round a ha! but a node may respond to any number of requests! • choose partners at random diameter a ha! allow nodes to learn addresses and use them

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Breaking the Logarithmic Bound • synchronous rounds • each node initiates one connection • number of informed nodes at most doubles each round • a ha! but a node may respond to any number of requests! • choose partners at random diameter a ha! allow nodes to learn addresses and use them

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Breaking the Logarithmic Bound • synchronous rounds • each node initiates one connection • number of informed nodes at most doubles each round • a ha! but a node may respond to any number of requests! • choose partners at random • graph of all interactions has diameter • a ha! allow nodes to learn addresses and use them (e.g., TCP/IP)

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A Hybrid Gossip Model • synchronous rounds • each node initiates one connection number of informed nodes at most doubles each round a ha! but a node may respond to any number of requests! • choose partners at random or by direct addressing • learn addresses from interaction • diameter • a ha! allow nodes to learn addresses and use them

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Breaking the Logarithmic Bound • GOSSIP with DA in O(√log n) rounds [Avin and Elsässer, DISC 2013] • Improved to O(loglog n) rounds [Haeupler and Malkhi, PODC 2014] • Use for Network Discovery – choose a leader in O(loglog n) rounds – One push/pull via leader to share topology information – O(log D loglog n) rounds [Haeupler and Malkhi, PODC 2015]

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