Linearizing PeertoPeer Systems with Oracles by Rizal Mohd

Linearizing Peer-to-Peer Systems with Oracles by Rizal Mohd Nor Mikhail Nesterenko Sébastien Tixeuil SSS 2013 Nov 13 -16, 2013

Introduction Fundamental limits on construction of structured peer-to-peer systems in asynchronous [1] § inability to connect a disconnected network § discard peer identifiers that are not present in the system Studying these limits § identify peer-to-peer system specific oracles and isolate the source of impossibility § show minimality of oracles and proving their necessity for solution existence § provide an oracle-based algorithm Why Linearization (topological sort) § foundational for most popular peer-to-peer systems as construction starts with topologically sorting the peers in the system § similar to consensus as one can observe how it pertain to all peer-to-peer systems [1] Rizal Mohd Nor, Mikhail Nesterenko, and Christian Scheideler. Corona: A stabilizing deterministic message-passing skip list. In 13 th. International Symposium on Stabilization, Safety and security of Distributed Systems (SSS) pages 356 -370, October 2011 2 SSS 2013

Outline § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 3 SSS 2013

Linearization Problem § Strict (SL) and eventual (EL) linearization variants § SL requires each process to output its neighbors exactly once and allows only correct output § EL’s computation contains a suffix where the output of each process is correct § Existing (EID) and non-existing (NID) identifiers within the linearization problem § EID prohibits the existence of non-existing identifiers § NID allows non-existing identifiers 4 SSS 2013

Outline § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 5 SSS 2013

Solution Oracles § weak connectivity oracle (WC) has a single action that: § selects a pair of processes p and q such that they are disconnected in the channel connectivity graph CC, then § connecting graph by adding q to the incoming channel of p creating a link (p, q) § participant detector (PD) oracle removes a non-existent identifier stored in p. § neighbor output oracle (NO) just output identifiers stored in left and right variables of p § consequent process detector (CD) § outputs the stored identifier only if it is consequent with p § the guard of CD mentions all the identifiers of the system 6 SSS 2013

Outline § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 7 SSS 2013

Necessary Conditions: Weak Connectivity Oracle § For a linearization algorithm A, p 1 has to eventually output p 2 p 1 will eventually add p 2, if the computation contains the actions of a weak connectivity oracle. Without WC, it will remain disconnected. p 1 p 2 p 3 p 4 p 5 Theorem 1. Every solution to the linearization problem requires a weak connectivity oracle. 8 SSS 2013

Subset Splittable § subset splittable § prevents a subset of processes from learning the state of the rest of the system § denoted as SS (subset spilttable) and NSS (non-subset spilttable) § neighborhood output oracle (NO) is subset spilttable (SS) § subset splittable since its guards only mention local variables § participant detector oracle (PD) is subset spilttable (SS) a b nid d S 2 S 1 Consider 2 sets of non-intersecting processes, S 1 and S 2 SSS 2013 e 9

Subset Splittable § weak connectivity oracle (WC) is subset spilttable (SS) a b c d e S 2 S 1 Consider 2 sets of non-intersecting processes, S 1 and S 2, where c and d are disconnected 10 SSS 2013

Non Subset Splittable § consequent process detector (CD) is not subset spilttable (NSS) a c b d e S 2 S 1 If a process has a consequent process, it is enabled. If b is added to set S 1, c would be enabled. This disables CD enabled in the previous state. Consider 2 sets of non-intersecting processes, S 1 and S 2 are totally ordered 11 SSS 2013

Necessary Conditions: Non-subset Splittable Oracle a b c d e S 2 S 1 Consider 2 sets of non-intersecting processes, S 1 and S 2 Theorem 2. A solution to the strict linearization problem requires a non-subset splittable oracle. 12 SSS 2013

Necessary Conditions: Non-subset Splittable Oracle a b c d e Since strict linearization requires each process to output only once, in this case, b or c cannot output without knowing the state of the rest of the system Theorem 2. A solution to the strict linearization problem requires a non-subset splittable oracle. 13 SSS 2013

Necessary Conditions: NID requires PD oracles § PD is required to remove the right id of process p 1 and left id of process p 2. Without PD, a process p 1 may be connected to a nonexisting identifier and result in a graph to be disconnected from the higher-id processes forever p 1 p 2 p 3 p 4 p 5 Theorem 3. A proper solution to the linearization problem that allows non-existing identifiers requires a participant detector oracle. 14 SSS 2013

Outline § oracles § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 15 SSS 2013

Necessary Oracles The necessary oracles to solve the 4 variants of the linearization problem. Eventual Linearization (EL) Strict Linearization (SL) Existing IDs (EID) WC WC+NSS Non-existing IDs (NID) WC+PD+NSS Linearization algorithm L with a combination of oracles to solve the linearization problem Eventual Linearization (EL) Strict Linearization (SL) Existing IDs (EID) L+WC+NO L+WC+CD Non-existing IDs (NID) L+WC+NO+PD L+WC+CD+PD 16 SSS 2013

Outline § oracles § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 17 SSS 2013

EL+EID Execution Example using L+WC § Starting from an arbitrary state where processes are weakly connected except process a a b c d e 18 SSS 2013

EL+EID Execution Example using L+WC § a b c d e 19 SSS 2013

EL+EID Execution Example using L+WC § Node a WC oracle is enabled and node a sends its ID to node e a a b c d e 20 SSS 2013

EL+EID Execution Example using L+WC § starting from this state, processes are weakly connected and WC is never enabled § the only actions enabled are the actions of the linearization algorithm L § actions § receive ID from right → set closer right neighbor or forward to right neighbor § receive ID from left → set closer left neighbor or forward to left neighbor § true (timeout) → send ID to right and left neighbor Node e receive a's ID, forward to c a b c d a e 21 SSS 2013

EL+EID Execution Example using L+WC § the only actions enabled are the actions of the linearization algorithm L § actions § receive ID from right → set closer right neighbor or forward to right neighbor § receive ID from left → set closer left neighbor or forward to left neighbor § true (timeout) → send ID to right and left neighbor Node c receive a's ID, forward to b a a b c d e 22 SSS 2013

EL+EID Execution Example using L+WC § the only actions enabled are the actions of the linearization algorithm L § actions § receive ID from right → set closer right neighbor or forward to right neighbor § receive ID from left → set closer left neighbor or forward to left neighbor § true (timeout) → send ID to right and left neighbor Node b receive a's ID, set its new left neighbor a b c d e 23 SSS 2013

EL+EID Execution Example using L+WC § the only actions enabled are the actions of the linearization algorithm L § actions § receive ID from right → set closer right neighbor or forward to right neighbor § receive ID from left → set closer left neighbor or forward to left neighbor § true (timeout) → send ID to right and left neighbor Node b sends it's ID, to Node a b c d e 24 SSS 2013

EL+EID Execution Example using L+WC § the only actions enabled are the actions of the linearization algorithm L § actions § receive ID from right → set closer right neighbor or forward to right neighbor § receive ID from left → set closer left neighbor or forward to left neighbor § true (timeout) → send ID to right and left neighbor Node a receive b's ID, set its new right neighbor a b c d e 25 SSS 2013

EL+EID Execution Example using L+WC processes take independent actions 1. c a b c d e 26 SSS 2013

EL+EID Execution Example using L+WC processes take independent actions 1. c a b c 2. d e d a b c 27 SSS 2013

EL+EID Execution Example using L+WC processes take independent actions 1. c a b c d e d 2. a b 3. d a b 28 SSS 2013

EL+EID Execution Example using L+WC processes take independent actions 1. c a b c d e d 2. a b 3. d a b c d e 4. 29 SSS 2013

EL+EID Execution Example using L+WC processes take independent actions 1. 5. c a b c d e d 2. a b 3. d a b c d e 4. 30 SSS 2013 e

EL+EID Execution Example using L+WC processes take independent actions 1. 5. c a b c d e d 2. a b 3. a b c 6. c d e e a b c e d d a b c d e 4. 31 SSS 2013 e

EL+EID Execution Example using L+WC processes take independent actions 1. 5. c a b c d e d 2. a b 3. c a b c 6. d e d b c d e a b c d e d e e a b c 4. 32 SSS 2013 e e 7. a d

Outline § oracles § linearization problem § solution oracles § necessary conditions § linearization solutions § execution example § oracle implementation 33 SSS 2013

Oracle Implementation Example: WC § WC repairs the network disconnections, an encapsulation of bootstrap service commonly found in peer-to-peer systems § Example: § One bootstrap process b is always present in the system § the responsibility of this process is to maintain the greatest and smallest identifier of the system § if process p is disconnected, it does not have a left or right neighbor, it assumes that its own identifier is the greatest or, respectively, smallest § process p then sends its identifier to b § process b then either confirms this assumption or sends p, its current smallest or greatest identifier § weak connectivity is restored b A disconnected peer will contact the bootstrap service to get smallest or greatest identifier q p q 34 SSS 2013

Oracle Implementation Example: PD § PD encapsulates the limits between relative process speeds and maximum message propagation delay and can be implemented using a heartbeat protocol § Example: § if process p contains an identifier q, p sends q a heartbeat message requesting a reply § if p does not receive this reply after the time above the maximum network delay, p considers q non-existent and discards it Discard left link No reply from q heartbeat p q 35 SSS 2013

Future Research § Narrowing the gap between necessary and sufficient § Strict linearization solution relies on CD, which is a specific kind of the necessary non-subset splittable detector § Narrowing the gap between necessary and sufficient conditions for the solution to the strict linearizability problem remains to be addressed in future research. § Thank you. Any Questions. 36 SSS 2013