1 / 77

A Combinatorial Characterization of the Distributed 1-Solvable Tasks

A Combinatorial Characterization of the Distributed 1-Solvable Tasks . Ofer Biran , Shlomo Moran, Shmuel Zaks Presented by Ami Paz Technion , Haifa, Israel January 2011. Roadmap. Roadmap. 1-Solvable Tasks – Informal. Asynchronies system. N processors p 1 ,…, p N .

crescent
Download Presentation

A Combinatorial Characterization of the Distributed 1-Solvable Tasks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A Combinatorial Characterization of the Distributed 1-Solvable Tasks OferBiran, Shlomo Moran, ShmuelZaks Presented by Ami Paz Technion, Haifa, Israel January 2011

  2. Roadmap

  3. Roadmap

  4. 1-Solvable Tasks – Informal • Asynchronies system. • N processors p1,…,pN. • All the processes are connected to one another. • There are no limitations on the messages (number or size). • But: One process may Crash during the run. • Only a fault crash is allowed. • Which problems can be solved?

  5. Example 1: Weak Consensus • Each processor has initial value of 0 or 1. • Should all decide on 0 or 1. 1 0 1 0 0 0 0 0 1 0

  6. Consensus Alg.: Take I • Algorithm: if someone has 0, all decide 0. 1 1 0

  7. Consensus Alg.: Take I • Algorithm: if someone has 0, all decide 0. 0 0 0

  8. Consensus Alg.: Take II • Algorithm: if someone has 0, all decide 0. 1 1

  9. Consensus Alg.: take II • Algorithm: if someone has 0, all decide 0. ? ?

  10. Consensus Alg.: Take II • Algorithm: if someone has 0, all decide 0. Let’s go for 1 He’s dead?

  11. Consensus Alg.: Take II • Algorithm: if someone has 0, all decide 0. 1 1

  12. Consensus Alg.: Take III • Algorithm: if someone has 0, all decide 0. 1 1 Zzz…

  13. Consensus Alg.: Take III • Algorithm: if someone has 0, all decide 0. ? ? Zzz…

  14. Consensus Alg.: Take III • Algorithm: if someone has 0, all decide 0. Let’s go for 1 He’s dead? Zzz…

  15. Consensus Alg.: Take III • Algorithm: if someone has 0, all decide 0. 1 1 0 Zzz… Guys?

  16. Example 2:Order Preserving Renaming • Each processor starts with a rational value. • Rename to values 1…10, while preserving order. 1 3 -3 1 1000 9 7 4 20 5

  17. OPR Alg.: Take I -3 12 16

  18. OPR Alg.: Take I 2 5 8

  19. OPR Alg.: Take II -2 12

  20. OPR Alg.: Take II ? ?

  21. OPR Alg.: Take II Never mind He’s dead?

  22. OPR Alg.: Take II 2 5

  23. OPR Alg.: Take III -3 12 Zzz… (16)

  24. OPR Alg.: Take III ? ? Zzz… (16)

  25. OPR Alg.: Take III Never mind He’s dead? Zzz… (16)

  26. OPR Alg.: Take III 2 5 16 Zzz… (16) I decide: 7

  27. OPR Alg.: Take IV -3 12 Zzz… (9)

  28. OPR Alg.: Take IV 2 5 What if we had to rename to values 1…3? Zzz… (9) I decide: 3 9 1...4?

  29. Roadmap

  30. Asynchronous Distributed Network • N–processes, N≥3. • Unique identities, Mutually known. • Also: may not be mutually known. • Massages pass along communication lines: • Arrive with no error. • Arrive in finite, unbounded and unpredictable time. • Complete network. • Also: bi-connected network (connected even after removing one process).

  31. Decision Task • A mapping of possible inputs to allowable outputs. • Let X be a set of input values. • Let D be a set of output values. • A distributed decision taskT is a function: • Where XT ⊆ XN is the set of allowable inputs of T. Maps each allowed vector to a set of legal outputs

  32. Example 1: Weak Consensus - Revised • Each processors has initial value of 0 or 1. • Should all decide on 0 or 1. • Task formulation: This does not rule out trivial solutions

  33. Example 1.5: Strong Consensus • Each processor has initial value of 0 or 1. • Should all decide on 0 or 1. • Starting with consensus, should end up with the same value. • Task formulation:

  34. Why Do We Care About Consensus? • Xoris a consensus task • Majority is a consensus task • Consensus can be done by leader election. • So, impossibility of consensus implies impossibility of leader election. Can leader election be done by consensus?

  35. Example 2:Order Preserving Renaming - Revised • Each processors start with a rational value. • Rename to values 1…k, while preserving order. • Task formulation:

  36. Protocols and Solvability • A Protocolα for a given network is a set ofN programs, each associated with a processor. • A protocol αsolves a task T if for any input vector x∈XT, it satisfies: • All execution are complete. • All Execution outputs are legal.

  37. Faults • A processor P is faulty in an execution e if all messages sent by P during e after a certain time are never received. This is called a crash fault. • No Byzantine faults here…

  38. 1-Solvability • A protocol α1-solves a Task T if: • If no processor is faulty, it solves the task. • If in an execution a processor is faulty, then all other processors eventually decide. • If there exist such a protocol, we say the task Tis 1-solvable.

  39. Roadmap

  40. Adjacency Graph • Let for some set A. • Two vectors in A are adjacent if they differ only in one entry. • The adjacency graph of S is denoted by G(S). G(S) S

  41. Connectivity Thm.[MW] • If, for a task T: • The input graph is connected • A protocol α 1-solves T Then: • The output graph of α is connected.

  42. Connectivity – Example 1:Weak Consensus • α for all Pi: decide “1”. Input graph Output graph α • (0,0,0) • (1,0,0) • (1,1,1) • (1,1,0) • (0,1,0) • (1,0,1) • (0,0,1) • (0,1,1) • (1,1,1)

  43. Connectivity – Example 2:Strong Consensus • Input graph is connected, • α 1-solves T, • But: • The output graph of αis not connected! • Is there a protocol α which 1-solves T? Input graph Output graph α • (0,0,0) • (1,0,0) • (1,1,1) • (1,1,0) • (0,1,0) • (1,0,1) • (0,0,0) • (0,0,1) • (0,1,1) • (1,1,1)

  44. The Connectivity Condition • Let T be a decision task. • For any set of allowable inputs C, if: • G(C) is a connected subgraph of the input graph • A protocol α 1-solves T Then: • The output graph of α induced by running α on the inputs in C is connected. • Easy to deduce from the last Thm.: • Define a new task with input C, and the output induced by α and C.

  45. Connectivity – Example 3:Strong Consensus with Majority • Can we solve Strong Consensus, if we have a promise that there is a majority? • The connectivity condition does not deny such a protocol Input graph • (0,0,0,1) α • (0,0,0,0) • (0,0,1,0) Output graph • (0,1,0,0) • (1,0,0,0) • (1,1,1,0) • (1,1,0,1) • (0,0,0,0) • (1,0,1,1) • (1,1,1,1) • (0,1,1,1) • (1,1,1,1)

  46. Induced Task Let α be a protocol that solves a task T. • We denote by Tα the task induced by α and XT. • Tα is defined by: • XT α = XT • The allowable outputs of Tα are all the possible outputs of α and XT.

  47. Example: Strong Consensus • Task formulation: • T: • Protocol α: every Pi chooses by majority. • Tα:

  48. Pointwise Connectivity • A task T is pointwise connected if G(T(x)) is connected for each x∈XT. • i.e., the output graph for each input vector is connected. So, from the connectivity condition we get the following corollary: • Let α be a protocol that 1-solves a task T, then Tα is pointwise connected.

  49. Roadmap

  50. Vectors and Executions • A partial vector is a vector in which one of its’ entries is not specified. • For a vector s=(s1,…,sN), • We denote: si=(s1,…,si-1,*,si+1,…,sN). • s is an extension of si. • Let xi be a partial input vector, and di a partial decision vector of a task T. • di is a covering vector of xi if: • For each extension of xi to a legal input vector x, there is an extension of di to a decision vector d,such that d is a legal decision vector of T on x.

More Related