Computability in Anonymous Networks - (TIK),(ETH Zurich)

Computability in Anonymous Networks: Revocable vs. Irrecovable Outputs

Yuval Emek 1 1 Technion 2 ETH

Jochen Seidel 2

Roger Wattenhofer 2

– Faculty of Industrial Engineering and Management – ie.technion.ac.il

Zurich – Distributed Computing Group – www.disco.ethz.ch

Anonymous Networks

Computability in Anonymous Networks


Computable

Not Computable

Computability in Anonymous Networks 3

2

2

3

2

Computable I

Degree

2

Not Computable


Computable I

Degree

I

Coloring

I

...

Not Computable

Computability in Anonymous Networks ⊥

⊥

⊥

⊥

Computable I

Degree

I

Coloring

I

...

⊥

Not Computable I

Leader Election

Computability in Anonymous Networks ⊥

⊥

⊥

⊥

Computable I

Degree

I

Coloring

I

...

⊥

Not Computable I

Leader Election

Computability in Anonymous Networks ?

?

?

?

?

Computable I

Degree

I

Coloring

I

...

?

Not Computable I

Leader Election


Computable I

Degree

I

Coloring

I

...

Not Computable I

Leader Election


Computable I

Degree

I

Coloring

I

...

Not Computable I

Leader Election


Computable I

Degree

I

Coloring

I

...

Not Computable I

Leader Election


Computable I

Not Computable

Degree

I

Leader Election

I

Coloring

I

Network Size

I

...

I

...

Revocable vs. Irrecovable Outputs Irrevocable ε

Revocable ε


Revocable ε


Revocable ε


Revocable ε ε

Revocable vs. Irrecovable Outputs Irrevocable

Revocable

ε

ε ε

Write-Once

Re-Write

I

WO-algorithms

I

RW-algorithm

I

Problem Class WO

I

Problem Class RW

WO ⊆ RW

Revocable vs. Irrecovable Outputs Irrevocable

Revocable

ε

ε ε

Write-Once

Re-Write

I

WO-algorithms

I

RW-algorithm

I

Problem Class WO

I

Problem Class RW

WO ( RW

RW: How to Specify the Network’s Output?

ε

a

a

b

x

y

ε

y

Time

Ready Configurations

ε

a

a

b

x

y

ε

y

Time

Correctness Notions

ε

a

a

b

x

y

ε

y

Time

2. Default

(Every ready configuration correct)

Correctness Notions

ε

a

a

b ...

x

y

ε

y

Time

2. Default 3. Loose

(Every ready configuration correct) (First ready configuration correct)

Correctness Notions

ε

a

a

a ...

x

y

y

y

Time

1. Sustainable 2. Default 3. Loose

(Output irrevocable after first ready configuration) (Every ready configuration correct) (First ready configuration correct)

Correctness Notions

Theorem (Sustainable vs. Loose Correctness) The following three correctness notions are equivalent for RW: 1. Sustainable 2. Default 3. Loose


Correctness Notions 1. sustainable-RW ⊆ RW ⊆ loose-RW



Correctness Notions 1. sustainable-RW ⊆ RW ⊆ loose-RW X



Correctness Notions 1. sustainable-RW ⊆ RW ⊆ loose-RW X 2. loose-RW ⊆ sustainable-RW ?



Proof Idea 1. sustainable-RW ⊆ RW ⊆ loose-RW X 2. loose-RW ⊆ sustainable-RW ?

ε


5 5

ε


5

5 5 ε

ε

Proof Idea 1. sustainable-RW ⊆ RW ⊆ loose-RW X 2. loose-RW ⊆ sustainable-RW ? 5 ε 5 ε

ε

Proof Idea 1. sustainable-RW ⊆ RW ⊆ loose-RW X 2. loose-RW ⊆ sustainable-RW ? 5 ε

ε

5 ε

I

Round 5 is inhibited.

I

Pick output from first non-inhibited round =⇒ sustainable correctness.

Proof Idea 1. sustainable-RW ⊆ RW ⊆ loose-RW X 2. loose-RW ⊆ sustainable-RW X



Logical Or is Computable With Revocable Outputs

0 0 0 0 0 0


1 0 1 1 1 0


0

RW-algorithm for Or Round 1: Output 0 if input = 0. Round 2: Output 1.

1 0




0 0

RW-algorithm for Or

ε 1 0

Round 1: Output 0 if input = 0. Round 2: Output 1.

0




1 0

RW-algorithm for Or 1 1 1

Round 1: Output 0 if input = 0. Round 2: Output 1.

0



Logical Or is Not Computable With Irrevocable Outputs

0

0 vs.

0 0

...

0

0

0

1 0

...

0

0

Problem Zoo I’ve got 21 problems . . .

CF

RW

WO


CF Min-Coloring (≥ 3)-Hop-Coloring (≥ 3)-Hop-MIS Factor-Multiplicity Size-Apx Leader-Election Min-Cut Uniqueness IDs Diameter Diameter-Apx

RW Factor-Diam Consensus Coordination Factor-Graph And Or

WO (∆ + 1)-Coloring 2-Hop-Coloring 2-Hop-MIS MIS


RW

Factor-Diam Consensus Coordination Factor-Graph And Or

Gr ap hTh eo re tic

Min-Coloring (≥ 3)-Hop-Coloring (≥ 3)-Hop-MIS Factor-Multiplicity Size-Apx Leader-Election Min-Cut Uniqueness IDs Diameter Diameter-Apx

Ch ar ac te riz at io n

CF

WO

(∆ + 1)-Coloring 2-Hop-Coloring 2-Hop-MIS MIS


CF

Gr ap hTh eo re tic

Ch ar ac te riz at io n

Min-Coloring W (≥ h 3)-Hop-Coloring t ar (≥ a 3)-Hop-MIS RW e th Factor-Multiplicity e Factor-Diam Size-Apx har Leader-Election des Consensus Min-Cut WO Coordination t p Uniqueness r o Factor-Graph blem IDs (∆ + 1)-Coloring And Diameter 2-Hop-Coloring s? Or Diameter-Apx 2-Hop-MIS MIS

An Oracle

An Oracle

1 Round

Accessing a Distributed Oracle

1 Round

In each round: I Set oracle input (for next round) I

Get oracle answer (from previous round)


0 1 1 Round 0 In each round: I Set oracle input (for next round) I



0

1

1 1 Round 1 0 In each round: I Set oracle input (for next round) I


1

Distributed Oracles

An oracle does not help if you could answer the question yourself.

Theorem Fix a class C ∈ {RW, WO}, and let Q ∈ C. P solvable by C-algorithm accessing Q-oracle =⇒ P solvable by C-algorithm (without any oracle access).

Hard and Complete Problems Hardness Fix two classes B ⊇ C, and a problem Q. Q ∈ B-hard C , if I

for every P ∈ B:

I

there is a C-algorithm solving P with access to a Q-oracle

“N P-hard = N P-hardP ”

Hard and Complete Problems Hardness Fix two classes B ⊇ C, and a problem Q. Q ∈ B-hard C , if I

for every P ∈ B:

I

there is a C-algorithm solving P with access to a Q-oracle

“N P-hard = N P-hardP ” I

CF-hard RW

I

CF-hard WO

I

RW-hard WO


CF

RW CF-hard RW

CF-hard WO

RW-hard WO

WO


Min-Coloring Min-Cut-Value (≥ 3)-Hop-Coloring (≥ 3)-Hop-MIS

CF

RW CF-hard RW Factor-Mult

Factor-Diam Consensus

RW-hard WO CF-hard WO WO (< 2)-Size-Apx (∆ + 1)-Coloring Leader-Election Coordination (≥ 2)-Size-Apx 2-Hop-Coloring Min-Cut Factor-Graph Diameter 2-Hop-MIS Uniqueness And Diameter-Apx MIS IDs Or Min-Cut-Partition



CF

?

RW

CF-hard RW Factor-Mult





CF

!

RW

CF-hard RW Factor-Mult



Theorem CF-hard WO CF-hard RW

RW-hard WO


RW-hard WO

Interesting Part: CF-hard WO ⊇ CF-hard RW ∩ RW-hard WO


P

RW-hard WO

Interesting Part: CF-hard WO ⊇ CF-hard RW ∩ RW-hard WO P ∈ CF-hard RW ∩ RW-hard WO


P

RW-hard WO

Interesting Part: CF-hard WO ⊇ CF-hard RW ∩ RW-hard WO P ∈ CF-hard RW ∩ RW-hard WO With P-oracle, I

RW-algorithm for Leader-Election


P

RW-hard WO



I

WO-algorithm for Or


P

RW-hard WO



I

WO-algorithm for Or

WO-algorithm

Summary

I

3 Correctness Notions for Revocable Outputs I

I

3 Hardness Classes I

I

Equivalent

3rd is Intersection of the Other Two

21 – 24 Problems I

Completely Characterized