Local Distributed Verification A. Balliu , G. DAngelo, P. Fraigniaud, - - PowerPoint PPT Presentation

▶
local distributed verification
SMART_READER_LITE
LIVE PREVIEW

Local Distributed Verification A. Balliu , G. DAngelo, P. Fraigniaud, - - PowerPoint PPT Presentation

Nov 07, 2022 672 likes •1.41k views

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions Local Distributed Verification A. Balliu , G. DAngelo, P. Fraigniaud, and D. Oliveti CNRS and University Paris Diderot GSSI LAquila Model Local

slide-1
SLIDE 1

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Distributed Verification

  • A. Balliu, G. D’Angelo, P. Fraigniaud, and D. Oliveti

CNRS and University Paris Diderot GSSI L’Aquila

slide-2
SLIDE 2

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Goal

Classify problems according to their difficulty, i.e., build a complexity theory in the distributed seting. Build a hierarchy of complexity classes in the context of the LOCAL model.

slide-3
SLIDE 3

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph.

slide-4
SLIDE 4

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model.

slide-5
SLIDE 5

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model.

slide-6
SLIDE 6

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model.

slide-7
SLIDE 7

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model. Equivalent to a model where each node sees the network up to distance t.

slide-8
SLIDE 8

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model. Equivalent to a model where each node sees the network up to distance t. The time complexity of a local algorithm A is determined by the range t that it needs to explore.

slide-9
SLIDE 9

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Model

The distributed network is represented by a graph. Synchronous model. Equivalent to a model where each node sees the network up to distance t. The time complexity of a local algorithm A is determined by the range t that it needs to explore. We want t to be constant.

slide-10
SLIDE 10

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property.

slide-11
SLIDE 11

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network;

slide-12
SLIDE 12

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

slide-13
SLIDE 13

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision:
slide-14
SLIDE 14

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision: ”accept”
slide-15
SLIDE 15

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision: ”accept” or ”reject”.
slide-16
SLIDE 16

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision: ”accept” or ”reject”.

global_output =

v∈V

local_output(v).

slide-17
SLIDE 17

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision: ”accept” or ”reject”.

global_output =

v∈V

local_output(v).

slide-18
SLIDE 18

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Decision Problems

Decision Problems: the aim is to decide whether a global input instance satisfies some specific property. Each node:

gathers its local information from the network; perform some local computation;

  • utput its local decision: ”accept” or ”reject”.

global_output =

v∈V

local_output(v).

slide-19
SLIDE 19

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: Proper Coloring

Node input: a color. Each node checks the colors of its neighbors.

”Reject”

slide-20
SLIDE 20

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: Proper Coloring

Node input: a color. Each node checks the colors of its neighbors.

”Reject”

Local Decision (LD) is the class of distributed languages that can be locally decided [NS ’95].

slide-21
SLIDE 21

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

LD Class

LD is the class of all distributed languages L for which there exists a local algorithm A satisfying the following: for every input instance (G, x), (G, x) ∈ L ⇒ ∀id ∈ ID(G), ∀u ∈ V(G), A(G, x, id, u) = accept (G, x) / ∈ L ⇒ ∀id ∈ ID(G), ∃u ∈ V(G), A(G, x, id, u) = reject

slide-22
SLIDE 22

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property.

slide-23
SLIDE 23

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment;

slide-24
SLIDE 24

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment; gathers its local information from the network;

slide-25
SLIDE 25

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment; gathers its local information from the network; perform some local computation;

slide-26
SLIDE 26

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment; gathers its local information from the network; perform some local computation;

  • utput its local decision, that is ether ”accept” or ”reject”.
slide-27
SLIDE 27

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment; gathers its local information from the network; perform some local computation;

  • utput its local decision, that is ether ”accept” or ”reject”.

global_output =

v∈V

local_output(v).

slide-28
SLIDE 28

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Verification Problems

Verification problem: the aim is to verify whether a global input instance satisfies some specific property. Each node:

has a certificate, unbounded size and independent from the id assignment; gathers its local information from the network; perform some local computation;

  • utput its local decision, that is ether ”accept” or ”reject”.

global_output =

v∈V

local_output(v). Similar to PLS, but with id-independent certificates.

slide-29
SLIDE 29

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: is the given graph a tree?

Not locally decidable, but locally verifiable.

slide-30
SLIDE 30

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: is the given graph a tree?

Not locally decidable, but locally verifiable. Choose a node to be the root.

r

slide-31
SLIDE 31

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: is the given graph a tree?

Not locally decidable, but locally verifiable. Choose a node to be the root. Certificate of a node v: its hop-distance from the chosen root.

1 2 3 4 3 2 1 2 3 3

slide-32
SLIDE 32

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: is the given graph a tree?

Not locally decidable, but locally verifiable. Choose a node to be the root. Certificate of a node v: its hop-distance from the chosen root.

1 2 3 4 3 2 1 2 3 3

slide-33
SLIDE 33

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Example: is the given graph a tree?

Not locally decidable, but locally verifiable. Choose a node to be the root. Certificate of a node v: its hop-distance from the chosen root.

1 2 3 4 3 2 1 2 3 3

Nondeterministic LD (NLD) is the class of distributed languages that can be locally verified [FKP ’11].

slide-34
SLIDE 34

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

NLD Class

NLD is the class of all distributed languages L for which there exists a local algorithm A satisfying the following: for every input instance (G, x), (G, x) ∈ L ⇒ ∃c ∈ C(G), ∀id ∈ ID(G), ∀u ∈ V(G), A(G, x, c, id, u) = accepts (G, x) / ∈ L ⇒ ∀c ∈ C(G), ∀id ∈ ID(G), ∃u ∈ V(G), A(G, x, c, id, u) = rejects

slide-35
SLIDE 35

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

NLD Class

NLD is the class of all distributed languages L for which there exists a local algorithm A satisfying the following: for every input instance (G, x), (G, x) ∈ L ⇒ ∃c ∈ C(G), ∀id ∈ ID(G), ∀u ∈ V(G), A(G, x, c, id, u) = accepts (G, x) / ∈ L ⇒ ∀c ∈ C(G), ∀id ∈ ID(G), ∃u ∈ V(G), A(G, x, c, id, u) = rejects L ∈ NP if there is a polynomial time algorithm A such that, x ∈ L ⇐ ⇒ ∃c s.t. A accepts x with c.

slide-36
SLIDE 36

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

More about NLD

NLD is the class of all problems closed under lif [FKP ’11]. Let (G, x) and (G′, x′) be two input instances. (G′, x′) is a lif of (G, x) if there exists a function f such that: f : V(G′) → V(G) preserving the local view of each node.

G G’

slide-37
SLIDE 37

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

NLD is Closed Under Lif

Let L be a language in NLD. If (G, x) ∈ L ∧ (G′, x′) is a lif of (G, x), then (G′, x′) ∈ L.

G G’

c1 c3 c2 c2 c1 c3 c1 c2 c3

slide-38
SLIDE 38

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Goal

Build a hierarchy of complexity classes in the distributed seting. Distributed hierarchies in other seting:

[Reiter ’14] in the context of automata; [FFH ’16] in a model inspired by the CONGEST one.

slide-39
SLIDE 39

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Complexity Classes

LD = Σloc = Πloc (similar to P in the sequential seting).

slide-40
SLIDE 40

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Complexity Classes

LD = Σloc = Πloc (similar to P in the sequential seting). NLD = Σloc

1

(similar to NP in the sequential seting).

slide-41
SLIDE 41

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Complexity Classes

LD = Σloc = Πloc (similar to P in the sequential seting). NLD = Σloc

1

(similar to NP in the sequential seting). Σloc

k : An input instance satisfies a certain property in Σloc k

iff ∃c1, ∀c2, . . . , Qck, all nodes accept.

slide-42
SLIDE 42

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Complexity Classes

LD = Σloc = Πloc (similar to P in the sequential seting). NLD = Σloc

1

(similar to NP in the sequential seting). Σloc

k : An input instance satisfies a certain property in Σloc k

iff ∃c1, ∀c2, . . . , Qck, all nodes accept. Πloc

k : An input instance satisfies a certain property in Πloc k

iff ∀c1, ∃c2, . . . , Qck, all nodes accept.

slide-43
SLIDE 43

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Complementary Classes

A globaly accepted input instance. A globaly rejected input instance. A globaly accepted input instance. A globaly rejected input instance.

In a class: In a complementary class:

slide-44
SLIDE 44

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Lever 0 of the Hierarchy

and : |{u ∈ V(G) : x(u) = 1}| = 0

  • r : |{u ∈ V(G) : x(u) = 1}| ≥ 1

LD co-LD diamk and

  • r
slide-45
SLIDE 45

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Πloc

1 : The Role of the Last Universal Qantifier

Πloc

1 :

(G, x) ∈ L ⇔ ∀c all nodes accept. LD: (G, x) ∈ L ⇔ all nodes accept

slide-46
SLIDE 46

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Πloc

1 : The Role of the Last Universal Qantifier

Πloc

1 :

(G, x) ∈ L ⇔ ∀c all nodes accept. LD: (G, x) ∈ L ⇔ all nodes accept Problems that can be solved only if a specific node knows (an upper bound of) the size of the network!

slide-47
SLIDE 47

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

Let f be a function and a and b two non-negative integers.

slide-48
SLIDE 48

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

Let f be a function and a and b two non-negative integers. A configuration in iter consists in a path P = LvR with a special node v (pivot).

v

slide-49
SLIDE 49

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

Let f be a function and a and b two non-negative integers. A configuration in iter consists in a path P = LvR with a special node v (pivot). Nodes in L (resp., in R) are given as input f , f i(a) (resp., f , f i(b)); to v is given in input f , a, b.

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

slide-50
SLIDE 50

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

Let f be a function and a and b two non-negative integers. A configuration in iter consists in a path P = LvR with a special node v (pivot). Nodes in L (resp., in R) are given as input f , f i(a) (resp., f , f i(b)); to v is given in input f , a, b. f is s.t. f (0) = 0 and f (1) = 1

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

slide-51
SLIDE 51

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

Let f be a function and a and b two non-negative integers. A configuration in iter consists in a path P = LvR with a special node v (pivot). Nodes in L (resp., in R) are given as input f , f i(a) (resp., f , f i(b)); to v is given in input f , a, b. f is s.t. f (0) = 0 and f (1) = 1 An input instance is in iter if and only if:

f |L|(a) ∈ {0, 1} and f |R|(b) ∈ {0, 1} f |L|(a) = 0 or f |R|(b) = 0

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

slide-52
SLIDE 52

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

7 1

An endpoint node rejects only if it has in input something different from 1 or 0; otherwise accepts. In this case, the lef endpoint node rejects.

slide-53
SLIDE 53

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

7 5 f(7) = 6

Nodes reject if they notice local inconsistencies.

slide-54
SLIDE 54

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

1 1

(G, x) / ∈ L ⇒ ∃c s.t. at least one node rejects. v rejects only if f |L|(a) = f |R|(b) = 1; otherwise accepts. Certificate of node v: un upper bound of the size of the network.

slide-55
SLIDE 55

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

iter

v f2(b) f3(b) f4(b) f(b) f,a ,b f( a ) f2( a ) f3( a ) f4( a ) f5( a ) f6( a ) f7( a ) f8( a )

1

(G, x) ∈ L ⇒ ∀c s.t. all nodes accept. Whatever certificate v has, it will never compute f |L|(a) = f |R|(b) = 1.

slide-56
SLIDE 56

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Hierarchy

LD co-LD diamk and

  • r

Πloc

1

co-Πloc

1

iter iter

slide-57
SLIDE 57

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Hierarchy

LD co-LD diamk and

  • r

Πloc

1

co-Πloc

1

iter iter NLD = Σloc

2

co-NLD tree alts amos miss↑ miss↑

slide-58
SLIDE 58

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Πloc

2 Class

Π2 class: An input instance satisfies a certain property in Π2 iff ∀c1, ∃c2, all nodes accept. Two party game between a disprover and a prover.

slide-59
SLIDE 59

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Exactly Two Selected

1 1

Input

slide-60
SLIDE 60

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Exactly Two Selected

1 0,4,3 1,3,2 2,3,1 1,4,2 4,2,1 2,2,2 3,2,0 3,1,1 4,1,1 5,1,2 1 4,0,2

Input Disprover

slide-61
SLIDE 61

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Exactly Two Selected

1 0,4,3 3 1,3,2 2 2,3,1 1 1,4,2 2 4,2,1 1 2,2,2 2 3,2,0 3,1,1 1 4,1,1 1 5,1,2 2 1 4,0,2 2

Input Prover Disprover

slide-62
SLIDE 62

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Local Hierarchy

LD co-LD diamk and

  • r

Πloc

1

co-Πloc

1

iter iter NLD = Σloc

2

co-NLD tree alts amos miss↑ miss↑ All = Πloc

2

exts miss

LD ⊂ Πloc

1

⊂ NLD = Σloc

2

⊂ Πloc

2

= All (all inclusions are strict).

slide-63
SLIDE 63

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

miss: a Πloc

2 -complete Problem

Every node u of (G, x) is given a family F(u) of input instances, each described by

An adjacency matrix representing a graph; array representing the inputs to the nodes of that graph.

slide-64
SLIDE 64

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

miss: a Πloc

2 -complete Problem

Every node u of (G, x) is given a family F(u) of input instances, each described by

An adjacency matrix representing a graph; array representing the inputs to the nodes of that graph.

Every node u has an input string x′(u) ∈ {0, 1}∗ (notice that (G, x′) is also an input instance).

slide-65
SLIDE 65

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

miss: a Πloc

2 -complete Problem

Every node u of (G, x) is given a family F(u) of input instances, each described by

An adjacency matrix representing a graph; array representing the inputs to the nodes of that graph.

Every node u has an input string x′(u) ∈ {0, 1}∗ (notice that (G, x′) is also an input instance). The current (G, x) is legal if (G, x′) is missing in all families F(u) for every u ∈ V(G). miss = {(G, x) : ∀u ∈ V(G), x(u) = (F(u), x′(u)) and (G, x′) / ∈ F}

slide-66
SLIDE 66

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

miss: a Πloc

2 -complete Problem

slide-67
SLIDE 67

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Reduction to miss

Each node u with identity id(u) and input x(u) computes its width ω(u) = 2|id(u)|+|x(u)|.

slide-68
SLIDE 68

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Reduction to miss

Each node u with identity id(u) and input x(u) computes its width ω(u) = 2|id(u)|+|x(u)|. Each node u generates F(u), i.e., all (H, y) / ∈ L

At most ω(u) nodes; y(v) has value at most ω(u).

slide-69
SLIDE 69

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Reduction to miss

Each node u with identity id(u) and input x(u) computes its width ω(u) = 2|id(u)|+|x(u)|. Each node u generates F(u), i.e., all (H, y) / ∈ L

At most ω(u) nodes; y(v) has value at most ω(u).

If (G, x) ∈ L

(G, x) / ∈ F since only illegal instances are in F; all nodes will accept.

slide-70
SLIDE 70

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Reduction to miss

Each node u with identity id(u) and input x(u) computes its width ω(u) = 2|id(u)|+|x(u)|. Each node u generates F(u), i.e., all (H, y) / ∈ L

At most ω(u) nodes; y(v) has value at most ω(u).

If (G, x) ∈ L

(G, x) / ∈ F since only illegal instances are in F; all nodes will accept.

If (G, x) / ∈ L

There exists u with id(u) or x(u) big enough, which guarantees that u generates the graph G, i.e., (G, x) ∈ F(u); at least one node will reject.

slide-71
SLIDE 71

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Open Problems

Unbounded size id-independent certificates:

find a complete problem for Πloc

1

and co-Πloc

1 ;

find a problem in the intersection between the classes Πloc

1

and co-Πloc

1 .

Bounded size (O(log n)) id-dependent certificates

we don’t know if the hierarchy collapses; there are no separating problems for Σloc

2

and Σloc

3

(neither for classes higher in the hierarchy).

slide-72
SLIDE 72

Model Local decision Local verification Local Hierarchy Complete Problems Conclusions

Thank you!