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Attack-Resilient Multitree Data Distribution Topologies

Sascha Grau1

Technische Universit¨ at Ilmenau

December 19th, 2012

1This work was supported by the Deutsche Forschungsgemeinschaft under grant number KU 658/10-2.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 1 / 18

Distribution Topologies

Multi-Tree Data Distribution

The Goal

Reliable broadcast of data from a resource-restricted data source s to a large audience of nodes V = {1, . . . , n} = [n].

The Approach

each block of data is split into k subblocks, to be distributed over a fixed set of k trees redundant encoding (e.g. multiple description coding

• r error-correcting coding) is applied

⇒ nodes are satisfied as long as they receive data in at least a certain share of trees

Data

s s s

The Applications

peer-to-peer live streaming systems reversed data flow: data aggregation tasks

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 2 / 18

Distribution Topologies

Multi-Tree Data Distribution

The Goal

Reliable broadcast of data from a resource-restricted data source s to a large audience of nodes V = {1, . . . , n} = [n].

The Approach

each block of data is split into k subblocks, to be distributed over a fixed set of k trees redundant encoding (e.g. multiple description coding

• r error-correcting coding) is applied

⇒ nodes are satisfied as long as they receive data in at least a certain share of trees

Data

s s s

The Applications

peer-to-peer live streaming systems reversed data flow: data aggregation tasks

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 2 / 18

Distribution Topologies

Multi-Tree Data Distribution Topologies

Definition (Distribution Topology)

A distribution topology with k trees over nodes V is a k-tuple T = (T1, . . . , Tk)

• f directed trees. For each i ∈ [k], the tree Ti has the same root s ∈ V and node

set {s} ∪ V .

Restrictions on Communication

possible tree edges {(u, v) | u ∈ {s} ∪ V , v ∈ V \ {u}} nodes can have degree-restrictions here: only source is restricted

s s s

The class T(n, C, k) is the set of distribution topologies with k trees over node set [n], in which source s has at most Ck incident edges (C ∈ N).

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 3 / 18

Distribution Topologies

Multi-Tree Data Distribution Topologies

Definition (Distribution Topology)

A distribution topology with k trees over nodes V is a k-tuple T = (T1, . . . , Tk)

• f directed trees. For each i ∈ [k], the tree Ti has the same root s ∈ V and node

set {s} ∪ V .

Restrictions on Communication

possible tree edges {(u, v) | u ∈ {s} ∪ V , v ∈ V \ {u}} nodes can have degree-restrictions here: only source is restricted

s s s

The class T(n, C, k) is the set of distribution topologies with k trees over node set [n], in which source s has at most Ck incident edges (C ∈ N).

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 3 / 18

Distribution Topologies

Stable Multi-Tree Distribution Topologies

Attacks ...

suddenly remove a set of nodes from all trees. are maliciously planned to maximize damage (worst-case model).

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 7 3 4 2 5 T1 T2 T3

Prior Work: damage = number of disturbed source-to-node paths

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 4 / 18

Distribution Topologies

Stable Multi-Tree Distribution Topologies

Attacks ...

suddenly remove a set of nodes from all trees. are maliciously planned to maximize damage (worst-case model).

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 7 3 4 2 5 T1 T2 T3

Prior Work: damage = number of disturbed source-to-node paths Nice, but user-centered notion of damage is more relevant in applications.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 4 / 18

Distribution Topologies

A User-Centered Measure of Damage

Given: distribution topology T ∈ T(n, C, k) quality threshold z ∈ [k] (depending on redundancy in data encoding) set X ⊆ [n] of removed nodes

Damage bT (X, z)

Number of nodes not satisfied, i.e., reachable from the source in at most k − z of the k trees. v

T1 T2 T3 T4 T5

X = {2, 5, 7} bT (X, 2) = 6

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 7 3 4 2 5 T1 T2 T3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 5 / 18

Attack-Resilience and Forward-Stability

Attack-Resilient Distribution Topologies

Goal

Determine topologies in T(n, C, k), that minimize the maximum possible damage for all attack sizes and quality thresholds.

Definition (Attack-Resilient Distribution Topology)

A topology T ∈ T(n, C, k) is called attack-resilient, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k): max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bC(X, z).

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 6 / 18

Attack-Resilience and Forward-Stability

Attack-Resilient Distribution Topologies

Goal

Determine topologies in T(n, C, k), that minimize the maximum possible damage for all attack sizes and quality thresholds.

Definition (Attack-Resilient Distribution Topology)

A topology T ∈ T(n, C, k) is called attack-resilient, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k): max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bC(X, z). Problem: direct analysis is hard Trick: study topologies optimizing a highly similar damage measure

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 6 / 18

Attack-Resilience and Forward-Stability

Forward-Damage

Forward-Damage bfT (X, z)

highly similar to damage measure bT (X, z) Difference: some directly attacked nodes possibly not counted as damage Consequence: bT (X, z) and bfT (X, z) differ by at most |X| bT ({3, 4, 5}, 2) = 5 bfT ({3, 4, 5}, 2) = 3

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 7 / 18

Attack-Resilience and Forward-Stability

Forward-Damage

Forward-Damage bfT (X, z)

highly similar to damage measure bT (X, z) Difference: some directly attacked nodes possibly not counted as damage Consequence: bT (X, z) and bfT (X, z) differ by at most |X| bT ({3, 4, 5}, 2) = 5 bfT ({3, 4, 5}, 2) = 3

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 7 / 18

Attack-Resilience and Forward-Stability

Dominance of Forward-Damage

For growing numbers of nodes in a topology (application-relevant cases), the worst-case forward-damage dominates the worst-case damage.

Theorem

For all T ∈ T(n, C, k), z ∈ [k], and x ∈ [n], it holds that max

X⊆V |X|=x

bfT (X, z) ≤ max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bfT (X, z) + min(Cz − 1, x).

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 8 / 18

Attack-Resilience and Forward-Stability

Dominance of Forward-Damage

For growing numbers of nodes in a topology (application-relevant cases), the worst-case forward-damage dominates the worst-case damage.

Theorem

For all T ∈ T(n, C, k), z ∈ [k], and x ∈ [n], it holds that max

X⊆V |X|=x

bfT (X, z) ≤ max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bfT (X, z) + min(Cz − 1, x). Distribution topologies that minimize forward-damage are: easier to analyze, give a good approximation of attack-resilient topologies for n ≫ Ck.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 8 / 18

Attack-Resilience and Forward-Stability

Restricted Attacks

Notions

“v is head in Ti”: v is adjacent to s in Ti (all heads of Ti: HT

i )

“v is forwarding in Ti”: v is head of or has children in Ti

Definition (t-restricted Attack)

An attack X is called t-restricted, if there is I ⊆ [k],|I| = t, such that every node v ∈ X is forwarding either in a tree Ti with i ∈ I or in no tree at all. All t-restricted attacks on topology T constitute the set χ(T , t). Special case: all nodes in X together are forwarding in at most t trees

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 9 / 18

Attack-Resilience and Forward-Stability

Restricted Attacks

Notions

“v is head in Ti”: v is adjacent to s in Ti (all heads of Ti: HT

i )

“v is forwarding in Ti”: v is head of or has children in Ti

Definition (t-restricted Attack)

An attack X is called t-restricted, if there is I ⊆ [k],|I| = t, such that every node v ∈ X is forwarding either in a tree Ti with i ∈ I or in no tree at all. All t-restricted attacks on topology T constitute the set χ(T , t). Special case: all nodes in X together are forwarding in at most t trees {2, 3, 6} ∈ χ(T , 2) {2, 3, 4} ∈ χ(T , 2)

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 9 / 18

Attack-Resilience and Forward-Stability

Restricted Attacks

Notions

“v is head in Ti”: v is adjacent to s in Ti (all heads of Ti: HT

i )

“v is forwarding in Ti”: v is head of or has children in Ti

Definition (t-restricted Attack)

An attack X is called t-restricted, if there is I ⊆ [k],|I| = t, such that every node v ∈ X is forwarding either in a tree Ti with i ∈ I or in no tree at all. All t-restricted attacks on topology T constitute the set χ(T , t). Special case: all nodes in X together are forwarding in at most t trees {2, 3, 6} ∈ χ(T , 2) {2, 3, 4} ∈ χ(T , 2)

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 9 / 18

Attack-Resilience and Forward-Stability

Attack-Resilient vs. Forward-Stable Topologies

Definition (Attack-Resilient Distribution Topology)

A topology T ∈ T(n, C, k) is called attack-resilient, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k): max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bC(X, z).

Definition ((t-)Forward-Stable Distribution Topologies)

A topology T ∈ T(n, C, k) is t-forward-stable, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k) : max

X∈χ(T ,t) |X|=x

bfT (X, z) ≤ max

X∈χ(C,t) |X|=x

bfC(X, z). T is called forward-stable, if it is t-forward-stable for all t ∈ [k]. fine-grained due to attack restrictions forward-stable topologies are similar to attack-resilient topologies

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 10 / 18

Attack-Resilience and Forward-Stability

Attack-Resilient vs. Forward-Stable Topologies

Definition (Attack-Resilient Distribution Topology)

A topology T ∈ T(n, C, k) is called attack-resilient, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k): max

X⊆V |X|=x

bT (X, z) ≤ max

X⊆V |X|=x

bC(X, z).

Definition ((t-)Forward-Stable Distribution Topologies)

A topology T ∈ T(n, C, k) is t-forward-stable, if it holds that ∀x ∈ [n], ∀z ∈ [k], ∀C ∈ T(n, C, k) : max

X∈χ(T ,t) |X|=x

bfT (X, z) ≤ max

X∈χ(C,t) |X|=x

bfC(X, z). T is called forward-stable, if it is t-forward-stable for all t ∈ [k]. fine-grained due to attack restrictions forward-stable topologies are similar to attack-resilient topologies

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 10 / 18

Attack-Resilience and Forward-Stability

Basic Properties of Forward-Stable Topologies

Lemma 1

For every node v in a t-forward-stable topology T , it holds that:

1

v is forwarding in at most one tree

2

in this tree, v has at most n

C

• successors

s 1 2 3 4 5 7 6 8 9 s 3 1 2 4 5 9 8 6 7

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 11 / 18

Attack-Resilience and Forward-Stability

Basic Properties of Forward-Stable Topologies

Lemma 1

For every node v in a t-forward-stable topology T , it holds that:

1

v is forwarding in at most one tree

2

in this tree, v has at most n

C

• successors

s 1 2 3 4 5 7 6 8 9 s 3 1 2 4 5 9 8 6 7 1 2 3

• 2

4

• 1

5 6 7 8 9

7

• 5

1 2 3 4 5

• 4
• 3

6 7 8

8

9

9

In topologies with this property, there is always an attack of maximum forward-damage targeting only heads. Topology stability can be characterized by a matrix representation of the successor sets of heads.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 11 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3,

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3,

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                 1 1 1                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                 1 1 1 1 1 2                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                 1 1 1 1 1 2 1 1 3                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                 1 1 1 1 1 2 1 1 3 2 2 1                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 k n

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

The Matrix MT and σ(X)

T1 T2 T3

1 2 3 1 2 3 1 2 3 s 1 2 3 4 5 6 7 8 9

10

s 2 1 3 5 4 8 9 6 7

10

s

10

1 4 7 8 2 5 3 6 9

Choose bijections σi : HT

i

→ [|HT |] for all i ∈ [k]. σ1(1) = 1, σ1(4) = 2, σ1(7) = 3, σ2(2) = 1, σ2(5) = 2, σ2(6) = 3, σ3(10) = 1, σ3(8) = 2, σ3(3) = 3 σ(X) = X1 × . . . × Xk with Xi =

• {0}

, if X ∩ HT

i

= ∅ {σi(v)|v ∈ X ∩ HT

i }

, else k n

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 12 / 18

Attack-Resilience and Forward-Stability

Forward-Damage in the Matrix Representation

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

X = {1, 5, 8} σ(X) = {(1, 2, 2)}

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 13 / 18

Attack-Resilience and Forward-Stability

Forward-Damage in the Matrix Representation

It holds that bfT (X, z) =

• {v ∈ V | ∃x ∈ σ(X): d(MT [v], x) ≤ k − z}
• .

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

X = {1, 5, 8} σ(X) = {(1, 2, 2)} z = 3 bfT (X, z) = 0

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 13 / 18

Attack-Resilience and Forward-Stability

Forward-Damage in the Matrix Representation

It holds that bfT (X, z) =

• {v ∈ V | ∃x ∈ σ(X): d(MT [v], x) ≤ k − z}
• .

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

X = {1, 5, 8} σ(X) = {(1, 2, 2)} z = 2 bfT (X, z) = 3

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 13 / 18

Attack-Resilience and Forward-Stability

Forward-Damage in the Matrix Representation

It holds that bfT (X, z) =

• {v ∈ V | ∃x ∈ σ(X): d(MT [v], x) ≤ k − z}
• .

MT =                 1 1 1 1 1 2 1 1 3 2 2 1 2 2 2 2 3 3 3 3 1 3 2 2 3 2 3 3 3 1                

X = {1, 5, 8} σ(X) = {(1, 2, 2)} z = 1 bfT (X, z) = 7

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 13 / 18

Attack-Resilience and Forward-Stability

Orthogonal Arrays

Definition (Orthogonal Array)

An n × k-matrix M with entries from an alphabet [C] is called Orthogonal Array of strength t, if for each choice of t columns of M, every possible row x ∈ [C]t appears exactly n/C t times.

M =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 14 / 18

Attack-Resilience and Forward-Stability

Orthogonal Arrays

Definition (Orthogonal Array)

An n × k-matrix M with entries from an alphabet [C] is called Orthogonal Array of strength t, if for each choice of t columns of M, every possible row x ∈ [C]t appears exactly n/C t times.

M =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

Theorem

A topology T with the properties given in Lemma 1, for which MT is an Orthogonal Array of strength t is t-forward-stable.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 14 / 18

Attack-Resilience and Forward-Stability

Orthogonal Arrays

Definition (Orthogonal Array)

An n × k-matrix M with entries from an alphabet [C] is called Orthogonal Array of strength t, if for each choice of t columns of M, every possible row x ∈ [C]t appears exactly n/C t times.

M =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

Theorem

A topology T with the properties given in Lemma 1, for which MT is an Orthogonal Array of strength t is t-forward-stable.

Theorem

For every forward-stable topology T , the matrix MT is an Orthogonal Array of maximum strength.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 14 / 18

Attack-Resilience and Forward-Stability

Consequences for Forward-Stable Topologies

Orthogonal Arrays

extremal parameters mostly unknown special case: “MDS Conjecture”, unsettled since 1955 existence of pseudopolynomial construction algorithm for Orthogonal Arrays

• f maximum strength unknown

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 15 / 18

Attack-Resilience and Forward-Stability

Consequences for Forward-Stable Topologies

Orthogonal Arrays

extremal parameters mostly unknown special case: “MDS Conjecture”, unsettled since 1955 existence of pseudopolynomial construction algorithm for Orthogonal Arrays

• f maximum strength unknown

Theorem

If there is an algorithm with pseudopolynomial runtime for the construction of forward-stable topologies, then there is a pseudopolynomial algorithm for the construction of Orthogonal Arrays of maximum strength. This result illustrates the hardness of constructing both forward-stable and attack-resilient topologies.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 15 / 18

Conclusion

Conclusion

forward-stable distribution topologies closely approximate attack-resilient topologies for n ≫ Ck (relevant scenarios)

s s s MT =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 16 / 18

Conclusion

Conclusion

forward-stable distribution topologies closely approximate attack-resilient topologies for n ≫ Ck (relevant scenarios) a forward-stable topology T

1

satisfies two basic requirements and

2

has a matrix MT of specific properties

s s s MT =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 16 / 18

Conclusion

Conclusion

forward-stable distribution topologies closely approximate attack-resilient topologies for n ≫ Ck (relevant scenarios) a forward-stable topology T

1

satisfies two basic requirements and

2

has a matrix MT of specific properties

if MT is an Orthogonal Array of strength t, then T is t-forward-stable if T is forward-stable, then MT is an Orthogonal Array

• f maximum strength

s s s MT =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 16 / 18

Conclusion

Conclusion

forward-stable distribution topologies closely approximate attack-resilient topologies for n ≫ Ck (relevant scenarios) a forward-stable topology T

1

satisfies two basic requirements and

2

has a matrix MT of specific properties

if MT is an Orthogonal Array of strength t, then T is t-forward-stable if T is forward-stable, then MT is an Orthogonal Array

• f maximum strength

implicates notion of computational hardness of constructing forward-stable or attack-resilient topoplogies

s s s MT =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 16 / 18

Outlook

There is still a lot to do . . .

Forward-Stable Topologies

the case C ∈ N: mixed-level Orthogonal Arrays if Orthogonal Arrays of maximum strength t are known:

1

t < k? Packing Arrays of minimum t-column subrow-frequency for each t ∈ [k]!

2

Are there further demands? Probably: Yes.

And More

attack-resilient topologies for small n considering restoration capabilities if forward error correcting codes are used (does not help in topologies of depth ≤ 2)

s s s MT =               1 1 1 1 2 3 1 3 2 2 1 3 2 2 2 2 3 1 3 1 2 3 2 1 3 3 3              

x y z

1 2 3 2 3 2 3

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 17 / 18

Thank you

Thank you for your attention.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 18 / 18

Thank you Forward-Stability

Forward-Damage

Damage

bT (X, z) :=

• I⊆{1,...,k},|I|=z
• i∈I
• v∈X succT

i (v)

• with

succT

i (v) = {w | s → v → w-path in Ti}.

bT ({3, 4, 5}, 2) = 5

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Forward-Damage

bfT (X, z) :=

• I⊆{1,...,k},|I|=z
• i∈I
• v∈X succT →

i

(v)

• with

succT →

i

(v) =

• succT

i (v)

, v forwarding in Ti ∅ , otherwise.

bfT ({3, 4, 5}, 2) = 3

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 19 / 18

Thank you Forward-Stability

Forward-Damage

Damage

bT (X, z) :=

• I⊆{1,...,k},|I|=z
• i∈I
• v∈X succT

i (v)

• with

succT

i (v) = {w | s → v → w-path in Ti}.

bT ({3, 4, 5}, 2) = 5

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Forward-Damage

bfT (X, z) :=

• I⊆{1,...,k},|I|=z
• i∈I
• v∈X succT →

i

(v)

• with

succT →

i

(v) =

• succT

i (v)

, v forwarding in Ti ∅ , otherwise.

bfT ({3, 4, 5}, 2) = 3

s 3 7 2 1 5 6 4 s 2 1 3 4 5 6 7 s 6 1 4 7 3 2 5

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 19 / 18

Thank you Topology Construction

Construction of Forward-Stable Distribution Topologies

Input: n, C, k ∈ N, n ≥ Ck Output: a forward-stable T ∈ T(n, C, k)

1

Determine suitable n × k matrix MT (e.g.: max. strength, min. row frequencies for arbitrary choice of columns). This is the hard part.

2

Determine feasible set of heads HT ⊆ [n] by solving assignment problem on auxiliary graph. Result: Bijections σ1, . . . , σk

3

For each tree i ∈ [k]: connect the s with each h ∈ HT

i

and position each node v ∈ V \ HT

i

as child of head σ−1

i

(MT [v]i).

4

For each subtree rooted in a forwarding head: arbitrary reorganization possible as long as trees remain pairwise inner-disjoint.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 20 / 18

Thank you References

References (1)

Michael Brinkmeier, Guenter Schaefer, and Thorsten Strufe. Optimally DoS Resistant P2P Topologies for Live Multimedia Streaming. IEEE Transactions on Parallel and Distributed Systems, 20(6):831–844, 2009. Sascha Grau, Mathias Fischer und G¨ unter Sch¨ afer. On the Dependencies between Source Neighbors in Optimally DoS-stable P2P Streaming Topologies. In IEEE International Conference on Distributed Computing Systems 2011, ICDCS, S. 121–130, Minneapolis, MN, Juni 2011. IEEE Computer Society. Sascha Grau. On the Stability of Distribution Topologies in Peer-to-Peer Live Streaming Systems. PhD Thesis, Technische Universit¨ at Ilmenau, 2012.

Sascha Grau (TU Ilmenau) Multitree Data Distribution Topologies December 19th, 2012 21 / 18