Attacker Independent Stability Guarantees for Peer-2-Peer-Live-Streaming Topologies Andreas Brieg, Michael Brinkmeier, Sascha Grau, Mathias Fischer, Guenter Schaefer This work was in part supported by the Deutsche Forschungsgemeinschaft under grant numbers KU658/10-1 and SCHA1533/1-1. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 1 / 21
P2P-Live-Streaming - What & Why? Goal Realtime distribution of continously generated multimedia-stream to varying and potentially large set of viewers. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 2 / 21
P2P-Live-Streaming - What & Why? Goal Realtime distribution of continously generated multimedia-stream to varying and potentially large set of viewers. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 2 / 21
P2P-Live-Streaming - What & Why? Goal Realtime distribution of continously generated multimedia-stream to varying and potentially large set of viewers. Key Idea Incorporate viewers’ resources for distribution to overcome problems of classical Client-Server approach: restricted bandwidth resources at server high hardware costs inefficient traffic patterns (all paths lead to server) Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 2 / 21
Packet distribution in trees Packets: s enter the system at source node s Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 3 / 21
Packet distribution in trees Packets: s enter the system at source node s distributed to direct neighbors v 1 v 2 v 3 Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 3 / 21
Packet distribution in trees Packets: s enter the system at source node s distributed to direct neighbors replicated and re-distributed to v 1 v 2 v 3 other nodes v 4 v 5 v 6 v 7 v 8 Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 3 / 21
Packet distribution in trees Packets: s enter the system at source node s distributed to direct neighbors replicated and re-distributed to v 1 v 2 v 3 other nodes . . . v 4 v 5 v 6 v 7 v 8 Packet distribution over Spanning Trees ! Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 3 / 21
Packet distribution in trees Packets: s enter the system at source node s distributed to direct neighbors replicated and re-distributed to v 1 v 2 v 3 other nodes . . . v 4 v 5 v 6 v 7 v 8 Packet distribution over Spanning Trees ! Service quality of peers in low levels of the tree depends on cooperation and health of all nodes in its path to the source. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 3 / 21
Problems of P2P-Live-Streaming systems But peers... constantly join and leave the system have small resources are vulnerable to attacks and have high failure probability Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 4 / 21
Problems of P2P-Live-Streaming systems But peers... constantly join and leave the system have small resources are vulnerable to attacks and have high failure probability A key idea Using multiple distribution trees with varying inner nodes decreases dependency on single nodes. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 4 / 21
Model of push-based P2P-Streaming systems (1) Basic model Stream is divided into k substreams called stripes Participants V = { s , v 1 , . . . , v n } are nodes of a graph G Stripe i is distributed using a directed spanning tree T i over V Streaming Topology T = { T 1 , . . . , T k } is set of these k distribution trees Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 5 / 21
Model of push-based P2P-Streaming systems (2) More definitions... Assumption: source has a maximum degree of C · k , for C ∈ N + Nodes receiving packets directly from s are called heads of T The successors succ i ( v ) of a node v ∈ V in T i ∈ T are all nodes of the maximal subtree of T i that is rooted in v Stripe 1 Stripe 2 s s C = 3 2 5 7 3 5 10 Heads H = { 2 , 3 , 5 , 7 , 10 } 1 3 6 8 8 2 4 7 9 11 11 succ 1 (2) = { 1 , 2 , 3 , 4 } 4 9 1 6 10 Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 6 / 21
What do we aim for? Goal Identify the class of all streaming topologies that are optimally 1 stable against node failures due to malicious DoS attacks. Provide rules for their efficient construction. 2 Design and implement distributed topology management 3 mechanisms realizing stable topologies in P2P-streaming systems. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 7 / 21
Attackers, damage and stability (1) Abstract attacker A map from T and x ∈ N to a set X ⊆ V \ { s } of x failing peers. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 8 / 21
Attackers, damage and stability (1) Abstract attacker A map from T and x ∈ N to a set X ⊆ V \ { s } of x failing peers. Why exclude source s? Source attack would always be optimal. Would disregard influence of distribution topology → seemingly equal stability of P2P and client-server approach Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 8 / 21
Attackers, damage and stability (1) Abstract attacker A map from T and x ∈ N to a set X ⊆ V \ { s } of x failing peers. Why exclude source s? Source attack would always be optimal. Would disregard influence of distribution topology → seemingly equal stability of P2P and client-server approach Damage function a T ( X ) The damage function a T : 2 V → R quantifies the damage incured on T by the failure of nodes. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 8 / 21
Attackers, damage and stability (2) In this work, we chose on the packet loss damage function, summing up the number of successors of nodes in X over all stripes. � � k � � � � a T ( X ) = succ i ( v ) � � � � i =1 � � v ∈ X Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 9 / 21
Attackers, damage and stability (2) In this work, we chose on the packet loss damage function, summing up the number of successors of nodes in X over all stripes. � � k � � � � a T ( X ) = succ i ( v ) � � � � i =1 � � v ∈ X Stripe 1 Stripe 2 Stripe 3 s s s 3 1 2 5 6 7 2 7 2 5 4 1 6 7 1 3 5 6 3 4 4 a T 1 ( X ) = 4 a T 2 ( X ) = 7 a T 3 ( X ) = 5 a T ( X ) = 16 Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 9 / 21
The complete class of optimally stable streaming topologies (1). The complete class of optimally stable streaming topologies is characterized by the damage incured by an optimal attacker. Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 10 / 21
The complete class of optimally stable streaming topologies (1). The complete class of optimally stable streaming topologies is characterized by the damage incured by an optimal attacker. For l = ( i div C ) and h = ( i mod C ), define �� n � + ( k − 2 l − 1) if h ≤ ( n mod C ) δ C , k C = � n i � + ( k − 2 l − 1) otherwise C δ C , k 1 δ C , k C +1 δ C , k 2 C +1 δ C , k Ck n mod C C Ck Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 10 / 21
The complete class of optimally stable streaming topologies (2). For l = ( i div C ) and h = ( i mod C ), define �� n � + ( k − 2 l − 1) if h ≤ ( n mod C ) δ C , k C = � n i � + ( k − 2 l − 1) otherwise C Optimally stable topologies [1] A topology T with parameters C , k , n is optimally stable if and only if a T ( O ( T , m )) = � m i =1 δ C , k for 1 ≤ m ≤ C · k . i [1] Brinkmeier et. al., ”Optimally DoS Resistant P2P Topologies for Live Multimedia Streaming”, IEEE Transactions on Parallel and Distributed Computing , 2009 Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 11 / 21
The bad news. Decision problem Decide whether any given streaming topology T is optimally stable. We have shown that this problem is co-NP-complete. Hence, without P=NP, it is not solvable in polynomial time . Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 12 / 21
The bad news. Decision problem Decide whether any given streaming topology T is optimally stable. We have shown that this problem is co-NP-complete. Hence, without P=NP, it is not solvable in polynomial time . All is not lost We can identify a large and easy-to-check subclass of optimally stable topologies! Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 12 / 21
Necessary properties of stable topologies (1) Stable topologies must follow a number of necessary rules. Not-Too-Many-Successors Rule � n Every peer has at most δ C , k � = + ( k − 1) successors. 1 C Stability Guarantees for Live-Streaming Topologies S. Grau ITI, TU Ilmenau, Germany Page 13 / 21
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