Deterministic Oblivious Local Broadcast in the SINR Model Tomasz - PowerPoint PPT Presentation

Deterministic Oblivious Local Broadcast in the SINR Model Tomasz Jurdzi´ nski, Michał Ró˙ za´ nski Institute of Computer Science University of Wrocław, Poland FCT 2017 Bordeaux, France FCT 2017 1 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 2 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 3 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 4 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 5 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 6 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 7 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 8 / 29

b b b b Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. FCT 2017 9 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. FCT 2017 10 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. Algorithm is executed in a distributed manner by all the nodes. FCT 2017 10 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. Algorithm is executed in a distributed manner by all the nodes. All stations start the execution of the algorithm at the same time. FCT 2017 10 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. Algorithm is executed in a distributed manner by all the nodes. All stations start the execution of the algorithm at the same time. Messages are exchanged in rounds, according to the SINR constraints. FCT 2017 10 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. Algorithm is executed in a distributed manner by all the nodes. All stations start the execution of the algorithm at the same time. Messages are exchanged in rounds, according to the SINR constraints. Each station has a unique identifier from the set { 1 , ..., N } . FCT 2017 10 / 29

Local Broadcast Definition A graph (network) is the input to the problem. Each node (station) has a message that has to be delivered to all the neighbors in the graph. Environment Nodes of the network are located on a plane. Algorithm is executed in a distributed manner by all the nodes. All stations start the execution of the algorithm at the same time. Messages are exchanged in rounds, according to the SINR constraints. Each station has a unique identifier from the set { 1 , ..., N } . The algorithm is oblivious – all actions of a node are predetermined by its identifier. FCT 2017 10 / 29

The model: signal power v u d ( u, v ) P P ( d ( u,v )) α sender: v receiver: u Power of signal from v received by u P v P ( v , u ) = d ( u , v ) α α > 2 – path loss exponent (environment dependent) P v – transmission power of station v , we assume that every station has the same power P FCT 2017 11 / 29

Signal to Interference and Noise Ratio (SINR) Interference � I T ( u ) = P ( w , u ) (1) w ∈T FCT 2017 12 / 29

Signal to Interference and Noise Ratio (SINR) Interference � I T ( u ) = P ( w , u ) (1) w ∈T SINR P ( v , u ) SINR ( v , u , T ) = (2) N + I T \{ v } ( u ) T – set of transmitters FCT 2017 12 / 29

Model: signal reception SINR, signal reception P ( v , u ) SINR ( v , u , T ) = N + I T \{ v } ( u ) ≥ β (3) Parameters Path Loss Exponent α > 2 Background Noise N > 0 Threshold β ≥ 1 FCT 2017 13 / 29

SINR example w 1 d ( u, w 1 ) v u d ( u, v ) d ( u, w 2 ) T = { v, w 1 , w 2 } w 2 Node u receives a message from v iff P ( v , u ) SINR ( v , u , { w 1 , w 2 } ) = N + P ( w 1 , u ) + P ( w 2 , u ) ≥ β. FCT 2017 14 / 29

SINR Diagrams Figure: Reception zones Blue dots - transmitting stations Yellow area - region of audibility Green area - too noisy to decode anything FCT 2017 15 / 29

The model: network topology Communication graph G = ( V , E ) V - the set of stations r - transmission range ( u , v ) ∈ E ⇔ d ( u , v ) ≤ ( 1 − ε ) r ε - sensitivity parameter FCT 2017 16 / 29

The model: network topology Communication graph G = ( V , E ) V - the set of stations r - transmission range ( u , v ) ∈ E ⇔ d ( u , v ) ≤ ( 1 − ε ) r ε - sensitivity parameter Without loss of generality we assume that r = 1. FCT 2017 16 / 29

The model: network topology Communication graph G = ( V , E ) V - the set of stations r - transmission range ( u , v ) ∈ E ⇔ d ( u , v ) ≤ ( 1 − ε ) r ε - sensitivity parameter Without loss of generality we assume that r = 1. Complexity parameters ∆ - density of the network = maximal number of nodes per unit disc n - number of stations FCT 2017 16 / 29

Local broadcast Previous results Randomized algorithm O (∆ log N ) [Goussevskaia, Moscibroda, Wattenhofer ’08] Randomized algorithm with the feedback mechanism O (∆ + log N log log N ) [Barenboim, Peleg ’15] Randomized algorithm without the knowledge of ∆ O (∆ log N + log 2 N ) [Yu, Hua, Wang, Lao ’12] Our results (deterministic) Oblivious algorithm – O (∆ 2 + 2 / ( α − 2 ) log N ) Semi-adaptive algorithm with the feedback mechanism – O (∆ log N ) FCT 2017 17 / 29

Oblivious algorithms Oblivious algorithm An oblivious algorithm A ∆ solving a local broadcast in m rounds in networks of density ∆ can be represented by a sequence ( R 1 , ..., R m ) , where R i ⊆ [ N ] is the set of nodes transmitting in round i . FCT 2017 18 / 29

First approach Observation 1 Let v be any node. There exists a constant I max such that, if maximal interference in B ( v , 1 − ε ) is less than I max then, if v transmits, all neighbors of v hear its message. FCT 2017 19 / 29

First approach Observation 1 Let v be any node. There exists a constant I max such that, if maximal interference in B ( v , 1 − ε ) is less than I max then, if v transmits, all neighbors of v hear its message. In other words, if interference in any point of B ( v , 1 − ε ) is at most I max , then for any neighbor u of v we have: SINR ( v , u ) = P ( v , u ) N + I ( v ) ≥ β, where � I ( v ) = P ( w , u ) ≤ I max . w − transmitting w � = v FCT 2017 19 / 29

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � First approach Observation 2 For all networks of density ∆ there exists d ∆ ∈ R such that if no node transmits in B ( v , d ∆ ) (apart from v ), then all neighbors of v hear its message. d Δ 1 − ε � v FCT 2017 20 / 29

First approach Observation 2 For all networks of density ∆ there exists d ∆ ∈ R such that if no node transmits in B ( v , d ∆ ) (apart from v ), then all neighbors of v hear its message. There are O ( i ∆) nodes within distance [ i , i + 1 ] from v . FCT 2017 21 / 29

First approach Observation 2 For all networks of density ∆ there exists d ∆ ∈ R such that if no node transmits in B ( v , d ∆ ) (apart from v ), then all neighbors of v hear its message. There are O ( i ∆) nodes within distance [ i , i + 1 ] from v . The interference from them, assuming that all are transmitting, is at most O ( i ∆ · P i α ) . FCT 2017 21 / 29