Centralized communication in radio networks with strong interference František Galˇ cík Institute of Computer Science P .J. Šafárik University, Faculty of Science Košice, Slovakia SIROCCO 2008 F. Galˇ cík Radio communication with a strong interference
What is a radio network ? a collection of receiver-transmitter devices - nodes nodes are autonomous communication via sending messages single shared communication frequency nodes work in globally synchronised time slots - rounds in each round, node makes a decision: acting as a receiver acting as a transmitter F. Galˇ cík Radio communication with a strong interference
Communication in radio network: standard model If a node transmits, then the signal goes to all nodes within its transmission range. F. Galˇ cík Radio communication with a strong interference
Communication in radio network: standard model If a node listen, then it receives a message if and only if it is in the transmission range of exactly one transmitting node. F. Galˇ cík Radio communication with a strong interference
Communication in radio network: standard model If a node listen and it is in the range of more than one transmitting node, then a collision occurs and no message is received. F. Galˇ cík Radio communication with a strong interference
Graph model and reachability graphs T ( v ) transmission range of the node v set of nodes that can receive a message transmitted by v radio network can be modelled by a directed reachability graph G : ( u , v ) ∈ E ( G ) ⇐ ⇒ v ∈ T ( u ) considered parameters of radio network: number of nodes n diameter of the reachability graph D maximum degree of the reachability graph ∆ G is assumed to be strongly connected undirected reachability graph - if transmission power of all nodes is the same F. Galˇ cík Radio communication with a strong interference
Graph model and reachability graphs T ( v ) transmission range of the node v set of nodes that can receive a message transmitted by v radio network can be modelled by a directed reachability graph G : ( u , v ) ∈ E ( G ) ⇐ ⇒ v ∈ T ( u ) considered parameters of radio network: number of nodes n diameter of the reachability graph D maximum degree of the reachability graph ∆ G is assumed to be strongly connected undirected reachability graph - if transmission power of all nodes is the same For some real-world settings this model is not appropriate F. Galˇ cík Radio communication with a strong interference
Extended interference in radio network Transmitted signal can reach a larger area, where it is too weak to be decoded as a message, but it is still strong enough . . . F. Galˇ cík Radio communication with a strong interference
Extended interference in radio network In the node D , the transmitted signal from A is still strong enough to cause an interference with the signal from C . F. Galˇ cík Radio communication with a strong interference
Interference reachability graph I ( v ) interference range of the node v set of nodes where a transmission of v causes an interference with other simultaneously incoming transmissions T ( v ) ⊆ I ( v ) (for standard model T ( v ) = I ( v ) ) such a radio network can be modelled by an interference reachability graph G = ( V , E T ∪ E I ) ⇒ v ∈ T ( u ) ( transmission edge ) ( u , v ) ∈ E T ⇐ ⇒ v ∈ I ( u ) \ T ( u ) ( interference edge ) ( u , v ) ∈ E I ⇐ E T ∩ E I = ∅ transmission spanning subgraph G ( E T ) is assumed to be strongly connected weaker variant of the model considered by Bermond et al. (PPL ’06) in the context of gathering problem F. Galˇ cík Radio communication with a strong interference
Our communication setting broadcasting information dissemination problem the goal is to distribute a message from one distinguished node, source, to all other network nodes centralized communication each node has a labelled copy of an underlying interference reachability graph can be considered as a process controlled by a central controller construction of polynomial time algorithm that produces efficient schedule of transmissions main efficiency measure is time (number of rounds, length of schedule) F. Galˇ cík Radio communication with a strong interference
Known results for the standard graph model broadcasting in general graphs: upper bound O ( D + log 2 n ) by Kowalski and Pelc (DC’07) lower bound Ω( log 2 n ) by Alon et al. (JCSS’91) broadcasting in special settings O ( D + ∆ . log n ) by Ga ¸sieniec at al. (PODC’05) 3 . D for planar graphs by Ga ¸sieniec et al. (PODC’05) F. Galˇ cík Radio communication with a strong interference
Difficulty of fast broadcasting G m = ( V m , E T ∪ E I ) V ( G m ) = { s , a 1 , . . . , a m , b 1 , . . . , b m } E T ( G m ) = { ( s , a i ) , ( a i , b i ) | 1 ≤ i ≤ m } E I ( G m ) = { ( a i , b j ) | 1 ≤ i � = j ≤ m } one transmission of the source s informs all a -nodes exactly one a -node can transmit in a round, otherwise no message is received by a b -node at least m + 1 = Ω( n ) rounds are necessary to complete broadcasting (in the graph G m having the constant diameter) F. Galˇ cík Radio communication with a strong interference
In the following ... another parameters of radio network have to be considered maximum over ratios of incident interference edges to incident transmission edges of a node maximum degree in an underlying interference reachability graph (IRG) bipartite IRG as key element of the layer-by-layer information dissemination approach senders - set of informed nodes with an uninformed neighbor receivers - set of uninformed nodes with an informed neighbor all senders have the same message schedule of transmissions of senders that informs receivers as soon as possible F. Galˇ cík Radio communication with a strong interference
Interference ad-hoc selective families (1) informed nodes: A, B, C, D, E uninformed nodes: X, Y X : ( T X , I X ) = ( { A } , { B } ) Y : ( T Y , I Y ) = ( { A , C } , { B , D , E } ) F = { ( T 1 , I 1 ) , ( T 2 , I 2 ) , . . . , ( T m , I m ) } describes informed (transmission/interference) neighbors for each uninformed node T i ∩ I i = ∅ and T i � = ∅ S = { S 1 , S 2 , . . . , S k } schedule of transmissions for initially informed nodes S i is the set of informed nodes that transmit in the i -th round goal is to construct short schedule of transmissions (i.e. minimize k ) after that all initially uninformed nodes become informed F. Galˇ cík Radio communication with a strong interference
Interference ad-hoc selective families (2) Definition Let F = { ( T 1 , I 1 ) , ( T 2 , I 2 ) , . . . , ( T m , I m ) } to be a collection of set-pairs such that T i ∩ I i = ∅ and T i � = ∅ , for all i = 1 , . . . , m . Denote U ( F ) = � m i = 1 T i ∪ I i . A family S = { S 1 , S 2 , . . . , S k } of subsets of U ( F ) is said to be selective for F if and only if for any ( T i , I i ) there is a set S j such that the following holds | T i ∩ S i | = 1 | I i ∩ S i | = 0. there are instances of F , |F| = | U ( F ) | = n that for each S it holds |S| ≥ n (nothing better than trivial construction works) F. Galˇ cík Radio communication with a strong interference
Interference ad-hoc selective families (3) Question How small S can be constructed if there is a constant r ( F ) ( interference ratio ) such that for all ( T i , I i ) ∈ F it holds | I i | ≤ r ( F ) · | T i | ? F. Galˇ cík Radio communication with a strong interference
Interference ad-hoc selective families (4) Theorem Let F = { ( T 1 , I 1 ) , ( T 2 , I 2 ) , . . . , ( T m , I m ) } to be a collection of set-pairs such that T i ∩ I i = ∅ , T i � = ∅ , and ∆ min ≤ | T i | + | I i | ≤ ∆ max , for all i = 1 , . . . , m . There is a deterministic polynomial-time algorithm that produces an interference ad-hoc selective family S of the size O (( 1 + r ( F )) · (( 1 + log (∆ max / ∆ min ))) · log |F| ) extension of the construction given by Clementi at al. [RANDOM’01] existence of interference ad-hoc selective families with an appropriate length is shown by probabilistic argument explicit construction by de-randomization method of conditional probabilities F. Galˇ cík Radio communication with a strong interference
Maximum degree as a parameter of the network (1) consider the maximum degree ∆ (over all network nodes) degree of a node is the sum of incident transmission and interference in-edges it is possible to inform all uninformed nodes (one of partitions) in bipartite IRG in O (∆ 2 ) rounds input G = ( V S ∪ V R , E T ∪ E I ) - directed bipartite IRG construct undirected graph G ′ = ( V S , E ′ ) such that ( u , v ) ∈ E ( G ′ ) ⇐ ⇒ ∃ w ∈ V R , ( u , w ) ∈ E T ∧ ( v , w ) ∈ E T ∪ E I ∆( G ′ ) < ∆ 2 greedy algorithm finds a proper vertex coloring of G ′ with at most ∆( G ′ ) + 1 ≤ ∆ 2 colors if nodes of V S with color i transmit in the i -th round then all nodes in V R become informed after at most ∆ 2 rounds F. Galˇ cík Radio communication with a strong interference
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