Centralized communication in radio networks with strong interference - - PowerPoint PPT Presentation

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, ET ∪ EI)

(u, v) ∈ ET ⇐ ⇒ v ∈ T(u) (transmission edge) (u, v) ∈ EI ⇐ ⇒ v ∈ I(u) \ T(u) (interference edge) ET ∩ EI = ∅

transmission spanning subgraph G(ET) 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

• f 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 + log2 n) by Kowalski and Pelc (DC’07) lower bound Ω(log2 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)

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cík Radio communication with a strong interference

Difficulty of fast broadcasting

Gm = (Vm, ET ∪ EI) V(Gm) = {s, a1, . . . , am, b1, . . . , bm} ET(Gm) = {(s, ai), (ai, bi)|1 ≤ i ≤ m} EI(Gm) = {(ai, bj)|1 ≤ i = j ≤ m}

• ne 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 Gm 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

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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 : (TX, IX) = ({A}, {B}) Y : (TY, IY) = ({A, C}, {B, D, E}) F = {(T1, I1), (T2, I2), . . . , (Tm, Im)}

describes informed (transmission/interference) neighbors for each uninformed node Ti ∩ Ii = ∅ and Ti = ∅

S = {S1, S2, . . . , Sk}

schedule of transmissions for initially informed nodes Si 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 = {(T1, I1), (T2, I2), . . . , (Tm, Im)} to be a collection of set-pairs such that Ti ∩ Ii = ∅ and Ti = ∅, for all i = 1, . . . , m. Denote U(F) = m

i=1 Ti ∪ Ii. A family S = {S1, S2, . . . , Sk} of

subsets of U(F) is said to be selective for F if and only if for any (Ti, Ii) there is a set Sj such that the following holds |Ti ∩ Si| = 1 |Ii ∩ Si| = 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 (Ti, Ii) ∈ F it holds |Ii| ≤ r(F) · |Ti| ?

• F. Galˇ

cík Radio communication with a strong interference

Interference ad-hoc selective families (4)

Theorem Let F = {(T1, I1), (T2, I2), . . . , (Tm, Im)} to be a collection of set-pairs such that Ti ∩ Ii = ∅, Ti = ∅, and ∆min ≤ |Ti| + |Ii| ≤ ∆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

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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 = (VS ∪ VR, ET ∪ EI) - directed bipartite IRG construct undirected graph G′ = (VS, E′) such that (u, v) ∈ E(G′) ⇐ ⇒ ∃w ∈ VR, (u, w) ∈ ET ∧ (v, w) ∈ ET ∪ EI ∆(G′) < ∆2 greedy algorithm finds a proper vertex coloring of G′ with at most ∆(G′) + 1 ≤ ∆2 colors if nodes of VS with color i transmit in the i-th round then all nodes in VR become informed after at most ∆2 rounds

• F. Galˇ

cík Radio communication with a strong interference

Maximum degree as a parameter of the network (2)

All uninformed nodes in bipartite IRG can be informed in O(∆2) rounds using previous method O(∆ · log ∆ · log n) rounds using interference ad-hoc selective families

interference ratio of the graph is upper-bounded by ∆ better in the case when ∆ = Ω(log2 n)

at most 2 · ∆ − 1 = O(∆) rounds, if each node is incident with exactly one transmission edge

• F. Galˇ

cík Radio communication with a strong interference

Maximum degree as a parameter of the network (3)

Consider (non-interference) bipartite graph (VS ∪ VR, E) AS - algorithm informing nodes in VR by transmissions of VS AF - algorithm informing nodes in VR by transmissions of VS if deg(v) = 1, for all v ∈ VS ∪ VR

Ga ¸sieniec et al., PODC’05 Let G = (V, E) to be a reachability graph. There is an algorithm (schema) that produces a radio broadcast schedule of the length O(AF(n, ∆) · D + AS(n, ∆) · log n)

• F. Galˇ

cík Radio communication with a strong interference

Maximum degree as a parameter of the network (4)

Broadcasting in IRG Let G = (V, ET ∪ EI) to be an IRG with the maximum degree ∆. There is a deterministic polynomial time algorithm that for a given source node s produces a radio broadcast schedule of the length O(∆D + min{∆, log ∆ · log n} · ∆ log n) apply the schema from Ga ¸sieniec et al. in transmission subgraph of G utilize algorithms for bipartite IRG (taking into account the parameter ∆) as AS and AF algorithms in order to avoid impact of interference edges

• F. Galˇ

cík Radio communication with a strong interference

Conclusion

newly proposed model of radio network for applications where the standard or weaker interference model are not appropriate first step towards the study of

information dissemination in the proposed model impact of "unrestricted" interference

• pen problems

better parameters of IRG for expressing the presence of interference edges shorter broadcasting schedules for IRGs lower bounds of the broadcasting time in IRGs approximation algorithms unknown network topology/other communication tasks

• F. Galˇ

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Thank you for your attention

• F. Galˇ

cík Radio communication with a strong interference