Université du Québec en Outaouais, Canada 1/46
Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 2/46
Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 3/46
Radio network A radio network is modeled as a graph whose nodes are stations or sensors that can transmit and receive. A directed edge (uv) means that v can be reached by the transmitter of u. Nodes send messages in synchronous rounds. In every round every node acts either as a transmitter or as a receiver. A transmitter sends a message to all its out-neighbors. 4/46
Radio network A node gets a message in a given round, if and only if, it acts as a receiver and exactly one of its in-neighbors transmits in this round. If at least two in-neighbors of a receiving node u transmit simultaneously in a given round, none of the messages is received by u in this round. In this case it is said that a collision occurred at u. 5/46
Collision detection If nodes can distinguish collision from silence, we say that collision detection is available. 6/46
Geometric radio network The set of nodes (sensors, stations) is modeled as points in the plane. Every node u has a range r u depending on the power of its transmitter, and it can reach all nodes at distance at most r u from it. The collection of nodes equipped with ranges determines a directed graph on the set of nodes, called a geometric radio network (GRN), in which a directed edge (uv) exists if node v can be reached from u. If the power of all transmitters is the same then all ranges are equal and the corresponding GRN is symmetric. 7/46
Geometric radio network The set of nodes (sensors, stations) is modeled as points in the plane. Every node u has a range r u depending on the power of its transmitter, and it can reach all nodes at distance at most r u from it. The collection of nodes equipped with ranges determines a directed graph on the set of nodes, called a geometric radio network (GRN), in which a directed edge (uv) exists if node v can be reached from u. If the power of all transmitters is the same then all ranges are equal and the corresponding GRN is symmetric. 7/46
Geometric radio network The set of nodes (sensors, stations) is modeled as points in the plane. Every node u has a range r u depending on the power of its transmitter, and it can reach all nodes at distance at most r u from it. The collection of nodes equipped with ranges determines a directed graph on the set of nodes, called a geometric radio network (GRN), in which a directed edge (uv) exists if node v can be reached from u. If the power of all transmitters is the same then all ranges are equal and the corresponding GRN is symmetric. 7/46
Geometric radio network The set of nodes (sensors, stations) is modeled as points in the plane. Every node u has a range r u depending on the power of its transmitter, and it can reach all nodes at distance at most r u from it. The collection of nodes equipped with ranges determines a directed graph on the set of nodes, called a geometric radio network (GRN), in which a directed edge (uv) exists if node v can be reached from u. If the power of all transmitters is the same then all ranges are equal and the corresponding GRN is symmetric. 7/46
Applications The radio network model is applicable to wireless networks using a single frequency. The model of geometric radio networks is applicable to wireless networks where stations are located in a flat region without large obstacles. In such a terrain, the signal of a transmitter reaches receivers at the same distance in all directions, i.e., the set of potential receivers of a transmitter is a disc. 8/46
Applications The radio network model is applicable to wireless networks using a single frequency. The model of geometric radio networks is applicable to wireless networks where stations are located in a flat region without large obstacles. In such a terrain, the signal of a transmitter reaches receivers at the same distance in all directions, i.e., the set of potential receivers of a transmitter is a disc. 8/46
Applications The radio network model is applicable to wireless networks using a single frequency. The model of geometric radio networks is applicable to wireless networks where stations are located in a flat region without large obstacles. In such a terrain, the signal of a transmitter reaches receivers at the same distance in all directions, i.e., the set of potential receivers of a transmitter is a disc. 8/46
Broadcasting Broadcasting is one of the fundamental tasks in network communication. Broadcasting One node of the network, called the source, has to transmit a message to all other nodes. Remote nodes are informed via intermediate nodes, along directed paths in the network. In this talk we restrict attention to deterministic broadcasting algorithms. 9/46
Performance measure One of the basic performance measures of a broadcasting scheme is the total time, i.e., the number of rounds it uses to perform the task. Parameters: n number of nodes, D eccentricity of the source. 10/46
Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 11/46
Centralized broadcasting Broadcasting in networks whose nodes have complete knowledge of the topology is equivalent to centralized broadcasting in which all transmissions are scheduled in advance by a central monitor. 12/46
Time of centralized broadcasting Two problems: Existence of short randomized schemes Given a graph G with a source s, prove the existence of a randomized broadcasting scheme with short expected time. Efficient construction of short deterministic schemes Construct a polynomial deterministic algorithm that, given a graph G with a source s, outputs a short deterministic broadcasting scheme for this graph. 13/46
Time of centralized broadcasting Chlamtac, Weinstein (IEEE Trans, Commun. 1991): polynomially constructible deterministic O(D log 2 (n/d)) scheme Gaber, Mansour (SODA 1995): randomized scheme with expected time O(D + log 5 n) polynomially constructible deterministic O(D + log 6 n) scheme Elkin, Kortsarz (SODA 2005): randomized scheme with expected time O(D + log 4 n) polynomially constructible deterministic O(D + log 5 n) scheme 14/46
Time of centralized broadcasting Chlamtac, Weinstein (IEEE Trans, Commun. 1991): polynomially constructible deterministic O(D log 2 (n/d)) scheme Gaber, Mansour (SODA 1995): randomized scheme with expected time O(D + log 5 n) polynomially constructible deterministic O(D + log 6 n) scheme Elkin, Kortsarz (SODA 2005): randomized scheme with expected time O(D + log 4 n) polynomially constructible deterministic O(D + log 5 n) scheme 14/46
Time of centralized broadcasting Chlamtac, Weinstein (IEEE Trans, Commun. 1991): polynomially constructible deterministic O(D log 2 (n/d)) scheme Gaber, Mansour (SODA 1995): randomized scheme with expected time O(D + log 5 n) polynomially constructible deterministic O(D + log 6 n) scheme Elkin, Kortsarz (SODA 2005): randomized scheme with expected time O(D + log 4 n) polynomially constructible deterministic O(D + log 5 n) scheme 14/46
Time of centralized broadcasting continued Gasieniec, Peleg, Xin (PODC 2005): randomized scheme with expected time O(D + log 2 n) polynomially constructible deterministic O(D + log 3 n) scheme Kowalski, Pelc (Distr. Comp. 2007) polynomially constructible deterministic O(D + log 2 n) scheme optimal in view of the lower bound Ω(log 2 n) by Alon, Bar-Noy, Linial, Peleg (JCSS 1991). 15/46
Time of centralized broadcasting continued Gasieniec, Peleg, Xin (PODC 2005): randomized scheme with expected time O(D + log 2 n) polynomially constructible deterministic O(D + log 3 n) scheme Kowalski, Pelc (Distr. Comp. 2007) polynomially constructible deterministic O(D + log 2 n) scheme optimal in view of the lower bound Ω(log 2 n) by Alon, Bar-Noy, Linial, Peleg (JCSS 1991). 15/46
Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 16/46
Symmetric networks with known neighborhood Assumptions: Every node knows its own label and labels of all neighbors The network is symmetric No collision detection [Bar-Yehuda, Goldreich, Itai: JCSS 1992] claimed a linear lower bound in this model for deterministic broadcasting in networks of constant diameter. This claim is incorrect. 17/46
Symmetric networks with known neighborhood Assumptions: Every node knows its own label and labels of all neighbors The network is symmetric No collision detection [Bar-Yehuda, Goldreich, Itai: JCSS 1992] claimed a linear lower bound in this model for deterministic broadcasting in networks of constant diameter. This claim is incorrect. 17/46
Symmetric networks with known neighborhood [Kowalski, Pelc: FOCS 2002] Theorem. There exists a deterministic broadcasting algorithm working in sublinear time in all symmetric networks of diameter o(log log n). However, Theorem. Every deterministic broadcasting algorithm requires time Ω( 4 n) on some n-node network of diameter 4. 18/46
Symmetric networks with known neighborhood [Kowalski, Pelc: FOCS 2002] Theorem. There exists a deterministic broadcasting algorithm working in sublinear time in all symmetric networks of diameter o(log log n). However, Theorem. Every deterministic broadcasting algorithm requires time Ω( 4 n) on some n-node network of diameter 4. 18/46
Symmetric networks with known neighborhood This lower bound has been strengthened by Brito, Gafni, Vaya: STACS 2004 Theorem. Every deterministic broadcasting algorithm requires time Ω( n) on some n-node network of diameter 4. 19/46
Symmetric ad hoc networks Assumptions: Every node knows only its own label The network is symmetric No collision detection 20/46
Symmetric ad hoc networks Here we have tight bounds on deterministic broadcasting time. [Chlebus, Gasieniec, Gibbons, Pelc, Rytter: SODA 2000] Theorem. There exists a deterministic broadcasting algorithm working in time O(n) in all symmetric n-node networks. [Kowalski, Pelc: SIROCCO 2003] Theorem. Every deterministic broadcasting algorithm requires time Ω(n) on some n-node network of constant diameter. 21/46
Arbitrary ad hoc networks Assumptions: Every node knows only its own label The network is arbitrary No collision detection Notation: n number of nodes, D eccentricity of the source 22/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Deterministic broadcasting in arbitrary ad hoc networks folklore: O(n 2 ) Chlebus, Gasieniec, Gibbons, Pelc, Rytter (SODA 2000): O(n 11/6 ) De Marco, Pelc (IPL 2001): O(n 5/3 log 1/3 n) Chlebus, Gasieniec, Ostlin, Robson (ICALP 2000): O(n 3/2 ) Chrobak, Gasieniec, Rytter (FOCS 2000): O(n log 2 n) Kowalski, Pelc (STACS 2003): O(n log n log D) Czumaj, Rytter (FOCS 2003): O(n log 2 D) De Marco (SODA 2008): O(n log n log log n) 23/46
Lower bounds Bruschi, Del Pinto (Distr. Comp. 1997): Ω(D log n) Clementi, Monti, Silvestri (SODA 2001): Ω(n log D) 24/46
Outline Intro Known Ad hoc GRN 1 Introduction 2 Networks with known topology 3 Ad hoc networks 4 Geometric radio networks 25/46
Knowledge radius For a fixed real s 0, called the knowledge radius, it is assumed that each node knows the part of the network within the circle of radius s centered at it, i.e., it knows the positions, labels and ranges of all nodes at distance at most s. Problem [Dessmark, Pelc: SPAA 2001] How the size of the knowledge radius influences deterministic broadcasting time in GRN? 26/46
Notation Intro Known Ad hoc GRN n - number of nodes D - eccentricity of the source, i.e., the maximum length of all shortest paths in the graph from the source to all other nodes. s - knowledge radius R - finite set of ranges labels of nodes are integers from the set 1,..., M, where M O(n) 27/46
Knowledge of nodes Every node knows the set of ranges the upper bound M on the labels positions, labels and ranges of all nodes within the knowledge radius no collision detection capability 28/46
Knowledge of nodes Every node knows the set of ranges the upper bound M on the labels positions, labels and ranges of all nodes within the knowledge radius no collision detection capability 28/46
Large knowledge radius Theorem. The minimum time to perform broadcasting in an arbitrary GRN with source eccentricity D and knowledge radius s > max(r) (or with global knowledge of the network) is Θ(D). This should be contrasted with broadcasting in non-geometric radio networks: there are networks with bounded D requiring time Ω(log 2 n) to broadcast, with full knowledge of the network. 29/46
Large knowledge radius Theorem. The minimum time to perform broadcasting in an arbitrary GRN with source eccentricity D and knowledge radius s > max(r) (or with global knowledge of the network) is Θ(D). This should be contrasted with broadcasting in non-geometric radio networks: there are networks with bounded D requiring time Ω(log 2 n) to broadcast, with full knowledge of the network. 29/46
Knowledge radius zero Theorem. It is possible to broadcast in arbitrary n-node GRN with knowledge radius zero in time O(n). The above upper bound for GRN should be contrasted with the lower bound [Bruschi, Del Pinto 1997] showing that some graphs require broadcasting time Ω(n log n). These graphs are not GRN. 30/46
Knowledge radius zero Theorem. It is possible to broadcast in arbitrary n-node GRN with knowledge radius zero in time O(n). The above upper bound for GRN should be contrasted with the lower bound [Bruschi, Del Pinto 1997] showing that some graphs require broadcasting time Ω(n log n). These graphs are not GRN. 30/46
Knowledge radius zero The reason for a longer broadcasting time is really not the topology of the graph but the difference in knowledge available to nodes. Theorem. If each node knows only its label and range but not its position then there exists a family of GRN requiring broadcasting time Ω(n log n). 31/46
Knowledge radius zero The reason for a longer broadcasting time is really not the topology of the graph but the difference in knowledge available to nodes. Theorem. If each node knows only its label and range but not its position then there exists a family of GRN requiring broadcasting time Ω(n log n). 31/46
The model with collision detection Here we have tight bounds for symmetric networks. Theorem. In the model with collision detection and knowledge radius zero it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n). Theorem. For any broadcasting algorithm with collision detection and knowledge radius zero, there exist n-node symmetric GRN of diameter 2 for which this algorithm requires time Ω(log n). 32/46
The model with collision detection Here we have tight bounds for symmetric networks. Theorem. In the model with collision detection and knowledge radius zero it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n). Theorem. For any broadcasting algorithm with collision detection and knowledge radius zero, there exist n-node symmetric GRN of diameter 2 for which this algorithm requires time Ω(log n). 32/46
The model without collision detection For the model without collision detection it is possible to maintain complexity O(D + log n) of broadcasting. However, we need a stronger assumption concerning knowledge radius: it is no longer 0, but positive, although arbitrarily small. Theorem. In the model without collision detection, it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n), for any positive constant knowledge radius. 33/46
The model without collision detection For the model without collision detection it is possible to maintain complexity O(D + log n) of broadcasting. However, we need a stronger assumption concerning knowledge radius: it is no longer 0, but positive, although arbitrarily small. Theorem. In the model without collision detection, it is possible to broadcast in any n-node symmetric GRN of diameter D in time O(D + log n), for any positive constant knowledge radius. 33/46
Unit Disc Graphs (UDG) [Emek, Gasieniec, Kantor, Pelc, Peleg, Su: PODC 2007] All ranges equal 1 (UDG networks are symmetric) Nodes don t have labels but each node knows its own coordinates All nodes know the minimum Euclidean distance d between all nodes no collision detection 34/46
Parameters Intro Known Ad hoc GRN Broadcasting time depends on two parameters: diameter D (same order of magnitude as source eccentricity) the granularity g = 1/d of the network. 35/46
Two broadcasting models In the conditional wake up model, the nodes other than the source are initially idle and cannot transmit until they receive the source message for the first time (and subsequently wake up). In the spontaneous wake up model, all nodes are assumed to be awake when the source transmits for the first time, and may contribute to the broadcasting process by transmitting control messages even before they received the source message. 36/46
Two broadcasting models In the conditional wake up model, the nodes other than the source are initially idle and cannot transmit until they receive the source message for the first time (and subsequently wake up). In the spontaneous wake up model, all nodes are assumed to be awake when the source transmits for the first time, and may contribute to the broadcasting process by transmitting control messages even before they received the source message. 36/46
Results for the conditional wake up Theorem. There exists an algorithm that completes broadcast in time O(Dg) in any UDG radio network of diameter D and granularity g. Theorem. Any broadcasting algorithm for UDG radio networks in the conditional wake up model has running time Ω(D g) on some network of diameter D and granularity g. 37/46
Results for the conditional wake up Theorem. There exists an algorithm that completes broadcast in time O(Dg) in any UDG radio network of diameter D and granularity g. Theorem. Any broadcasting algorithm for UDG radio networks in the conditional wake up model has running time Ω(D g) on some network of diameter D and granularity g. 37/46
Results for the spontaneous wake up We show two broadcasting algorithms: one working in time O(D + g 2 ) and the other in time O(D log g). These algorithms are based on completely different ideas and, depending on parameter values, one or the other may be more efficient. The combined algorithm obtained by interleaving these two algorithms completes broadcast in time O (min {D + g 2, D log g}). This time turns out to be optimal. 38/46
Results for the spontaneous wake up We show two broadcasting algorithms: one working in time O(D + g 2 ) and the other in time O(D log g). These algorithms are based on completely different ideas and, depending on parameter values, one or the other may be more efficient. The combined algorithm obtained by interleaving these two algorithms completes broadcast in time O (min {D + g 2, D log g}). This time turns out to be optimal. 38/46
Results for the spontaneous wake up Theorem. Any broadcasting algorithm for UDG radio networks in the spontaneous wake up model has running time Ω (min {D + g 2, D log g}) on some network of diameter D and granularity g. 39/46
Separation of models Our results give a provable separation between the conditional and the spontaneous wake up models for broadcasting in UDG radio networks: for networks of small diameter (e.g., D bounded or polylogarithmic in g) the lower bound for the conditional wake up model is significantly larger than the upper bound for the spontaneous wake up model. 40/46
Separation of models Our results give a provable separation between the conditional and the spontaneous wake up models for broadcasting in UDG radio networks: for networks of small diameter (e.g., D bounded or polylogarithmic in g) the lower bound for the conditional wake up model is significantly larger than the upper bound for the spontaneous wake up model. 40/46
Closing the gap [Emek, Kantor, Peleg: PODC 2008] Theorem. Any broadcasting algorithm for UDG radio networks in the conditional wake up model has running time Ω(Dg) on some network of diameter D and granularity g. Hence the opimal broadcasting time in the conditional wake up model is Θ(Dg). 41/46
Missing and inaccurate information The assumptions that each node knows the density of the network knows exactly its own position in the plane. are unrealistic in many situations. Does removing them influence broadcasting time? 42/46
Missing and inaccurate information The assumptions that each node knows the density of the network knows exactly its own position in the plane. are unrealistic in many situations. Does removing them influence broadcasting time? 42/46
Missing and inaccurate information [Fusco, Pelc: DISC 2008] Assume that density is unknown and nodes perceive their position with some unknown error margin ɛ. This combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario... 43/46
Missing and inaccurate information [Fusco, Pelc: DISC 2008] Assume that density is unknown and nodes perceive their position with some unknown error margin ɛ. This combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario... 43/46
Missing and inaccurate information [Fusco, Pelc: DISC 2008] Assume that density is unknown and nodes perceive their position with some unknown error margin ɛ. This combination of missing and inaccurate information substantially changes the problem: the main new challenge becomes fast broadcasting in sparse networks (with constant density), when optimal time is O(D). Nevertheless, under our very weak scenario... 43/46
Missing and inaccurate information Theorem There exists a broadcasting algorithm that maintains optimal time O (min {D + g 2, D log g}) for all networks of unknown diameter D and granularity g (previously obtained with exact positions and known granularity), if each node perceives its position with error margin ɛ = αd, for any (unknown) constant α < 1, and the source knows that it is not the only 2 node in the network. 44/46
Missing and inaccurate information If the source does not have this information then the minimum time of an algorithm working correctly for all networks, and hence stopping if the source is alone, turns out to be Θ(D + g 2 ). Thus, somewhat surprisingly, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, if ɛ 1 d, then broadcasting is impossible. 2 45/46
Missing and inaccurate information If the source does not have this information then the minimum time of an algorithm working correctly for all networks, and hence stopping if the source is alone, turns out to be Θ(D + g 2 ). Thus, somewhat surprisingly, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, if ɛ 1 d, then broadcasting is impossible. 2 45/46
Missing and inaccurate information If the source does not have this information then the minimum time of an algorithm working correctly for all networks, and hence stopping if the source is alone, turns out to be Θ(D + g 2 ). Thus, somewhat surprisingly, the mere stopping requirement for the special case of the lonely source causes an exponential increase in broadcasting time, for networks of any density and any small diameter. Finally, if ɛ 1 d, then broadcasting is impossible. 2 45/46
Open problems 1 What is the optimal broadcasting and gossiping time in arbitrary ad hoc radio networks? 2 Is it possible to broadcast in time o(n) in arbitrary n-node GRN with eccentricity D sublinear in n, for knowledge radius zero? Note: in view of our results it is possible to broadcast in time O(n). 46/46