Broadcasting Protocol for an Amorphous Computer Lukáš Petrů MFF UK, Prague Jiří Wiedermann ICS AS CR

Outline Motivation Model of an Amorphous computer Protocol Send Protocol Broadcast Conclusion

Device Miniaturization Technology allows producing very small energy efficient devices (MEMS) They can be built in large numbers Can communicate via wireless radio → Wireless sensor networks → Amorphous computers

Wireless Sensor Network A large number of devices is randomly distributed in an area of interest Each device is self-contained employing sensors, CPU, memory, controller, wireless transceiver and power source Sensors measuring light intensity, temperature, humidity, …, or intrusion detectors, etc. Data from all devices on the network are gathered in base station

Amorphous Computer (pict.) A bag containing a large number of computing elements

Amorphous Computer

Amorphous Computer (AC) Idea of a computing machine built from a large number of identical parts Differences from wireless networks: – only CPU with a severely limited memory – Random topology – All devices are identical (no id numbers) – No signal collision detection

Model of an AC No synchronization No collision detection (no message is received in case of collision) Special node IO-port No addresses

Node Registers Size O(log N) bits Control unit … Random number Input Output

Model of an AC Amorphous computer A=(N, P, A, r, T) N – number of nodes (=processors); each node is a RAM with registers of size O(log N) bits; has random number generator P, A – a process P randomly distributes nodes in a planar area A r – radio range; nodes within distance r are neighbours T – duration of one radio transmission

Random Topology When the node topology is random, what are the consequences on the communication possibility? Is communication possible? Under what conditions?

Random Topology Low density

Random Topology Medium density

Random Topology High density

Assumptions Assume that network connectivity graph contains one large component – Some nodes may not be part of the component N = size of the graph component Q = maximum neighbourhood size D = diameter of the graph component

Sending to Neighbours

Protocol Send An algorithm that determines how a node transmits a message to its neighbours Operates in an uncoordinated network of undistinguishable nodes that are not synchronized Assumptions: given Q – An upper bound on the number of node’s neighbours; given ε – the maximum allowed probability of failed transmission is known

Protocol Send Let p = 1/(Q+1); k = O(Q · log(1/ε)) procedure Send(m : message, p : probability) { For i := 1 to k do { Wait for time 2T; With probability p do Send message m; }

Protocol Send Theorem: If all nodes of the network use algorithm Send, then the probability that a message fails to be delivered from a sender to any of its neighbours is at most ε. The protocol works in time O( Q log (1/ε) ).

Broadcasting

Protocol Broadcast is used to deliver the same message to all nodes in the network A simple protocol for node-to-node communication when no routing information is available Assumptions: given N – the network size; Q – node neighbourhood size upper bound; ε – the maximum allowed probability of broadcast algorithm error

Protocol Broadcast procedure Broadcast(N : integer) { var m, m_last : message Loop forever { receive(m); If m != m_last { Send(m, ε/N); m_last = m; }

Protocol Broadcast Theorem: On a network of diameter D, algorithm Broadcast will run in time O(D · Q · log (N/ε)). The probability of algorithm failure is at most ε>0.

Conclusion We have … defined a formal model of AC developed a randomized broadcasting algorithm derived its time complexity

Future work Describe simulation of other theoretical models (Turing machine, RAM) Consider moving nodes – flying amorphous computer