1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Algorithms for Radio Networks Winter Term 2005/ Oct nd Lecture Christian Schindelhauer

Search Algorithms, WS 2004/05 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Theory of Wireless Routing

Search Algorithms, WS 2004/05 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer (Omni-) Directed Antennae classical radio communication for sending and receiving omni-directed communication directed communication

Search Algorithms, WS 2004/05 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Problems for the MAC-Protocol  Hidden Terminal Problem  Exposed Terminal Problem  Asymmetry (var. transmission power) ABC ABCD D AB C

Search Algorithms, WS 2004/05 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A Simple Physical Network Model  Homogenous Network of –n radio stations s 1,..,s n on the plane  Radio transmission –One frequency –Adjustable transmission range Maximum range > maximum distance of radio stations Inside the transmission area of sender: clear signal or radio interference Outside: no signal –Packets of unit length

Search Algorithms, WS 2004/05 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer The Routing Problem  Given: –n points in the plane, V=(v 1,..,v n ) representing mobile nodes of a mobile ad hoc network –the complete undirected graph G = (V,E) as possible communication network representing a MANET where every connection can be established  Routing problem: –f : V  V  N, where f(u,v) packets have to be sent from u to v, for al u,v  V –Find a path for each packet of this routing problem in the complete graph –Let  The union of all path systems is called the Link Network or Communication Network

Search Algorithms, WS 2004/05 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Link network Path system l (e) = load of edge e = # paths(messages) through this link Load of an Edge  The load of an edge e is defined as the number of message paths going through e –we add the number of messages in both directions

Search Algorithms, WS 2004/05 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer transmitted message acknowledgment interference Radio Interferences  Links (edges) interfere, if simultaneous transmission is not possible because radio signals interfere  Links are symmetric (because of acknowledgments)  Interference may be asymmetric

Search Algorithms, WS 2004/05 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Formal Definition of Interference  Let D r (u) the disk of radius u with center u in the plane  Define for an edge e={u,v} D(e) = D r (u)  D r (v)  The set of edges interfering with an edge e = {u,v} of a communication network N is defined as:  The Interference Number of an edge is given by |Int(e)|  The Interference Number of the Network is max{|Int(e} | e  E}

Search Algorithms, WS 2004/05 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Load: Congestion: [MSVG02] interfering edges (depends on transmission model) Load + Interferences = (Radio) Congestion

Search Algorithms, WS 2004/05 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Formal Definition of Congestion  The Congestion of an edge e is defined as:  The Congestion of the path system P is defined as  The Dilation D(P) of a path system is the length of the longest path.

Search Algorithms, WS 2004/05 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Energy  The energy for transmission of a message can be modeled by a power over the distance d between sender and transceiver  Two energy models: –Unit energy accounts only the energy for upholding an edge Idea: messages can be aggregated and sent as one packet –Flow Energy Model: every message is counted separately

Search Algorithms, WS 2004/05 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Three Parts of the Routing Problem  Path Selection –select a path system P for the routing problem  Interference handling: –Design a strategy that can handle the transmission problem of a packet along a link  Packet switching –Decide when and in which order packets are sent along a link

Search Algorithms, WS 2004/05 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A Lower Bound for the Routing Time  A routing schedule is a timeline which describes for each message when it is passed along its path in the path system. –A routing schedule is valid if no interferences occur  The routing time is the number of steps of a routing schedule.  The optimal routing time for a given demand is the number of steps of the minimal valid routing schedule.  Theorem 1

Search Algorithms, WS 2004/05 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof  1. Case: T≥D –Since D is the length of the longest path for a message, the minimal time for sending a message is at least D  2. Case: T≥C/(2 c d ) –Consider the edge e with maximum congestion C –Partition the two-dimensional space into regions R1,..., R12. –Every edge interfering with e=(u,v) is in at least one of these regions and interferes with u or v. –Let Rx be the region with the largest number of such inter- fering messages. –Then this number is at least C/12

Search Algorithms, WS 2004/05 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof continued  If C/12 messages interfere with one node in region Rx –then there is an ordering of these message according to the distance to the node u resp v –such that the node closest to u is disturbed by all the other messages.  So, all messages in Rx have to be transmitted sequentially  Therefore, the overall routing time is at least C/12.  The case for three dimensions can be extended if one knows how to partition the globe into 20 cones with maximum angle of 60º (which is not trivial) –see Hardin, Sloane, Smith, Spherical Codes.

Search Algorithms, WS 2004/05 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer A short preview to MAC  The Problem of Medium Access Protocols is to decide when to send a message over the radio channel.  If the congestion of an edge is known one can use the following simple probabilistic protocol:  Activate link e with probability  (e) where  Lemma

Search Algorithms, WS 2004/05 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Proof  Lemma Proof:

Search Algorithms, WS 2004/05 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer An Upper Bound for Routing  This Lemma can be used to prove the following Theorem  Theorem  Proof omitted here.

20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Christian Schindelhauer Thanks for your attention End of 2nd lecture Next lecture:Mo 02 Nov 2005, 4pm, F1.110 Next exercise class: Mo 26 Oct 2005, 1.15 pm, F2.211 or Tu 01 Nov 2005, 1.15 pm, F1.110