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Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies

Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies. Thomas Moscibroda Roger Wattenhofer Aaron Zollinger MOBIHOC 2006 Modified by Kyung-Joon Park CS 598JH Fall 2007, UIUC. Topology Control Meets SINR. Topology Control Meets SINR:

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Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies

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  1. Topology Control Meets SINR:The Scheduling Complexity of Arbitrary Topologies Thomas Moscibroda Roger Wattenhofer Aaron Zollinger MOBIHOC 2006 Modified by Kyung-Joon Park CS 598JH Fall 2007, UIUC

  2. Topology Control Meets SINR Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies Thomas Moscibroda, MOBIHOC 2006

  3. Topology Control Protocol Topology Control Meets SINR What is topology control ? Given node location, find a (static) communication graph with desirable properties Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies • Idea:Drop links if possible • Goal: Reduces energy and interference! But still stay connectedandsatisfies other properties: • Low node degree • Low static interference • Etc… • Static problem! No node should be interfered by too many other nodes. Thomas Moscibroda, MOBIHOC 2006

  4. Topology Control Meets SINR What decides SINR?  Physical scheduling = who send when Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies • A schedule to actually realize selected links (transmission requests), to successfully transmit message over them Received signal power from sender Power level of sender u Path-loss exponent Minimum signal-to-interference ratio Noise Distance between two nodes Received signal power from all other nodes (=interference) Thomas Moscibroda, MOBIHOC 2006

  5. Topology Control Meets SINR What is relation between topology control and physical scheduling? Topology Control Meets SINR: The Scheduling Complexity of Arbitrary Topologies • Which topologies can be scheduled efficiently? • How should requests/topologies be scheduled? Thomas Moscibroda, MOBIHOC 2006

  6. Outline • Graph-based protocol design vs. physical interference! • The scheduling complexity of wireless networks • Intuitive, but inefficient scheduling protocols • A note on the energy metric • Our efficient O(Iin·log2(n)) protocol • Topologies with low Iin • Symmetric versus asymmetric links • Conclusions Thomas Moscibroda, MOBIHOC 2006

  7. Relation Graph-based Topology vs. Physical Scheduling? Fundamenal question: What is the relationship between topology control and physical scheduling? structure of topology (set of comm. requests) difficulty of scheduling Thomas Moscibroda, MOBIHOC 2006

  8. Good topology or bad topology…? A wants to sent to B, C wants to send to D C D A B 4m 1m 2m Can A and C send simultaneously...? No, they cannot! D is inside A's transmission range! Interference causes a collision at D! it seems… Thomas Moscibroda, MOBIHOC 2006

  9. Simultaneous transmission is possible! Good topology or bad topology…? • Let =3, =3, and N=0.01μW • Set the transmission powers as follows PC= -15 dBm and PA= 1 dBm • SINR at B is: • SINR at D is: A wants to sent to B, C wants to send to D C D A B 4m 1m 2m Thomas Moscibroda, MOBIHOC 2006

  10. Scheduling in Wireless Networks Relationship between a topology and scheduling is not trivial! • Big discrepancy between graph-based and SINR-based models • Interference created by simultaneous senders cumulates • Power may not be chosen uniformly Not clear whether topology control helps in scheduling! We need a measure that captures how quickly a topology can be scheduled Thomas Moscibroda, MOBIHOC 2006

  11. Iin: Measuring a topology's interference [von Rickenbach et al., WMAN‘05] • Given a topology (or a set of communication requests) T • Iin is the maximum number of nodes by which a receiver can potentially be interfered. • Formally, • Node u may disturb all nodes closer than its farthest neighbor Draw a disk around each node with radius = longest outgoing link • Interferenceof node u = #nodes whose distance to u is at most the distance to their farthest neighbors #disks by which u is covered - 1 • Iin Interference of topology or set of requests T = maximum interference over all nodes Interference arises at the receiver! Coverage of Node u Interference 2 Thomas Moscibroda, MOBIHOC 2006

  12. Outline • Graph-based protocol design vs. physical interference! • The scheduling complexity of wireless networks • Intuitive, but inefficient scheduling protocols • A note on the energy metric • Our efficient O(Iin·log2(n)) protocol • Topologies with low Iin • Symmetric versus asymmetric links • Conclusions Thomas Moscibroda, MOBIHOC 2006

  13. The Scheduling Complexity of Wireless Networks • n nodes in 2D Euclidean plane(arbitrary, possibly worst-case position) • An arbitrary topology T (analogous: a set of communication requests) • Nodes can choose power levels • Message successfully received if SINR at receiver sufficient Scheduling Complexity S(T) The minimum number of time slots required until all links in T have been successfully scheduled at least once! Moscibroda, Wattenhofer, Infocom 2006 Scheduling Complexity of Strong Connectivity: S(T) ∈ O(log4n) General Topologies? What is known… Thomas Moscibroda, MOBIHOC 2006

  14. Scheduling Complexity – Example Consider topology T: 8 4 2 1 7 5 3 Time-SlotLinks: t1: 12, 45, 67 t2: 31, 54, 76 t3: 78, 35 t4: 84 6  Scheduling complexity of T is at most 4 ! Do good topologies have a small scheduling complexity ? Static-graph-based topology control SINR-based scheduling Thomas Moscibroda, MOBIHOC 2006

  15. Our Results In the paper we prove the following theorem: Theorem: Scheduling Complexity of any topology T with in-interference Iin is at most S(T) ∈O(Iin·log2n) This result hold in every (even worst-case) networks Theoretically, good static topologies can be scheduled eficiently  no fundamental scaling problem in scheduling This implies that topology control (reducing Iin) helps! But, achieving this result requires highly non-trivial power assignments and scheduling ! A centralized scheduler is devised, which achieves the bound Thomas Moscibroda, MOBIHOC 2006

  16. Bad Scheduling in SINR [Moscibroda, Wattenhofer, Infocom 2006] • Consider the exponential chain: Thomas Moscibroda, MOBIHOC 2006

  17. By a factor (n) slower! Bad Scheduling in SINR (importance of power assignment) [Moscibroda, Wattenhofer, Infocom 2006] • Consider the exponential chain: 1 25 2 26 27 28 29 210 22 23 24 • This topology has interference Iin = 1 • All links can be scheduled in O(1) time! • But, it can be shown that: • Any protocol with uniform power assignment has time (n) • Any protocol with power according to has time (n) Not trivial… Transmitting according to energy-metric implies slow scheduling! Energy-Metric ! Thomas Moscibroda, MOBIHOC 2006

  18. Our Protocol • How can we break the (n) barrier...? • Observation: Scheduling a set of links of roughly the same length is easy... • Partition the set of links in length-classes • Schedule each length-class independently one after the other... • The problem is...  there may be up to n different length-classes  We must schedule links of different lengths simultaneously! • How can we assign powers to nodes?  Making the transmission power dependent on the length of link is bad! • We must make the power assigned to simultaneous links dependent on their relative position of the length class! e.g. uniform and ~d examples before Thomas Moscibroda, MOBIHOC 2006

  19. Our Protocol – overview Each point is a node A box is a length class Node xj in a length class Si: 2i <= longest link of xj < 2i+1 • Partition the set of nodes into sets, according to their longest link • Consider nodes in sets (0 <= k<log(3n)): Sk, Slog(3n)+k, S2log(3n)+k , … , Sxlog(3n)+k • Schedule all links belonging to nodes in these sets. • Assign power levels (somewhat tricky part) • Schedule as many as possible nodes to transmit simultaneously nodeseither have almost same radii or their radii differ significantly Thomas Moscibroda, MOBIHOC 2006

  20. Our Protocol – Power Assignment • A node v in length-class  and a link of length d transmit roughly with a power of P(v) ~(3nb)·d • But now, short links disturb distant long links!!! • Therefore, we also need to carefully select transmitting nodes! Intuitively, nodes with small links must overpower their receivers! This would be assignment Ooops, now it gets complicated...! Thomas Moscibroda, MOBIHOC 2006

  21. Our Protocol – Scheduling Links • Short links are “overpowered” • create much more interference • this precludes simple geometric arguments! • In each time slot T, consider all nodes in decreasing order of longest link in Fk = SkU Slog(3n)+kU…U Sxlog(3n)+k • Add a node to ET if allowed() evaluates to true • T = T +1; Fk = Fk \ ET To bound the interference Please find details in the paper... Thomas Moscibroda, MOBIHOC 2006

  22. Our Protocol – Analysis • 1. The resulting schedule is feasible (Theorem 6.3) • Interference from other nodes are bounded • With power~(3nb)·d, SINR >  will be satisfied • 2. The number of slots required does not exceed O(Iin · log2(n)) (Theorem 6.9) • For each node, there are at most O(Iin · log(n)) blocking nodes • So for each k, at most O(Iin · log(n)) slots are requires • Totally, O(Iin · log(n)·log(3n)) = O(Iin · log2(n)) • Requires  >1 • Not a practical protocol • Complicated • Centralized information needed Please find detailsin the paper... Thomas Moscibroda, MOBIHOC 2006

  23. Outline • Graph-based protocol design vs. physical interference! • The scheduling complexity of wireless networks • Intuitive, but inefficient scheduling protocols • A note on the energy metric • Our efficient O(Iin · log2(n)) protocol • Topologies with low Iin • Symmetric versus asymmetric links • Conclusions Thomas Moscibroda, MOBIHOC 2006

  24. What is the value of Iin ? Theorem: Scheduling Complexity of a topology T with in-interference Iin is at most S(T) ∈O(Iin·log2n) All current MAC protocols Topology Iin our protocol uniform power energy-metric nearest neighbor ~5 S(T)∈O(log2n) S(T)∈(n) exponential chain 1 S(T)∈O(log2n) S(T)∈(n)(directed) strong connectivity - asymmetric links O(log n) S(T)∈O(log3n) S(T)∈(n) Improves the scheduling complexity of connectivity! Thomas Moscibroda, MOBIHOC 2006

  25. What is the value of Iin ? Theorem: Scheduling Complexity of a topology T with in-interference Iin is at most S(T) ∈O(Iin·log2n) All current MAC protocols Topology Iin our protocol uniform power energy-metric nearest neighbor ~5 S(T)∈O(log2n) S(T)∈(n) exponential chain 1 S(T)∈O(log2n) S(T)∈(n)(directed) strong connectivity - asymmetric links O(log n) S(T)∈O(log3n) S(T)∈(n) - symmetric links S(T)∈(n) Scheduling asymmetric vs. symmetric links! Thomas Moscibroda, MOBIHOC 2006

  26. Outline • Graph-based protocol design vs. physical interference! • The scheduling complexity of wireless networks • Intuitive, but inefficient scheduling protocols • A note on the energy metric • Our efficient O(Iin · log2(n)) protocol • Topologies with low Iin • Symmetric versus asymmetric links • Conclusions Thomas Moscibroda, MOBIHOC 2006

  27. Conclusion - Our Contributions 1) Improved “scheduling complexity of connectivity”  from O(log4n) [Moscibroda, Wattenhofer, Infocom 2006] to O(log3n) 2) Scheduling symmetric vs. asymmetric links in topologies  using symmetric links has numerous practical advantages (ACK, ..)  but, asymmetric topologies can be scheduled much faster! • Power assignment is crucial  uniform power assignment leads to extremely slow schedules!  “energy-metric” power assignment P»d, too! 4) Bridge gap between information theoretic world (SINR) and protocol design (graph-based, topology control)  fundamental justification for topology control Thomas Moscibroda, MOBIHOC 2006

  28. C A B Graph-based Protocol Design vs. SINR Scheduling? Fundamenal question: What is the relationship between topology control and physical scheduling? This paper! Graph-based topologies • Protocol designers use(various) graph models • e.g. Topology control protocols SINR Scheduling • Information theoreticians useSINR (physical) models • e.g. capacity of wireless networks Topology Control helps in scheduling! but, only if scheduling is done right! Thomas Moscibroda, MOBIHOC 2006

  29. Comments • Derived bound on scheduling complexity may be further tightened • Only upper bound is derived by carefully devising a centralized scheduler • If we find a better scheduler, the bound can be tightened • Where is link connectivity (transmission range) constraint? • Link connectivity imposes lower bound on transmit power • Energy-metric power assignment comes from this constraint • Power assignment by scheduler (not by topology control) may be problematic (In fact, proposed algorithm is joint power control and scheduling) Thomas Moscibroda, MOBIHOC 2006

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