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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems. Spring 2009 Prof. Jennifer Welch. Lecture 5. Topics: Multiple Access Channels Sources: Gallager paper Komlos & Greenberg paper MIT 6.885 Fall 2008 slides. S. S. R. S.

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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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  1. CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

  2. Lecture 5 • Topics: • Multiple Access Channels • Sources: • Gallager paper • Komlos & Greenberg paper • MIT 6.885 Fall 2008 slides Discrete Algs for Mobile Wireless Sys

  3. S S R S Gallager, A Perspective on Multi-access Channels, 1985. • Classical paper for multi-access channels. • Discusses: • Coding techniques and limitations on their achievable reliability. • Collision resolution • Simpler setting than general mobile ad hoc networks: • Static, several senders and one receiver. • Problems: Noise, interference. • Messages arrive at senders at random times. • Can be used to model: • Uplink for satellite. • Traffic to a central node in a telephone network. • Traffic to one receiver in a fully-connected wireless radio network. • Focus on collision resolution, emphasizing the modeling assumptions.

  4. S S R S Gallager’s Assumptions • Assumes messages arrive from external sources randomly, sometimes in bursts. • Assumes (typical for theory papers): • If exactly one sender transmits at some time (slot), the message succeeds. • If more than one transmits, a collision happens and communication fails. • This oversimplifies: • Collisions aren't the only cause of communication failure: this ignores noise and other aspects of real communication. • Sometimes colliding messages can be recovered.

  5. S S R S Gallager’s Formal Assumptions Assumption 1:Slotted system, with message transmission time = slot length. • Senders synchronized, on slot boundaries. • Relies on synchronized clocks. • Precludes considering strategies involving combinations of long and short packets. • Precludes carrier-sense strategies in which sender could start transmitting at any time (even in the middle of a slot).

  6. S S R S Gallager's Formal Assumptions Assumption 2: Receiver-side information: collision or perfect reception, (0,message,c). • If no one sends, receiver learns this (0). • If exactly one sends, receiver receives message with no errors (message). • If more than one sends, a collision occurs; the receiver gets no information about the messages sent, but learns that a collision happened (receives special collision indicator c). • In particular, receiver can distinguish idle slot from collision. • These assumptions hide noise and communication aspects of the model/problem. • Not completely realistic: • Not always clear how a receiver can distinguish idle slot from collision. • Receiver sometimes receives a packet when two senders transmit. • Can lead to unrealistic solutions.

  7. S S R Gallager's Formal Assumptions Assumption 3: Infinite set of senders. • Each new packet arrives at a completely new sender. • Avoids queueing issues. • Precludes use of TDM. Assumption 4. Poisson packet arrivals, rate . Assumption 5. Sender-side information: (0,1,c) immediate feedback. • Each sender learns immediately whether 0, 1, or >1 packets were sent. • Assumes transmitters are always listening for feedback, when they are transmitting and when when they are idle. • Many algorithms designed for this assumption can be extended to the case where feedback is delayed.

  8. Nonadaptive Conflict Resolution • [Komlos & Greenberg, 1985] • Multiple-access channel: way for geographically dispersed computing entities to communicate • coaxial cable (Ethernet) • fiber optic • packet radio • satellite transmission Discrete Algs for Mobile Wireless Sys

  9. Mathematical Model • n stations (computing nodes) • synchronized steps (times when stations can transmit data) • if k > 0 stations transmit at same step: • if k = 1, then every station gets the data • if k > 1 (collision), then no station gets the data • at end of step, each station gets feedback 0, 1 or 2+, indicating how many stations tried to transmit => collision detection Discrete Algs for Mobile Wireless Sys

  10. General Approach • Schedule (re)transmissions so that each station that wants to transmit eventually gets a step where it is the sole transmitter • Algorithm identifies a query set at each step, a subset of the stations • At each step, a station transmits if • it is in the query set and • it was involved in the collision trying to be resolved but has not yet been successful Discrete Algs for Mobile Wireless Sys

  11. Categorizing Such Algorithms • Does k, the number of stations involved in the collision, need to be known (hard-coded into the algorithm)? • Does the query set at each step depend on the feedback from the previous set (adaptive) or not (nonadaptive)? • adaptive: each station must monitor feedback at each step • nonadaptive: each station only needs to monitor feedback at steps when it transmits, to see if it can "drop out" Discrete Algs for Mobile Wireless Sys

  12. Previous Work • Tree algorithm [2,3,11,17] • deterministic • resolves conflicts among k nodes in (k + k log(n/k)) steps • does not require knowledge of k • is adaptive • k steps are obviously necessary • If k is (n), then bound is (k) = (n) • If k is (1), then bound is (log n) Discrete Algs for Mobile Wireless Sys

  13. Previous Work • [10] gave a lower bound on worst-case time for any deterministic conflict resolution scheme of (k (log n)/(log k)) • Shows that the tree algorithm is approximately time-optimal • Can we do as well with a nonadaptive algorithm? Discrete Algs for Mobile Wireless Sys

  14. This Paper • Gives a nonadaptive algorithm with time  (k + k log(n/k)) • Requires k be known (hard-coded into algorithm) • Non-constructive result: • prove that such an algorithm must exist • but do not explicitly describe the algorithm Discrete Algs for Mobile Wireless Sys

  15. Nonconstructive Results • A drawback from a practical perspective • Still of some interest: • proves such an algorithm exists, so it is not pointless to keep searching for an explicit one • there may be ways to convert non-constructive proofs into constructive ones • ideas of proof may be helpful in constructing a randomized algorithm that has the good running time with high probability Discrete Algs for Mobile Wireless Sys

  16. Adaptiveness and Knowledge of k • Suppose you don't know exactly how many nodes are contending, but you know an upper bound on this number • k' : number actually contending • k : upper bound on number contending • Can use any nonadaptive algorithm for fixed k to construct an adaptive algorithm for (unknown) k'. • If k' << k, this can be useful. Discrete Algs for Mobile Wireless Sys

  17. Adaptiveness and Knowledge of k • Algorithm: • Run fixed-k algorithm with k = 2 • Finish with a step in which the query set is all the stations. • If all the conflicting stations succeeded, then there are no transmissions at the last step and we are done • Otherwise, run the fixed-k algorithm with k = 4 (doubling k) • Continue at most log n times (until k = n) • Adaptive because stations must listen at the end of each "subroutine call" to see if they need to continue with next value of k • Running time is asymptotically same as subroutine's. Discrete Algs for Mobile Wireless Sys

  18. The Challenge • Goal is a list of queries (sets of stations allowed to transmit at each step) that isolates every conflicting station (there is a step at which it is the only one to transmit) • Difficulties: • don't know in advance which subset of stations want to transmit; must handle all possibilities • Try to be time-efficient, so that if number contending is small, then number of steps to isolate all is small Discrete Algs for Mobile Wireless Sys

  19. Overview of Nonconstructive Proof • consider a list of k/2 queries chosen randomly • prove that with high probability the list isolates at least a constant fraction of any input of size k • use this result to prove that certain lists of desired length must exist • use such lists to construct the desired list Discrete Algs for Mobile Wireless Sys

  20. Key Notation • Q1, Q2, Qt is a list of queries (subsets of the n stations) • I0 = I • original set of colliding stations • I1 = I0 - Q1 if I0 and Q1 intersect in exactly one id; I1 = I0 otherwise • set of stations still contending after contending stations in Q1 transmit simultaneously • I2 = I1 - Q2 if I1 and Q2 intersect in exactly one id; I2 = I1 otherwise • set of stations still contending after contending stations in Q2 transmit simultaneously • Etc. Discrete Algs for Mobile Wireless Sys

  21. Key Notation • A list of queries is (,k,n)-universal if for all inputs of size k, the list isolates at least *k of the colliding stations • We want a list with  = 1, isolates all the stations • Proof will use as building blocks lists with smaller values of  • Assume k divides n Discrete Algs for Mobile Wireless Sys

  22. Random Queries • Let Q1, …, Qp be a list of queries where • p is approx k/2 • each query is of size n/k • each query is chosen uniformly at random from the C(n,n/k) possibilities • Lemma 1: Each Qj isolates one member of the input with probability > 1/2e2, no matter the result of previous queries. Discrete Algs for Mobile Wireless Sys

  23. Proof of Lemma 1 • Suppose k = 1. • Then we have one query, of size n/1 = n. • Since only one station is contending, this query isolates it. • Suppose k = n. • Then we have n/2 queries, each of size n/n = 1, so each query consists of one station. • At least half the stations are still contending at each query. • So probability that the station chosen in Qj is still contending is at least 1/2 > 1/2e2. Discrete Algs for Mobile Wireless Sys

  24. Proof of Lemma 1 • Intermediate values of k: Pick some Qj. Let x = |Ij-1|, number of colliding stations still to be resolved. • worst case: x = k • best case: This is last query and all the previous ones isolated a station: x = k - (p-1) • What is probability that Qj isolates a member of Ij-1 (stations that are still contending)? Discrete Algs for Mobile Wireless Sys

  25. Proof of Lemma 1 • Consider a particular element z of Ij-1. • Number of queries (sets of size n/k) that contain z but no other element of Ij-1 is C(n-x,n/k-1), the number of ways to choose the remaining n/k-1 elements of the query (after choosing z) from the n-x stations not in Ij-1. • There are x (size of Ij-1) different choices for z. • Probability that a randomly chosen query is one of these is x*C(n-x,n/k-1)/C(n,n/k). • Calculations show this expression is > 1/2e2. Discrete Algs for Mobile Wireless Sys

  26. Behavior of Series of Queries • Lemma 2: The list of queries isolates at least c*k members of the input with probability > 1 - 1/ebk • c is a constant strictly between 0 and 1 • b is a positive constant • Proof relies on repeated invocations of Lemma 1 plus more probability and calculations. Discrete Algs for Mobile Wireless Sys

  27. Converting Probabilities Into Certainties • Theorem 1: For all k and n, there is a (c,k,n)-universal list of length (k + k*log(n/k)) • c is the constant from Lemma 2 • Proof: Suppose we have a list Q1,…,Qt of queries, each of size n/k, each chosen uniformly at random. • Let random variable X be the number of inputs on which the list isolates < c*k members. • Show that EX < 1: there exists a list that isolates < c*k members on less than 1 (i.e., no) inputs. • This is the desired (c,k,n)-universal list!! Discrete Algs for Mobile Wireless Sys

  28. Proof of Theorem 1 • Represent X as the sum of indicator random variables, one for each input: • XI = 0 if list isolates ≥ c*k members of I • XI = 1 if list isolates < c*k members of I • Claim: If number of queries is large enough, then EXI < 1/C(n,k) for all inputs I • EXI = Pr(list isolates < c*k members of I) • probability goes to 0 as list length increases • so there is some value, call it t, such that EXI < 1/C(n,k) Discrete Algs for Mobile Wireless Sys

  29. Proof of Theorem 1 • EX = ∑I EXI < ∑I 1/C(n,k) = C(n,k)*1/C(n,k) = 1 • Recall there are C(n,k) choices for I. • How big does t (list length) have to be? • Break query list into m groups of size p (same p from before, about k/2). • Apply Lemma 2 to each group: • prob of isolating < c*k members at the end of each group is < 1/ebk • actually probability is even smaller, since some members have already succeeded and dropped out Discrete Algs for Mobile Wireless Sys

  30. Proof of Theorem 1 • Since groups are independent, prob that all m groups isolate < c*k members is < 1/ebkm • To ensure 1/ebkm < 1/C(n,k), set m = ln(C(n,k))/bk + 1 • t = m*p = ln(C(n,k))/(2b) + k/2 = (k + k*log(n/k)) • use Stirling's formula for last step Discrete Algs for Mobile Wireless Sys

  31. Creating Final List • Use lists from Theorem 1 as building blocks and combine them together to get desired list. • Theorem 2: For all k and n, there is a (1,k,n)-universal list of length (k+k*log(n/k)). • Proof: For appropriate value of p (TBD), apply Theorem 1 p times to get p query lists L0, …, Lp, where • Li is (c,k(1-c)i,n)-universal • Li has length (k(1-c)i + k(1-c)i log(n/(k(1-c)i)) • Let the final list be L = L0, L1, …, Lp-1. Discrete Algs for Mobile Wireless Sys

  32. Final List Isolates All Stations • Show L is (1,k,n)-universal. • Consider any input I of size k. • Applying L0 to I isolates at least c*k elements, by Theorem 1, leaving a set J1 of size at most k - c*k = k(1-c). • Applying L1 to J1 isolates at least c*k(1-c) elements, leaving a set J2 of size at most k(1-c) - c*k(1-c) = k(1-c)2. • … • Applying Lp-1 to Jp-1 isolates at least k*c(1-c)p-1 elements, leaving a set Jp of size at most k(1-c)p. • To ensure that size of Jp < 1, choose p so that k(1-c)p ≤ 1-c, i.e., p is log, base 1-c, of (1-c)/k. Discrete Algs for Mobile Wireless Sys

  33. How Long is Final List? • It is the sum of the lengths of L0, L1, …, Lp. • Do some algebra to verify that the sum is (k + k log(n/k)). k/8 k/16 k k/2 k/4 Lengths of the lists decrease in a geometric progression Sum of the lengths is only a constant multiple greater than length of first list Concatenated list isolates enough stations so that < 1 remains, meaning it isolates all stations Discrete Algs for Mobile Wireless Sys

  34. Observations • Can extend the previous analysis to show that a random list of c(k + k log(n/k)) queries resolves k conflicts with high probability, for appropriate constant c. • This provides a nonadaptive algorithm: just choose that number of queries at random, regardless of the feedback. • Future work: • constructive proof (find an algorithm) • close gap between upper and lower bounds Discrete Algs for Mobile Wireless Sys

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