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Packet Aggregation (Online Control Message Aggregation in Chain Networks)

Packet Aggregation (Online Control Message Aggregation in Chain Networks). Marcin Bieńkowski 1 Jarosław Byrka 1 Marek Chrobak 2 Łukasz Jeż 1 Jiří Sgall 3 Grzegorz Stachowiak 1 1 University of Wrocław 2 University of California at Riverside

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Packet Aggregation (Online Control Message Aggregation in Chain Networks)

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  1. Packet Aggregation(Online Control Message Aggregation in Chain Networks) Marcin Bieńkowski1 Jarosław Byrka1 Marek Chrobak2 Łukasz Jeż1JiříSgall3GrzegorzStachowiak1 1University of Wrocław 2University of California atRiverside 3Charles University Prague

  2. Packet aggregation problem Packets appear at various tree nodes and have to be forwarded to the root.[Khanna, Naor, Raz; ICALP 02] • Input: packet arrivals over (continuous) time. • Onlinealgorithm: a some time points chooses a subtree. • All packets from this subtree are transmitted. • Transmission is immediate (takes zero time).

  3. Costs Two types of costs: • Transmission costs: • Edges have lengths / weights • Transmission of a subtree costs the weight of this subtree(independently of the number of packets) • Waiting costs: • Each packets that is delayed time t pays t. Notes: • We consider centralized, global-knowledge algorithms • Competitive analysis: we compare the total cost of online algorithm to optimal offline algorithm (OPT) • Edge e has weight w(e). • Algorithm chooses when and what to transmit: at time t transmit a subtree and pays its weight. • Motivation: acknowledging a multicast transmission.

  4. Motivations: • Acknowledgement aggregation for multicast transmissions • Packets = small control messages, e.g., acknowledgements transmitted in response for the messages multicast by the root. • Acknowledgements carry little information in comparison with the overhead of sending them. • Information flow in organization networks • One more motivation to come later

  5. Rent-or-buystrategies“If there is a subset of packets whose accumulated delay is equal to the cost of their transmission, transmit them.”

  6. How good are rent-or-buy strategies? • For a single-edge tree: • Packet aggregation = TCP acknowledgement problem • Rent-or-buy strategy is optimal (2-competitive) [Dooly, Goldman, Scott; STOC 98] • For an arbitrary tree: • Under assumption that each packet waits at least 1 at the origin, rent-or-buy strategy is -competitive.[Khanna, Naor, Raz; ICALP 02] • In general, it is -competitive [Brito, Koutsoupias, Vaya; SODA 04]

  7. Focus of this paper: special case: chain network

  8. Simplification: Half-line Instead of a fixed chain network, we consider half-line with 0 denoting the root • Packets may appear at any positive point • Algorithm may transmit from an arbitrary point

  9. What is known for chain? tdt(a,b] = total delay at time t of packets waiting in interval (a,b] • Rent-or-buy strategy = transmit from x when td(0,x] = x competitive ratio is [Brito, Koutsoupias, Vaya; SODA 04] • Slight modification: “transmit from 2x when td(0,x] = x” is 8-competitive [Brito, Koutsoupias, Vaya; SODA 04] This paper: • “transmit from 2j when td(0,2j] = 2j-2” is 5-competitive • Lower bounds: • for any algorithm • for algorithms that transmit from powers of two • Extra: polynomial-time offline algorithm

  10. 5-competitive algorithm (1) Algorithm transmits from when ALG OPT time ALG cost = . We charge it to OPT’s actions of cost • and hence • This amount is paid by OPT as well • Additionally, charge to OPT’s transmission from to • Total OPT’s cost = Longest OPT’s transmission among those unobstructed by the cover sequence

  11. 5-competitive algorithm (2) Algorithm transmits from when time Corner case 1: no unobstructed adversarial transmission • OPT pays for the delay.

  12. 5-competitive algorithm (3) Algorithm transmits from when time Corner case 2:unobstructed OPT’s transmission is longer than • Set (ignore upper part) • Charge to OPT’s transmission from to

  13. 5-competitive algorithm (4) ALG transmissions at t and t’ charge to disjoint OPT actions: • Charges to OPT waiting cost • and charge to different sets of packets • Charges to OPT transmissions a) b)

  14. Lower bound for chain network (1) Single-phase game: • All packets appear at time 0 • At some time, adversary ends the sequence • OPT and ALG are charged for the waiting time of unsent packets

  15. Lower bound for chain network (2) Adversarial strategy for a single phase: • ALG behaves “nicely” if it transmits consecutive packets from after having waited . • If it does not (and its last transmission is from , then adversary ends the phase immediately • OPT transmits from and pays waiting cost of • For such cases, we want to enforce competitive ratio => this gives recurrence on and • If , then ‘s eventually stop growing => there is a strategy even if ALG always behaves nicely!

  16. Lower bound for chain network (3) From a single phase to the lower bound: • Compressed phase = normal phase with faster flow of time. • Waiting cost of packets remaining from previous round is negligible. phase 0 phase 1 phase k OPT transmits all remaining packets at this time

  17. Outlook • Chain networks: the competitive ratio is between 3.618 and 5. • Arbitrary trees: ??? • 2-level trees • The problem is called joint-replenishment problem • The competitive ratio is between 2.64 and 3.[Buchbinder, Kimbrel, Levi, Makarychev,Sviridenko; SODA 08] • We’re currently working on improving the bounds to 2.75 and 2.78, respectively.

  18. Thank you for you attention!

  19. Khanna et al. algorithm (1) O(log w(T))-competitive rent-or-buy approach: “If there is a subset of packets whose accumulated delay is equal to the cost of their transmission, transmit them.” Analysis: • OPT transmits sets over trees at times • Fix single OPT transmission at • Goal: bound ratio • What is ? packet arrivals time

  20. Khanna et al. algorithm (2) • Fix OPT transmission (set over tree at time ): • What is ? ALG has at mostpackets from here packet arrivals time If “eachpacket waits at least 1 at the origin”

  21. Lower bound for rent-or-buy strategies The proof for -competitiveness uses technical assumption: each packet waits at least 1 at the origin. • Prevents adversary to inject many packets at once Lower bound without technical assumption above: • Chain network of edges with weight 1 • OPT sends all at the beginning: total cost = • ALG transmits from all nodes separately: delay =

  22. Previous work: two-node networks (2) TCP acknowledgement = well-studied online problem: • 2-competitive deterministic algorithm[Dooly, Goldman, Scott; STOC 98] Rent-or-buy approach:wait till accumulated delay is equal to the cost of transmission • Randomized: e/(e-1)-competitive [Karlin, Keynyon, Randall; STOC 02] • Randomized via LP: e/(e-1)-competitive [Buchbinder, Jain, Naor; ESA 07] time ALG transmissions:

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