Packet Aggregation

(Online Control Message Aggregation in Chain Networks)

Marcin Bieńkowski1 Jarosław Byrka1 Marek Chrobak2 Łukasz Jeż1 Jirí Sgall3 Grzegorz Stachowiak1

1 University of Wrocław2 University of California at Riverside3 Charles University Prague

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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. Online algorithm: a some time points chooses a subtree.

All packets from this subtree are transmitted. Transmission is immediate (takes zero time).

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Costs

Two types of costs:

1. Transmission costs: Edges have lengths / weights Transmission of a subtree costs the weight of this subtree

(independently of the number of packets)2. 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.

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Motivations:

1. 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.

2. Information flow in organization networks

3. One more motivation to come later

Rent-or-buy strategies

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

Marcin Bieńkowski Acknowledgement Aggregation

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]

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Focus of this paper: special case: chain network

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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

Marcin Bieńkowski Acknowledgement Aggregation

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 algorithm9

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5-competitive algorithm (1)Algorithm transmits from when

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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

ALG

OPT

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Algorithm transmits from when

5-competitive algorithm (2)

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time

Corner case 1: no unobstructed adversarial transmission OPT pays for the delay.

Marcin Bieńkowski Acknowledgement Aggregation

Algorithm transmits from when

5-competitive algorithm (3)

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time

Corner case 2: unobstructed OPT’s transmission is longer than Set (ignore upper part) Charge to OPT’s transmission from to

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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)

5-competitive algorithm (4)

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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

Lower bound for chain network (1)

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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!

Adversarial strategy for a single phase: ALG behaves “nicely” if it transmits consecutive packets from

after having waited .

Lower bound for chain network (2)

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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.

Lower bound for chain network (3)

phase 0 phase 1 phase k

OPT transmits all remaining packets at this time

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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.

Thank you for you attention!

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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

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Khanna et al. algorithm (2) Fix OPT transmission (set over tree at time ): What is ?

packet arrivals

ALG has at mostpackets from here

time

If “each packet waits at least 1 at the origin”

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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 =

Lower bound for rent-or-buy strategies

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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: