1. Institute of Computer Science University of Wroclaw Geometric Aspects of Online Packet Buffering An Optimal Randomized Algorithm for Two Buffers Marcin Bienkowski Institute of Computer Science, University of Wroclaw, Poland Aleksander Mądry CSAIL, MIT, US

2. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 2 Network switch (1)  Discrete time divided into rounds  In one round any number of packets arrive  We may transmit one of them network switchoutputm input queues (buffers) Round 1Round 3Round 2

3. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 3 Network switch (2)  Each buffer has size B  No place in the buffer packets get lost Goal: maximize throughput, i.e. the number of sent packets switchoutputm input buffers Round 4

4. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 4 Online problem, algorithm does not know the future  Adversary: adds packets to buffers = creates input  Algorithm: decides from which buffer to transmit Performance ratio on : Competitive ratio: Competitive analysis throughput of the optimal offline algorithm throughput of online algorithm

5. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 5 Previous results Competitive ratio : Deterministic algorithms  Any reasonable algorithm: [Azar, Richter ’03]  B = 1: [Azar, Richter ’03]  Any B, large m: [Albers, Schmidt ’04]  Semi-Greedy alg., any m: [Albers, Schmidt ’04] Randomized algorithms  Random permutation algorithm, [Schmidt ’05] any B,m:

6. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 6, m = 2, : [Schmidt ’05] Most related results For any randomized algorithm, for any B: [Albers, Schmidt ’04] M234… h(m)…

7. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 7 Our results We consider the two-buffer case (m = 2) and any B Algorithm deterministic 16/13-competitive algorithm for fractional model (dividing packets possible) Algorithm randomized 16/13-competitive algorithm for standard model Two-dimensional randomized rounding Optimal competitiveness THIS TALK

8. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 8 Input description Round in which adversary adds packets to buffer 0 and packets to buffer 1: Round in which adversary does not add packets:

9. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 9 Bad input for GREEDY The competitive ratio of GREEDY is in the fractional model Input sequence incurring competitive ratio 9/7: Greedy: [Schmidt ’05] OPT: Buffer 0 Buffer 1 Greedy policy go to the anti-diagonal loss = 1/3 Bloss = 2/3 B In total: packets added

10. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 10 How can we improve GREEDY? Input sequence: Buffer 0 Buffer 1 GREEDY state OPT state set of possible OPT states (computed by the algorithm)

11. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 11 How can we improve GREEDY? Input sequence: Buffer 0 Buffer 1 OPT does not lose packets as long a part of is within the square Algorithm PB: stay as close to the Perpendicular Bisector of L as possible GREEDY state OPT state set of possible OPT states (computed by the algorithm) PB stateloss = 1/6 B loss = 1/3 B

12. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 12 Main Theorem The competitive ratio of PB is at most i.e. the performance ratio on any sequence is at most Main idea: find hardest (in terms of competitive ratio), but regular sequences and prove the bound on the performance ratio for them. all sequences proper sequences Proper sequences: (i)start with full buffers, (ii)L is always above the main diagonal i.e. the total number of OPT packets >= B

13. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 13 Proof outline 1.Proper sequences are hardest ones: 2.On proper sequences 3.For any proper sequence, On a proper sequence, always looks like this: perpendicular bisector = main anti-diagonal

14. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 14 Proper sequences are hardest Idea: step-wise transformation non-proper proper preserving A) spatial relations between L and state of PB B) length of L throughput on is the same as throughput on

15. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 15 “Properisation” preserving spatial relations Step of step of Case 1:, = const do nothing Case 2:, decreases in Case 3:, = const do nothing Case 4:, increases in Case 5:, decreases in Assume that already starts with

16. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 16 Proof outline 1.Proper sequences are hardest ones: 2.On proper sequences 3.For any proper sequence,

17. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 17 Nemesis proper sequence for GREEDY? Buffer 0 Buffer 1 This is the worst possible behavior of the adversary (potential-like proof) kpackets addedGREEDY loss

18. Institute of Computer Science University of Wroclaw Marcin Bieńkowski: Online Packet Buffering 18 Outlook  We show an optimal randomized algorithm for two buffers and any buffer size B  For we can get the same ratio for deterministic variant of PB  Geometry is neat, actual technical details are gory Open questions:  Is it possible to extend this approach to m > 2?  How well the deterministic version performs for small B?

19. Institute of Computer Science University of Wroclaw Thank you for your attention!