1. Competitive On-Line Admission Control and Routing By: Gabi Kliot Presentation version

2. Admission control and virtual circuit routing Competitive analysis AAP algorithm Lower bound on unlimited deterministic alg (Routing on the line) Randomized competitive algorithms Randomized Base Tree call-control algorithm “Classify and Randomly Select” algorithm Applications of “CRS” on different parameters Outline:

3. Virtual Circuit Routing Virtual Circuit Routing: In order to use the network (transmit video..) the user requests a “virtual” connection to be established between source and destination. This guarantee is critical for real-time applications. Although the rate of information through such a connection might vary in time, the network has to guarantee that the connection will support at least the bit rate that was agreed upon during the connection establishment. This guarantee is critical for real-time applications. In other words, establishing a connection corresponds to reserving the requested bandwidth along some path on the network. Admission Control Admission Control: Algorithm that decides which requests to satisfy and which to reject and determines the route.

4. Competitive analysis Cost problem: Benefit problem:

5. Network Model G=(V,E)|V|=n Capacity represents edge maximum bandwidth Call Each call can be admitted or rejected No preemption of the calls The goal is to maximize the benefit of the admitted calls of the admitted calls

6. AAP can deal with arbitrary parameters, but requires the rate of each call to be a “small fraction” of edge capacity AAP algorithm ( AAP algorithm (Awerbuch, Azar, Plotkin ) Goal: Goal:

7. AAP algorithm – Definitions:

8. AAP algorithm Intuition: we route only only on edges, whose relative load is small – only on edges not close to be saturated; That way we spread the bandwidth of admitted calls “equally”. rjrj

9. What should be proved: What should be proved: Advantages of AAP: Advantages of AAP: – Applies to all networks – Deterministic Disadvantages of AAP: Disadvantages of AAP: – No single communication request can require more then 1/log(n) of the capacity of a single edge.

10. There is a lower bound of on the competitive ratio of general deterministic on-line call-control algorithm (no assumptions) How do we measure lower bounds? How do we measure lower bounds? adversary algorithm We use an adversary algorithm, that presents the worst to the ALG, making him act badly, while OPT can still act well. Lower bound on AAP: Lower bound on AAP: It can be proved that any on-line algorithm has competitive ratio of even if all requested rates are very small (like AAP) Lower bounds

11. Routing on the line: I. II. n123n-1 There is a need for randomized algorithm !!!

12. expected benefit When considering a randomized on-line algorithm, the benefit gained by algorithm is the expected benefit, where the expectation is taken over the random choices of the algorithm. We analyze our algorithm against adversary (worst). Randomized on-line algorithm

13. Base Tree Algorithm: Goal: Goal: maximize the number of edge-disjoint admitted calls The Base Tree Alg is O(log(n)) competitive Assumptions: Assumptions:

14. The Randomized Base Tree alg. Classify all edges into maximum log(n) classes; Let N be the maximum assigned level to the edge of the tree. Select uniformly with probability 1/N one of the classes Apply greedy algorithm to the calls from that class, ignoring all others

15. Classification Algorithm:

16. 3 2 1 3 11 3 2 44 3 33 2 1 4 3 33 2 3 3 33 2 The above algorithm classifies the calls into O(log(n)) disjoint classes Example: Classification Algorithm:

17. Claim 1: Claim 1: Any 2 nodes in the same class are separated by a node of a lower-numbered level Claim 2: Claim 2: Let p be a call of class l. Then there are at most 2 edges of level l included in p. Proof: Proof: jklilllllm SD lll Lemma Lemma: The on-line alg applied to the calls of a single class is 2-competitive.

18. Claim 3: Claim 3: Let P be a set of paths of the same level l sharing an edge. Then, all paths of P share a level l edge. Proof: Proof: Vp l l Vq p q l l p q V m

19. Proof of the lemma 1 Proof of the lemma 1: Let p be a call not accepted by our alg, that is, when arrives there is already a call q that was accepted and intersects p. Then p and q share an edge of level l. There can be at most 2 such edges, so OPT can accept at most 2 calls instead of p, while ALG may accept only one. Lemma Lemma: The on-line alg applied to the calls of a single class is 2-competitive.

20. The Randomized Base Tree alg. Theorem: Theorem: The Base algorithm is O(log(n)) competitive. Proof: Proof: Again, It can be shown using the structure called “adversary tree” the lower bound of on the competitive ratio of any online randomized alg.

21. Bibliography: “Throughput-Competitive On-Line Routing”, B. Awerbuch, Y. Azar, S. Plotkin “Competitive Non-Preemptive Call Control”, B. Awerbuch, Y. Bartal, A. Fiat, A. Rosen. “Online Computation and Competitive Analysis”, Allan Borodin & Ran El-Yaniv

22. Goal Goal : deal with arbitrary call benefits, rates and durations. “ Classify and Randomly Select ” Let A’ be C-competitive alg tolerating values of some parameter p that differs by a factor at most 2 We define a randomized alg A’’. A’’ is O(C*log(M)) competitive The introduction of each additional varying parameter increases the competitive ratio by a factor of O(log(M))

23. A’’ creates in on-line manner at most L classes, selects one and uses A’ while presenting to it all calls of the selected class “ Classify and Randomly Select ” parameter p of A’’: parameter p of A’’: Define L=log(M)

24. Theorem: Proof: “ Classify and Randomly Select ” A’ is C-competitive

25. arbitrary benefits Any C-competitive call control alg for equal benefits can be transformed into O(C*log(M)) competitive randomized alg for the call control problem with arbitrary benefits. arbitrary durations Any C-competitive call control alg that handles infinite durations can be transformed into 2C competitive randomized alg to handle durations that vary by a factor of at most 2, and can be transformed into O(C*log(T)) competitive randomized alg for arbitrary durations. rates Same for rates. Applications of “ Classify and Randomly Select ” We improve now our base alg to be O(log(n)) competitive, even if rates vary by a factor of at most 2.

26. Lemma Lemma: If all calls have rate r such that, for some, then greedy on-line alg applied to the calls of single class l defined in the base alg, is 6 competitive. Proof Proof: - Let p be a call of rate r nor accepted by greedy alg. That is, there is an edge such that when p arrives the sum of the rates of the accepted calls that include e is greater than 1-r. -Consider the accepted calls that include e and p; they all of the same level l and share a level l edge. -ALG has accumulated at least benefit till far due to the calls passing on e (the minimum load that can overload e). -On the other hand OPT can accept at most additional calls including this level l edge.

27. Cont. Proof Cont. Proof: - OPT can accept at most what ALG accepts + on each level l edge of p. There are at most 2 level l edges included in p, so: