Throughput Competitive Online Routing Baruch Awerbuch Yossi Azar Serge Plotkin

Focus Basic Concepts Problem Statement Proofs Origin of Idea Earlier work and Results Impact Created Open Problems

Some Basic Terms … Throughput - > Maximize On-Line Competitive Ratio (profit, load, congestion) High Speed Networks

Basic terms … Amortized Throughput Relative Load

Problem at Hand Route given requests in an online manner, maximizing the overall profit. Duration and Source - Dest pair No Preemption or Re-Routing General Network Topology Within Capacity Constraints

Definitions Requests Given Assigned Paths (Pi) Relative Load on an Edge before the kth request. T :max duration

Assumptions … Normalizing the profit. Requested Rates are smaller than the edge capacity. No Interruption Cost exponentially with current load Edge cost monotonically increasing

The Route_OR_Block Algorithm Contribution of all edges of a potential path. Update relative load. Cost is bounded by profit.

Analysis No violation of the Capacity Contraints. Profit accrued is within logarithmic factor of the optimal off-line algorithm.

Proof of Lemma's Let A be the set of indices of all accepted requests. Sum of the link costs to lower bound the profit accrued by online. here k is the index of the last connection

Proof of Lemma's Sum of the link costs is the maximum profit that can be obtained by optimal off-line algorithm. The given algorithm accrues at least fraction of the profit accrued by the optimal offline algorithm. Profit (online) <= P (offline)

Lower Bound Unit Capacities of edges. G(n) a line of n edges. (Vo….Vn+1). 1.Any online algorithm for G(n) has a CR Phases and Groups Construction 2.Any online algorithm has CR of for a single link. 3.Any online algorithm has throughput CR

Origin of the idea… On-Line Load Balancing of Temporary Tasks. Tasks are assigned to machines. Load Vector per job “Limited Duration ” Related vs. Unrelated vs. Identical Minimize the maximum load.

Azar, Plotkin, Waarts, Kalyansundaram and Pruhs Non-Preemptive. O (log nT) Competitive Ratio in terms of load. Subset of machines capable, and increase in load depends on the task only. Improvised CR to from earlier by Azar, Broader and Karlin for unknown duration case.

Unrelated Case History … O(n) CR wrt Congestion using Greedy. This was improved to O (log n) for special case. (Azar, Naor and Rom 92). Finally to general unrelated machine case having O (log n). ( Aspenes, Azar, Fiat, Plotkins, Waarts)

Yet another scenario … Online Call Control in Communication Network. Preemption a boon for telecom companies but there may be loss of revenue. Garay and Gopal. “ Unknown Holding Times => unbounded CR.” Penalty for preemption. Different types of penalties …holding time, path length, constant.

Call Control Algorithms

Impact of this paper …! Randomized Non Preemptive Call Control for trees by Yair Bartal, Awerbuch, Fiat and Rosen. (O (log n) ) Imp: Without Rates Limitation. Classifying calls into classes. Infinite Call Duration, Uniform Rates and profits. O (log M X CR )

Other Extensions Allow Rerouting a finite number of times for unknown duration. O (log n) reroutes per call gets O (log n) CR wrt congestion. ( with Waarts ) Multicast Requests (Plotkin and Goel) Cost Functions independent of A/R decisions made in past.

References Online Throughput-Competitive Algorithm for Multicast Routing and Admission Control by Goel, Henzinger and Plotkins. Competitive Routing of Virtual Circuits with Unknown Duration by Awerbuch, Azar, Plotkin and Waarts. Competitive Non-Preemptive Call Control by Awerbuch, Bartal, Fiat and Rosen. Efficient On-Line Call Control Algorithms by Garay and Gopal.

References … Online Routing of Virtual Circuits with Applications to Load Balancing and Machine Scheduling by Aspenes, Azar, Fiat, Plotkin and Waarts. Online Load Balancing of Temporary Tasks by Azar, Kalyanasundram, Plotkin, Pruhs and Waarts.

Thank you Suggestions.. ? Questions.. ? Presented By: Varun Nayyar