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Connection Preemption in Multi-Class Networks Fahad Rafique Dogar Carnegie Mellon University, USA Collaborators: Laeeq Aslam and Zartash Uzmi (LUMS, Pakistan) Sarmad Abbasi (NUCES, Pakistan) Young-Chon Kim (Chonbuk National University, Korea)
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2 Agenda Preemption Problem Earlier Work Our Contribution Conclusion
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3 Problem Scenario 1. New connection request (R1,R8,bw,class) 2. Makes an admission control decision If enough bandwidth is available then accept the request; otherwise reject the request A third possibility: accept the request by preempting lower priority connections 5. Preemption decision for R6->R7 6. Preemption decision for R7->R4 7. Preemption decision for R4->R8 We consider the problem of which connections to preempt!!! 3. Makes a constraint based routing decision Say route ={R1->R6->R7->R4->R8} 4.Makes a preemption decision for R1->R6
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4 Preemption Problem: Constraint and Objectives What is the constraint while making the preemption decision? available bw + preempted bw bw of new request Some possible objectives? 1.Minimize the number of preempted connections 2.Minimize the preempted bandwidth 3.Minimize the priority of preempted connections We consider 1 and 2, in that order
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5 Earlier Work MinnConn [Peyravian et al. Infocom99] Enhanced version of our problem Considers priority as the third objective, so tries to achieve objectives 1,2, and 3, in that order Let’s assume that priority of preemptable connections is the same i.e., we only consider bronze class traffic for preemption. So MinnConn=Our Problem Authors’ claim: MinnConn solves the problem optimally in polynomial time MinnConn runs in polynomial time but is not optimal
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6 Our Contribution We show that solving this problem optimally in polynomial time is highly unlikely Prove that this problem is NP-complete by reducing it to the subset sum problem Propose exact and approximate algorithms to solve this problem Exact algorithm is optimal and runs in exponential time Polynomial time approximation algorithm gives a bounded difference from the optimal
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7 NP-completeness Proof Subset Sum (SS) Problem Given a set V={a 1,…,a n } of n positive integers and a number t, is there any subset S of V, such that How is it different from our problem? Yes/No problem (rather than finding a set) Sum is made equal to threshold (rather than overshoot) No restriction on the cardinality of the solution subset This difference is the key to reducing our problem to the subset sum problem 3 differences
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8 Proof (Contd.) How to solve the SS problem using the solution to our problem? Basic idea: Pre-Processing SS Input Our Problem Solver Input to our problem Post-Processing Output of our problem Output of SS
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9 Proof (Contd.) SS Input: V={a 1,…,a n } and t We construct V’={c 1,b 1 …,c n,b n } and t’ Ensures that exactly n elements are chosen Ensures that either c i or the corresponding b i (but not both) is selected Ensures that those c i s are chosen that minimize the overshoot from the threshold
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10 Proof --- Putting it together SS Input: V={a 1,…,a n } and t Check whether the sum of all elements exceed threshold (if not then no solution subset exists) Construct V’={c i s, b i s} and t’ Our Problem Solver V’ and t’ S’{c i s, b i s} Discard the dummy elements (b i s) from S’ Keep the l most significant bits of c i s If their sum equals threshold then output YES else NO SS Output: YES/NO Polynomial Complexity iff SS problem can be solved in polynomial time Pre-Processing Post-Processing
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11 Exact Algorithm (V,t,K) In any iteration i, the length of L can be as long as 2 i
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12 Approximate Algo. Similar to the exact algorithm but uses a trim function to reduce the length of L in each iteration Trimming: If two values are quite close (within some factor (1+ δ)) then we can keep the larger one and discard the smaller value Keeping the larger value ensures that our solution is feasible though not optimal But solution is within (1+ δ) K of the optimal simulation results show that actual difference is much less
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13 Conclusion Our contribution Proof of NP-Completeness Exact algorithm Approximate Algorithm Other applications of this problem Process preemption in OS Job preemption in scheduling systems
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14 Questions?
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