Tuesday Seminar 1

Connection Preemption in Multi-Class Networks

Fahad Rafique DogarWork done while at LUMS, Pakistan

Tuesday Seminar 2

Agenda

• Preemption Problem

• Earlier Work

• Our Contribution

• Conclusion

Tuesday Seminar 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

Tuesday Seminar 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

Tuesday Seminar 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

Let’s verify the above claims!

Tuesday Seminar 6

MinnConnBandwidth demand of the new request Available

bandwidth

While bw of new request is greater than available bw

Bandwidth required for preemptionFinding the minimum bw connection that is greater than the bw required for preemption --- i≠0 if such connection is found

If no SINGLE connection can fulfill the preemption request

Finding the connection with the largest bw

Removing the chosen connection

Including it in the preemption set

Updating the available bandwidth

Tuesday Seminar 7

MinnConn [contd.]

• Does it run in polynomial time? Inner loops (steps 4 and 11) run k times (where k is the number

of connections in the preemptable set) Outer loop can also run a maximum of k times since in every

iteration at least one connection is chosen for preemption So complexity is O(k2)

• Is it optimal? Consider C={70,50,50,20} Bp =100 and aj =0

MinnConn result= {70,50} while optimal result={50,50} Greedy approach of selecting the largest connection is sub-

optimal

Tuesday Seminar 8

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

Tuesday Seminar 9

NP-completeness Proof

• Subset Sum (SS) Problem Given a set V={a1,…,an} 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

Tuesday Seminar 10

Proof (Contd.)

• How to solve the subset sum problem using the solution to our problem? Basic idea: Introduce n dummy connections, each corresponding to a valid

connection Choose n connections from 2n options: either an actual

connection or its dummy counterpart (but not both) is selected• Dummy values are selected for those connections that are not part

of the actual solution From the resulting set of cardinality n, discard the dummy values

• if solution sum equal to threshold (and not greater) then a subset exists whose sum is equal to threshold; otherwise no subset exists

Tuesday Seminar 11

Proof (Contd.)

• Examples: (threshold=100 in each case; V’ and S’ are the input and output of our algorithm respectively) V’={30,60,80,10,D1,D2,D3,D4} S’={30,60,10,D3} SS solution --- YES V’={100,110,130,150,D1,D2,D3,D4} S’={100,D2,D3,D4} Solution ---YES V’={50,60,80,10,D1,D2,D3,D4} S’={50,60,D3,D4} Solution --- NO

• Challenge: How to modify the input and the threshold value such that the

solution to our problem can be used (as described above) to solve SS problem

Tuesday Seminar 12

Proof (Contd.)• SS Input: V={a1,…,an} and t

• We construct V’={c1,b1 …,cn,bn } and t’

Ensures that exactly n elements are chosen

Ensures that either ci or the corresponding bi (but not both) is selected

Ensures that those cis are chosen that minimize the overshoot from the threshold

Tuesday Seminar 13

Proof --- Putting it together

Given SS Input: V={a1,…,an} and t

• Check whether the sum of all elements exceed threshold (if not then no solution subset exists)

• Construct V’{cis, bis} and t’ and pass it to our problem• Discard the dummy elements from the solution set• Keep the l most significant bits of cis • If their sum equals threshold then a subset exists whose sum is

equal to the threshold else no subset exists• Steps 1,3,4,and 5 can be performed in polynomial time• If our problem solver in step 2 can run in polynomial time then

subset sum problem can be solved in polynomial time as well

Tuesday Seminar 14

Exact Algorithm (V,t,K)

• In any iteration i, the length of L can be as long as 2i

• So algorithm has exponential complexity

Tuesday Seminar 15

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

Tuesday Seminar 16

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

• Take home message Don’t blindly trust INFOCOM papers