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)
Agenda
•
•
•
•
Preemption Problem
Earlier Work
Our Contribution
Conclusion
2
Problem Scenario
7. Preemption decision
for R4->R8
1. New connection
request (R1,R8,bw,class)
5. Preemption decision for
R6->R7
2.3.Makes
an
admissionbased
control
decision
aa
constraint
routing
4. Makes
Makes
preemption decision
for
If
enough
bandwidth
is
available
then
decision
R1->R6
accept the request; otherwise reject the
Say route
={R1->R6->R7->R4->R8}
request
A third possibility: accept
the request by preempting
lower
priority connections
We
consider
the problem
6. Preemption decision for
R7->R4
of which connections to preempt!!!
3
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
4
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
5
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
6
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?
3 differences
 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
7
Proof (Contd.)
• How to solve the SS problem using the solution to our
problem? Basic idea:
SS Input
Pre-Processing
Input to our problem
Our Problem Solver
Output of our problem
Post-Processing
Output of SS
8
Proof (Contd.)
• SS Input: V={a1,…,an} and t
• We construct V’={c1,b1 …,cn,bn } and t’
Ensures that those cis
are chosen that
minimize the overshoot
from the threshold
Ensures that
exactly n
elements are
chosen
Ensures that either ci or the
corresponding bi (but not
both) is selected
9
Proof --- Putting it together
SS Input: V={a1,…,an} and t
Polynomial Complexity
•Check whether the sum of all elements exceed threshold
Pre-Processing (if not then no solution subset exists)
•Construct V’={cis, bis} and t’
V’ and t’
Our Problem Solver
S’{cis, bis}
iff SS
problem can
be solved in
polynomial
time
Post-Processing •Discard the dummy elements (bis) from S’
•Keep the l most significant bits of cis
•If their sum equals threshold then output YES else NO
SS Output: YES/NO
10
Exact Algorithm (V,t,K)
• In any iteration i, the length of L can be as long as 2i
11
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
12
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
13
Questions?
14