Online Multi-Commodity Flow with High Demands
Guy EvenEE School, Tel-Aviv University
Moti MedinaEE School, Tel-Aviv University
WAOA 2012
Big vs. Small
Problem Definition ONMCF: Online Multi-Commodity Flow with High Demands• The Network – – set of nodes ()– – set of directed edges ().– Every edge has a capacity
• The Online Input – Sequence of flow requests.
– - source and target nodes.– – flow demand.– – benefit.
• The Output – – a multi-commodity flow.– For each , is a flow from to .
• The Objective – An all-or-nothing ONMCF– That maximizes the total
benefit Of the served requests.
The requests arrive one-by-one. No information is
known about a request before its arrival.
Each request is either fully served or rejected.
We are credited for fully serving
• We consider an online maximization problem:– Competitive analysis.
• For every input sequence σ, |Alg(σ)| ≥ 1/ρ•|OPT(σ)|.
– Alg – deterministic online algorithm.– OPT – offline optimum.– |Alg| - total benefit of algorithm Alg.– ρ – competitive ratio
Competitive Analysis
Previous Work• Mostly studied in the context of single path routing.• Throughput maximization (TM)
– Maximizing the total benefit gained by flow requests that are served [AAP93, BN06,EMSS12].
• Load minimization (LM)– .– Routing all requests while minimizing the maximum load of the edges [AAF+97, AAPW01,
BN06,BLNZ11].
• The following variants are considered:– Permanent routing [AAF+97, BN06, EMSS12]
• AAF+97 : augmentation. (LM)• BN06: using the Primal-Dual framework augmentation. Can be extended to high demands. (LM)
– Unknown durations [AAPW01]– Known durations [AAP93, EMSS12, BLNZ11]
• AAP93: comp. (TM) . Requires
• BN06– Primal-Dual– For the case of unit demands and caps: - comp.
• EMSS12– Primal-Dual– Embedding of traffic patterns I the context of VNETs
• BLNZ11: augmentation. (LM)
The Main Result• An online algorithm for the ONMCF problem:– Centralized and deterministic,– There is no limitation on demands• In particular, may exceed
– All-or-Nothing,– Competes with an all-or-nothing offline optimal
algorithm,– – competitive, for a constant ,– Violates capacities by a factor of ,– Non-preemptive and monotone.
Assuming that caps and benefits
are
Flow is never retracted
Approaches for ONMCF with high demands
• [AAP93, BN06]– Route each request along a single path.• Requires that
– Augment capacities in advance?• Might be augmentation vs. the required
– Split requests into sub-requests• Demands are small• Some of the sub-requests might be rejected.
Not high demand
Not augmentation
Not All-or-Nothing
Approaches for ONMCF with high , demands, cont.
• Granularity of a flow– Smallest positive flow along an edge in the network.
• [BN09]– Formulating ONMCF as a packing LP– Apply Primal-Dual– The caps. Augmentation of BN09 depends on , might be
unbounded!
Not augmentation
An (simple) Example• We want:
– Accept ALL the requests.– Augmenting caps by a factor of at most 1.25.– Granularity of 0.75.
• The Network:– Caps = 1.
• Two requests:– , )– ,
• So…we need:– To route along multiple paths.– To reflow “small” flows, while augmenting (again) the edge caps
• Might affect the competitive ratio, i.e., the chosen flow is not the “lightest” one.→
• Tri-criteria oracle (approximated, augmentation, granularity).
0.75 0.75
11+0.25
0.25
1
1
1.25
1.25
Techniques• Main Technique– Extension of [AAP93] and [BN06] • Integrally packing paths by a centralized online algorithm.• log n – competitive.• Edge costs: exponential in the load of the edge.• Oracle: Finds a shortest path.• Alg :
– If the cost of the path is higher then its benefit, then reject,– Otherwise, accept.
Resource Manager Oraclejjr }{ Oracle
Oracle
Techniques• Main Technique– The Reduction• Now, the requests are flow requests.• Every request increases the load of edges that it uses.
– The edge cost is updated.
• The Oracle finds a “min-cost flow” that fully serves the request.• The Oracle
– Is an offline tri-criteria oracle.» Approximated, augmenting, granular.
Resource Manager Oraclejjr }{ Oracle
Oracle
Extending the Framework• Formally,
-Approximation -Augmentation -Granular
The Tri-Criteria Oracle
2-Approximation 2-Augmentation -Granular
The Main Result
Mixed Demands• Splitting a stream of packets along multiple
paths should be avoided, if possible →• One may require not to split requests with
low demand, i.e., • How? We employ two oracles:– Tri-criteria oracle for high demands (as before).– An exact (shortest path) oracle for low demands.
– Serves a request by a single path.
• This algorithm has the same properties!– – comp., caps aug., monotone, all-or-nothing.
Further Extensions
• Requests with known durations– Each request, upon arrival has an end time.– This talk was on requests that “stay forever”.– The same algorithm can be adapted to known
durations [AAP93, BN06, EMSS12].– Again, this algorithm has (almost) the same
properties!• Cap. augmentation of
– Where is the longest duration.