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.