network design with (dis)economies of scale lisa zhang joint with matthew andrews, spyridon...

Network Design with (Dis)economies of Scale

Lisa Zhang

Joint with Matthew Andrews, Spyridon Antonakopoulos and Steve

Fortune

2 | Presentation Title | Month 2006 All Rights Reserved © Alcatel-Lucent 2006, #####

Power: New Resource in Networking

Network power efficiency is a new metric

Mismatch in growth, 2020 projection– Traffic x30-100– CMOS power efficiency x3-6

Goal: Power proportional - power follows load

Current situation:

– Power consumption largely independent of traffic

– Under-utilized network elements (routers, switches) due to over-provisioned network for traffic fluctuation and redundancy

Opportunity for optimization

Routing: example of global optimization for a network of servers


Routing

Input

Network topology and matrix of traffic demands

Cost function f that models power consumption

Question

Find routing paths that satisfy the demands and minimize total power consumption of network.

– Min e f ( xe ) where xe is traffic load on e

Well studied problem for a range of f( )

What is appropriate f( ) in the context of power?

A B C …

A 0 5 3 …

B 5 0 1 …

C 3 1 0 …⋮ ⋮ ⋮ ⋮ ⋱

A B

CD

E

F


Some simple cost functions

Perfectly linear cost function shortest-paths routing for each traffic demand

Solvable in polynomial time [Dijkstra]

Step-function Steiner forest problem

2-approximation: can find a solution at most twice the optimal cost in polynomial time [Agrawal-Klein-Ravi, Goemans-Williamson]

link load

cost

link load

cost

perfectly linear step-function


Buy-at-bulk cost functions

Sub-additive cost function buy-at-bulk network design problem

Sub-additive: f(x1 + x2) ≤ f(x1) + f(x2); models economies of scale

Can find O(log n) approx for uniform case [Awerbuch-Azar]; polylog(n) approx for non-uniform case [Chekuri-Hajiaghayi-Kortsarz-Salavatipour]

Cannot approximated better than (log ¼ n) [Andrews]

link loadco

st

sub-additive

avera

ge


Energy costs and (dis)economies of scale

What if the cost function has the form f(x) = + x for x > 0, f(0) = 0?

Motivation: describes power consumption of CMOS circuits with dynamic frequency and voltage scaling

Reflects a combination of economies and diseconomies of scale

Similarly-shaped cost curves commonly encountered in many industries potential for wide model applicability

link loadco

stavera

ge


Results

If the cost function has the form f(x) = + x for x > 0, f(0) = 0

If = 0, constant approx

If > 0, – Polylog(n) approx– For every , (log ¼ n) lower bound

What is the opportunity for power savings in real networks?

Experimental study that motivates power-aware optimization


Polylog ApproximationJoint with M. Andrews and S. Antonakopoulos

link load

cost


Connection to Capacitated Network Design (CND)

f(x) (3 – 2)

(2 – 1)

2

…

31 / 21 / 1 /

1st edge

2nd edge

3rd edge


Algorithm outline

Goal: (polylog(n), polylog(n))-approx to Capacitated Network Design (CND)

Link capacity violated by polylog(n) factor

Total cost within polylog(n) times OPTLP , optimal fractional solution of CND

Implies polylog(n)-approx to integral routing under with diseconomy of scale

Simplify assumption: unit demands, unit link capacity

Algorithm outline for approximating CND:

While there exist unrouted demands:1. Solve LP relaxation and decompose fractional solution into well-cut-linked flows2. Construct an expander graph as virtual network topology3. Route some fraction of demands via edge-disjoint paths on the virtual topology

Output the union of all partial routings

We need to ensure that each partial routing (i) serves at least a polylog(n) fraction of the demands, and (ii) has cost at most polylog(n) times OPTLP.


LP relaxation

Solve an LP relaxation of CND Min e Ce i Xi,e

s.t.{flow conservation on Xi,e }i Xi,e ≤ 10 ≤ Xi,e ≤ 1

In the fractional solution OPTLP, each demand

may be routed along more than one paths, each carrying only part of the demand’s bandwidth.


Well-cut-linked decomposition

Well-cut-linked decompostion [Chekuri-Khanna-Shepherd] A partition of original graph into node-disjoint subgraphs, and a

function (·) on the terminals.

(·) represents surviving demand flows within each subgraph– At least a polylog fraction of total demand survives.– Each surviving demand has at least polylog fraction of surviving flow

Property of subgraph under decomposition:

– Any equal partition of any subgraph has large cut: fractional flow in OPTLP of

edges in A-B cut ≥ min{(A), (B)}

Large cut

A B


Min-cost flow

Each subgraph is well-cut-linked as a result of decomposition

Every equal partition has large cut using fractional flow in OPTLP

Cost of fractional links along fractional paths for the cut is bounded by OPTLP

Integrality theorem converts fractional flows into integral flows

Round up fractional links into integral links

Cost of integral links along integral flow paths is bounded by polylog * OPTLP

Large cut Large cut


Constructing an expander topology

When is a graph G′ = (V′, E′) an expander?

Expansion: for any S V′ with |S| |V′| / 2, the number of edges in the cut (S, V′ - S) is at least c|S|, for a constant c > 0.

Similar to the property of well-cut-linked terminals we saw earlier.

How to construct an expander?

Suppose we are given a routine that for any balanced partition (A, B) of a node set V* produces a perfect matching. Then, we can construct an expander by calling this routine O(log2 |V*|) times. [Khandekar-Rao-Vazirani]


Constructing an expander topology

How to build an expander on the terminals of each well-cut-linked subgraph of the decomposition?

Perfect matchings consist of entire paths joining terminals, not just edges; cost of these path matchings can be bounded polylog OPTLP

Well-cut-linkedness ensures the existence of such a path-matching.

Result: a virtual expander topology that uses each edge of the real topology at most a polylog number of times.


Edge-disjoint routing in the virtual expander topology

Given an expander and a set of node pairs, we can route at least a polylog fraction of those pairs via edge-disjoint paths.

In the real topology we can route at least a polylog fraction of the demands, while the load on every edge is at most polylog larger compared to the load under OPTLP.

We apply the same process on the remaining demands. No more than a polylog number of iterations required.

Theorem. (polylog(n), polylog(n))-approx to uniform Capacitated Network Design, where link cap is violated by polylog(n) factor and total cost within polylog(n) times OPTLP .

Theorem. Uniform network design with (dis)economies of scale is polylog(n)-approximable.


Open questions

What happens to non-uniform CND, where links have different capacities?

What happens to non-uniform routing with (dis)economies of scale, where each link e has a distinct cost function fe( )?

What happens to protection version of the problem?


Experimental motivationJoint with S. Antonakopoulos and S. Fortune


Goal: Quantification of opportunity for saving power as realistically as possible

• Opportunity for power saving Optimizing traffic routes (global optimization) Rate-adaptive network elements (local optimization)

– Adaptivity characterized by Energy Proportional Index (EPI): ratio of base power to full traffic power

– Full power at full link rate C– Base power to keep network element operational– Linearization

Both approaches have engineering challenges Opportunity for each method alone and in combination

perfect rate adaptivity

c c

100% EPI 50% EPI 25% EPI 0% EPI•traffic

• pow

er

•traffic

• pow

er

c•traffic

• pow

er

c

75% EPI•traffic

• pow

er

c•traffic

• pow

er

always onc•traffic

• pow

er


Data sets

Focus on service-provider networks: Rocketfuel datasets by U. Washington packet-tracing technology to discover router-level graphsrouters labeled with cities, multiple routers/city


Routing Heuristic

• “Iterative greedy least-power routing” motivated by [Charikar-Karagiazova]

• Randomly permute demands• Route each demand along route that incurs least marginal cost

• First use incurs base power• Subsequent use incurs increment cost

• Iterate until no improvement is observed

• •


Plots for x% EPI in rate-power curve

•0% EPI (perfect rate adaptivity)

•Min-hop routing optimal

always on (current state)• p

ow

er

•traffic multiplier •0% •100%

c•traffic

• pow

er

c•traffic

• pow

er

Overprovision in capacity


Plots for x% EPI in rate-power curve

•0% EPI (perfect rate adaptivity)

always on (current state)• s

avin

g d

ue t

o

• rate

adapta

tion

•traffic multiplier •0% •100%

Min-hop routing

Power-aware

routing

c•traffic

• pow

er

• savin

g d

ue t

o

• routi

ng


AS 1775

•1755 city (75% EPI)

•0

•5000

•10000

•15000

•20000

•25000

•30000

•35000

•40000

•45000

•0% •20% •40% •60% •80% •100% •120%

•always on

•min hop routing

•heuristic routing

•perf. prop. cost

c

75% EPI•traffic

• pow

er


AS 1755

•1755 city (100% EPI)

•0

•5000

•10000

•15000

•20000

•25000

•30000

•35000

•40000

•45000

•0% •20% •40% •60% •80% •100% •120%

•always on

•min hop routing


•perf. prop. cost

•1755 city (75% EPI)

•0

•5000

•10000

•15000

•20000

•25000

•30000

•35000

•40000

•45000

•0% •20% •40% •60% •80% •100% •120%

•always on

•min hop routing


•perf. prop. cost

•1755 city (50% EPI)

•0

•5000

•10000

•15000

•20000

•25000

•30000

•35000

•40000

•45000

•0% •20% •40% •60% •80% •100% •120%

•always on

•min hop routing


•perf. prop. cost

•1755 city (25% EPI)

•0

•5000

•10000

•15000

•20000

•25000

•30000

•35000

•40000

•45000

•0% •20% •40% •60% •80% •100% •120%

•always on

•min hop routing


•perf. prop. cost

For lightly loaded networks, power saving opportunities are there.

For small rate adaptivity, opportunity for routing optimization is there.


Open Problems

What happens to non-uniform CND, where links have different capacities?

What happens to non-uniform routing with (dis)economies of scale, where each link e has a distinct cost function fe( )?

What happens to protection version of the problem?

What is analogue of Charikar—Karagiazova for routing with (dis)economies of scale?


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