QoS Routing with Performance-
Dependent Costs
Funda Ergun ;Rakesh Sinha ;Lisa Zhang
INFOCOM 2000.Nineteenth Annual Joint Conference of the IEEE
Computer and Communications Societies.Proceedings.IEEE
Volume:1,2000,Page(s):137~146 vol.1
Outline
Introduction
Preliminaries
Polynomial-time approximation
Heuristics for partition
Conclusion
Abstract
Present approximation algorithms guarantee
to produce solutions that are within 1+ of
the optimal
The running times are polynomial in the input
size and 1/
Present good approximations algorithms and
heuristics which apply to general cost
functions
Introduction
Today’s Internet deploys best effort routing
without any assurance of service quality
Future Internet will support various QoS
classes,and each class has its own set of
service guarantee and associated costs
Packets will be given higher priority with aim
of satisfying performance requirements
Stringent requirements should be charged a
higher fee
Introduction(cont.)
QoS routing is to identify a routing path
based on an application’s Qos requirements
and resource avalilability
QoS requirements are specified either as
Set of path constraints
Set of link constraints
A feasible path is a path with sufficient
resources to satisfy QoS requirements
Optimal criteria narrow the selection among
feasible paths
Introduction(cont.)
This paper considers a model in which an
application is charged a per link price
depending on delay guarantee requested
Service provider provides multiple service
classes with different price
Problems
A network with n nodes and m links
Each link has a cost function ce(d) to
represent cost incurred by delay d on link e
Given source s and destination t,and end-to-
end delay constraint D needs to be satisfied
There are following two problems:
Constrained minimum cost path (PATH)
Constrained minimum cost partition (PARTITION)
Problems(cont.)
PATH problem
Chose an s-t path and minimize sum of link costs
along the path subject to delay constraint D
PARTITION problem
s-t path is already chosen
Determine delay to be imposed on every link
along path such that cost is minimized subject to
the end-to-end delay constraint D
Preliminaries
Network N has n nodes and m links
Given an s-t path P, let p be the number of
links on the path
Each link has associated cost function ce(d),
and it is non-increasing
Work with integral delays and costs
Use de(c) to denote the “inverse” of ce(d) and
it returns smallest delay that incurs cost at
most c
Preliminaries(cont.)
Given link e and delay d,we can retrieve cost
ce(d) in constant time
Given link e and cost c,we can compute delay
in logD time using binary search
OPT denote the cost of the optimal s-t path
subject to the delay constraint
C denote the maximum possible cost on any
link ,i.e. C=maxece(1)
Polynomial-time approximation
Approximation algorithms for PATH and
PARTITION are based on approximation
algorithm for RSP
RSP is a restricted version of PATH with each link
has fixed cost and delay
Theorem 1:the result is given by Hassin
RSP has an -approximation algorithm with
running time O( 1 mnloglog U)
U is upper bound of OPT
Polynomial-time approximation
Theorem 2:
PATH has an -approximation algorithm with
1
running time O(X mnloglog U),where X =min{D,
n
log C
+logD, +logD}
Theorem 3:
PARTITION has an -approximation algorithm
1 2
with running time O(X p loglog U),where X=min{
p
log C
D, +logD, +logD }
Algorithm 1
Derive from approximation algorithm for RSP
Transform network N(for PATH) into a
network N1(for RSP) such that optimal RSP
solution is equivalent to optimal PATH solution
Replace each link e of N by D links e1,…,eD
Each link ei has cost cei(i) with delay i
Apply Hassin’s approximation algorithm for RSP to
N1
Map resulting s-t path to a path in N
Replace link ei with link e with delay i and cost ce(i)
Algorithm 1(cont.)
Running time includes two components:
Time of creating N1
Time of applying approximation algorithm for RSP
Time for creating N1 is mD
Time for running Hassin’s algorithm is
O( D mnloglog U)
1
Lemma:
Algorithm 1 is an -approximation of PATH with
1
running time O(mD+D 1 mnloglog U)
Algorithm 2
Key idea of to achieve weaker guarantee with
far fewer links transformation
Goal of algorithm 1 is to capture all possible
choices of cost and delay assignments
Subdivide range of cost [1,ce(1)] into
log 1 ce 1 sub-ranges and pick one
representative from each sub-range
Reduce the linear “blow-up” to logarithmic “blow-
up”
Disadvantage is the approximation
Algorithm 2(cont.)
Precise description:
Consider a link e in N
Each semi-open sub-range:
ce 1 ce 1
0i
1 i 1 , 1 i ,
log1 ce 1
Find minimum delay di that incurs a cost within
above range,and create a link in N2 with delay di
and ce(di)
Transformation example
Algorithm 2(cont.)
Each link in N is replaced by
log1 ce 1 log1 C links
Creating N2 requires at most m log1 C and
computations of de(.) function,resulting in
running time Om log1 C log D
Lemma
Algorithm 2 is an - approximation of PATH with
running time
mn
O m log 1 C log D log 1 C log log U
Algorithm 3
High-level idea
Use TEST procedure to determine whether OPT is
greater than some given value V
Then start with some upper bound U and use
EXACT procedure for exact value of OPT
TEST procedure will determine whether OPT V or
OPT (1+ )V
Maintain a range,[L,(1+ )U],for OPT
To approximate OPT ,we repeatedly narrow the
gap
An Exact solution
Let gv(c) be the minimum delay from s to v
with total cost c
Minimum delay path s-v goes through some
intermediate node u
Obtain gv(c) by minimizing all possible
intermediate nodes u and all possible costs bD
The smallest c such that gt(c) D is equal to OPT
An Exact solution(cont.)
An Exact solution(cont.)
Running time for one iteration is
O(mlogD+mc)
For loop has OPT iterations,so overall running
time is O( mOPT logD+mOPT2)
The TEST procedure
Approximately determine if a given value is
greater than OPT
TEST resembles EXACT,except:
Costs ce(d) are scaled down by a factor V / n
n
For loop is executed for c=1,2,…,
If a path of delay at most D is found for some
c ,TEST outputs OPT (1+ )V
n
Scaled down cost and its inverse are denoted
by ce d and d e c
ˆ ˆ
The TEST procedure(cont.)
n n2
• running time is O m log D 2 m
The Approximation algorithm
n n2
• Total running time is O m log D 2 m log log U
Heuristics for PARTITION
Greedy algorithms
One unit of delay is initially assigned to each link
along path P
One unit of delay is added to the link that causes
largest reduction in total cost
Greedy algorithm(cont.)
Heuristics
Greedy algorithm always makes locally
optimal choice
For non-convex functions,make some locally
non-optimal choices to reach globally optimal
Two heuristics:
Greedy heuristic with Rollback
Continues adding delay to one link
Variable Step Size Heuristic
Adding delay in chunks of various sizes
Greedy Heuristic with Rollback
Proceeds greedily by always adding one unit
of delay to the link that offers largest cost
reduction
If cost reduction(g) due to current delay is
greater than previous reduction,checks every
link and performs a rollback
Rollback consists of removing delay units from
each link until a unit reach a unit whose reduction
was at least g
Greedy Heuristic with Rollback
Variable step size heuristic
Allow allocation of delay in sizes greater than
single unit during a iteration
Pick best link and best delay increment
Motivation
If a curve offers large per delay cost reduction,we
are not stuck in earlier part of that curve
Has additional advantage of making the heuristic
run faster
Two variants
Delay allocations are all possible powers of 2
All possible delay allocation between 1 and D
Variable step size heuristic(cont.)
Simulation results
Experiments are run on a path of 30 links
with D=250
Each group of experiments aims to test cost
functions with different shapes
Earlier groups tend to have more alternating
convex/concave regions
Actual cost functions are generated randomly
Each number in table shows percentage error
for corresponding heuristic
Difference between initial cost and optimal cost
Simulation results(cont.)
Conclusion
Present polynomial-time -approximations
for PATH and PARTITION problems with
general cost functions
Applying results to more complicated
structures such as multicast trees is left for
future research