Multiconstrained QoS Routing: Simple Approximations to Hard Problems

flangeeasyMobile - Wireless

Nov 21, 2013 (3 years and 8 months ago)

72 views

1

Multiconstrained QoS Routing: Simple
Approximations to Hard Problems

Guoliang (Larry) Xue


Arizona State University

Research Supported by
ARO

and
NSF

Collaborators: W. Zhang, J. Tang, A. Sen, and
K. Thulasiraman

2

Outline/Progress of the Talk


Problem Definitions



Related Works



Simple K
-
Approximation Algorithms



Faster Approximation Schemes



Conclusions

3

Multi
-
Constrained QoS Routing


Given a network where each link
e

has a
cost
c(e)

and a
delay
d(e),

we are interested in finding a source
-
destination path whose
cost is within a given cost
tolerance
C

and whose
delay is within a given delay
tolerance
D
.



This problem is NP
-
hard. There are many
heuristic

algorithms which have
no performance guarantees
, and
sophisticated
approximation schemes

which are
too
complicated

for protocol implementation
.



We have designed the
fastest approximation schemes
, as
well as
very simple hop
-
by
-
hop routing algorithms

that
have good performance guarantees.

4

Multi
-
Constrained QoS Routing


We study the
general

problem where there are
K

QoS
parameters
,
for any constant
K
≥2
.



We are given an undirected graph G(V, E) where each edge
e

E is associated with K nonnegative weights

1
(e),

2
(e),
…,

K
(e). We are also given a
source s

and
destination t
,
and K positive constants W
1
, …, W
K
.



The multi
-
constrained QoS routing problem asks for
an s

t
path p

such that

k
(p) ≤ W
k
, for k=1, 2, …, K.



For simplicity
, we assume K=2 for the most part of this talk.
In this case, we will talk about
cost
and

delay
.

5

Illustration of the Problem (C=W
1
, D=W
2
)

s

x

y

z

(2, 5)

(12, 5)

(10, 0)

K
= 2


W
1

= 16,
W
2

= 8

The shortest path with regard to the
1
st

edge weight

is
(s, z)

(14, 1)

The shortest path with regard to the
2
nd

edge weight

is
(s, y, z)

Neither of them is a feasible solution

!

Path
(s, x, y, z) is a feasible path.

6

Outline/Progress of the Talk


Problem Definitions



Related Works



Simple K
-
Approximation Algorithms



Faster Approximation Schemes



Conclusions

7

Related Works


J.M. Jaffe,
Algorithms for finding paths with multiple constraints
,
Networks
, 1984.


S. Chen and K. Nahrstedt,
On finding multi
-
constrained paths
,
IEEE
International Conference on Communications
, 1998.


X. Yuan,
Heuristic algorithms for multiconstrained quality of service
routing
,
IEEE/ACM Transactions on Networking
, 2002.


R. Hassin,
Approximation schemes for the restricted shortest path
problems
,
Mathematics of Operations Research
, 1992.


D.H. Lorenz and D. Raz,
A simple efficient approximation scheme for the
restricted shortest path problem
,
Operations Research Letters
, 2001.


G. Xue, A. Sen, W. Zhang, J. Tang, K. Thulasiraman;
Finding a path
subject to many additive QoS constraints
;
IEEE/ACM Transactions on
Networking
,
2007.


G. Xue, W. Zhang, J. Tang, K. Thulasiraman;
Polynomial time
approximation algorithms for multi
-
constrained QoS routing
;
IEEE/ACM
Transactions on Networking
,
2008.

8

Related Works


G. Xue;
Minimum cost QoS multicast and unicast routing in
communication networks
;
IPCCC’2000/IEEE Transactions on
Communications,
2003.


A. Junttner et al.,
Lagrange relaxation based method for the QoS routing
problems
,
IEEE INFOCOM,

2001.


A. Goel et al.,
Efficient computation of delay
-
sensitive routes from one
source to all destinations
,
IEEE INFOCOM,

2001.


T. Korkmaz and M. Krunz,
A randomized algorithm for finding a path
subject to multiple QoS requirements
,
Computer Networks
, 2001.


P. Van Mieghem et al.,
Concepts of exact QoS routing algorithms
,
IEEE/ACM Transactions on Networking
, 2004.


F.A. Kuipers et al.,
A comparison of exact and eps
-
approximation
algorithms for constrained routing
,
IFIP NETWORKING
, 2006.


A. Orda and A. Sprintson.,
Efficient algorithms for computing disjoint QoS
paths
,
IEEE INFOCOM
, 2004.

9

Outline/Progress of the Talk


Problem Definitions



Related Works



Simple K
-
Approximation Algorithms



Faster Approximation Schemes



Conclusions

10

A
Simple

Idea


The
decision problem

is to find a path p such that c(p)

C
and d(p)

D.


The
optimization problem

is to find a path p such that
max {c(p)/C, d(p)/D}

is minimized.


Define
l(p) = max {c(p)/C, d(p)/D} as a new path length.


The original problem has a feasible solution if and only if
there is a path p such that l(p)

1.


The optimization problem is NP
-
hard as well.


The Idea
: For each link e,
define
a new link weight

w(e) = max{c(e)/C, d(e)/D}.



The shortest path with respect to w(e) can be computed
easily, and is
within a factor of 2 from the optimal solution
.

11

Illustration of the Concepts (C=W
1
, D=W
2
)

s

x

y

z

(2, 5)

(12, 5)

(10, 0)

K
= 2


W
1

= 16,
W
2

= 8

The shortest path with regard to the
1
st

edge weight

is
(s, z), l(p)=
20/8
.

(14, 1)

The shortest path with regard to the
2
nd

edge weight

is
(s, y, z), l(p)=
11/8
.

Neither of them is a feasible/optimal solution !

The optimal path is
(s, x, y, z), l(p)=
7/8

12

A Simple 2
-
Approximation Algorithm

s

x

y

z

(2, 5)

(12, 5)

(10, 0)

K
= 2


W
1

= 16,
W
2

= 8

(14, 1)

The shortest path with regard to the new edge weight is
(s, y, z)
whose path length is 11/8.

This path has a length that is guaranteed to be within a factor of 2 from the optimal value
.

In this case, we have 11/8


2
×
7/8.

(2/16,
5/8
)

5/8

14/16

12/16

10/16

13

A
Better

Greedy 2
-
Approximation Algorithm

s

x

y

z

(2, 5)

(12, 5)

(10, 0)

K
= 2


W
1

= 16,
W
2

= 8

(14, 1)

The path found by Greedy is
(s, x, z)
with path length 1

[0,0]

[2/16,
5/8
]

A path from
s

to
x

with path weights [2/16, 5/8] is stored at node

x
. The
path length

is 5/8

[12/16,
20/8
]

[
12/16
, 5/8]

The path at node

x
is chosen because it has the minimum
path length

[4/16,
7/8
]

[
16/16
, 6/8]

The path at node

y
is chosen because it has the minimum
path length
among the unmarked nodes

[
22/16
, 5/8]

The optimal solution is
(s, x, y, z)


with path length 7/8

14

Proof of Correctness


K
-
Approx:


The central idea used in the proof of K
-
Approx relies on
the following simple fact.


Let
x be a point in the K
-
dimensional Euclidean plane
.
Then
||x||

≤||x||
1
≤K

||x||





Greedy:


Greedy never violates the upper
-
bound on path length
used in the proof of K
-
Approx.

15

Numerical Results


Algorithms compared


Greedy


Previously best known
K
-
approximation algorithm


FPTAS for the OMCP problem


K

= 3,
W = W
1

= W
2

= W
3


Networks


well
-
known Internet topologies


ArpaNet (20 nodes and 32 edges) and ItalianNET (33 nodes, 67 edges)


randomly generated topologies


BRITE
[BRITE]


Waxman model
[WaxJSAC88]

, and have the default parameters set by BRITE


the edge weights were uniformly generated in a given range (we used the
range [1,10]).


Three scenarios


Infeasible W = 5

Tight W = 10

Loose W = 20

[BRITE] BRITE; http://www.cs.bu.edu/brite/.

[WaxJSAC88] B.M. Waxman; Routing of multipoint connections;
IEEE Journal on Selected Areas in Communications
; Vol. (1988).

(
ε

= 0.1)

16

On ArpaNet Topology

The number of
better

paths: path
p
1 is better than path
p
2 if l(p1) < l(p2)

For any path
p
, its
relative error
is calculated as (l(p)
-

l(p
OMCP
))/
l(p
OMCP
) ,

where
p
OMCP

is the path found by
OMCP

for the source
-
destination pair.

17

On Large Random Network Topologies

Path quality, eps = 0.1, 100 nodes, 390 links.

Scalability of the algorithms, eps=0.5.

80x314, 210x474, 140x560, 160x634.

18

Outline/Progress of the Talk


Problem Definitions



Related Works



Simple K
-
Approximation Algorithm



Faster Approximation Schemes



Conclusions

19

Approximation Scheme for SMCP


We are given an undirected graph G(V, E) where each edge
e

E is associated with K nonnegative weights

1
(e),

2
(e),
…,

K
(e). We are also given a
source s

and
destination t
,
and K positive constants W
1
, …, W
K
. We want to find an s
-
t
path p s.t max{

k
(p)/ W
k
, 1
≤k≤K} is minimized.



In a paper published in TON’2007, we designed an algorithm
that can find a
(1+

)
-
approximation in O(m(n/

)
K
-
1
) time
.



This is the first FPTAS for the general SMCP problem (K

2).

20

Approximation Scheme for SMCP


The idea follows that used by other researchers in this field.


Find an initial pair of lower and upper bounds not too far away from each
other.


Use scaling/rounding/approximate testing to refine the bounds to within a
constant factor


Compute an (1+

)
-
approximation.



The difference is that we got a pair of lower and upper bounds with a
constant (K) factor
in a single step
, using our K
-
Approx. This leads to
faster running time.



O(mn(loglogn+1/

))


O(mn/

) for K=2
.



However, the problem is
slightly different

from the DCLC problem.
None
of the constraints is enforced
. Motivation for the second TON paper.

21

Faster Approximation Schemes for OMCP


All previous approximation schemes for OMCP are based on


Initial bounds


Scaling and rounding, and approximate testing


Final solution



Hassin rounds to
floor
. Lorenz and Raz round to
floor plus
one,

and showed its advantage over that of Hassin.



A
simple combination of the two

techniques leads to an
approximation scheme that is
better than both
.

22

Basic Definitions Again

23

Decision Version

of the Basic Problem

24

A
Restricted Decision Version

(for comparison)

25

Optimization Version

of the Problem

26

The DCLC Problem (used as a subproblem)

27

MCPN:
non
-
negative integers

(2 weights)

28

MCPP:
positive integers

(K weights)

29

Solving MCPP in O(mC
K
-
1
) Time

30

Solving MCPP in O(mC
K
-
1
) Time

31

Illustration of the idea (acyclic graph)

32

Solving MCPN in O((m+nlogn)C) Time

33

Solving MCPN in O((m+nlogn)C) Time

34

Solving MCPN in O((m+nlogn)C) Time

35

Non
-
negative
rounding

and

approximate testing

36

Non
-
negative
rounding

and

approximate testing

37

Positive
rounding

and

approximate testing

38

Positive
rounding

and

approximate testing

39

The Power of Approximate Testing


Assume UB

2(1+

)LB
.
Set
C
=sqrt(LB

UB/(1+

)).

Run
TEST(C,

).



If TEST(C,

)

YES,
then

DCLC
<C(1+

).

Decrease UB to
C(1+

).



If TEST(C,

)

NO,
then

DCLC
>C.

Increase LB to C.




In both cases,
UB/LB

is reduced to
sqrt((1+

)(UB/LB)).



We will have UB≤2(1+

)LB, after
loglog(initial UB/LB ratio)
iterations.

40

Faster Approximation Scheme for DCLC


Use the technique of Lorenz and Raz to compute LB and UB of

DCLC

so
that
LB ≤

DCLC
≤UB≤n

LB.

This takes
O((m+nlogn)logn))

time:
logn
shortest path computations
.



Set

N

to (logn)
2
, and apply TEST
N

to refine LB and UB so that
LB≤

DCLC
≤UB≤2(1+
(logn)
2
)

LB.

This takes
O(mn)

time:
loglog(n) TEST
N
,
each requires O((m+nlogn)n/

N
) time
.



Set

P

to 1
, and apply TEST
P

to refine LB and UB so that
LB≤

DCLC
≤UB≤2(1+
1
)

LB.

This takes
O(mnlogloglogn)

time:
loglog(logn)
TEST
P
, each requires O(mn/

P
) time
.



Solve MCPP with
scaling factor

=(n
-
1)/(LB

).

This takes
O(mn/

)

time.



O(mn(loglogn+1/

))



O(mn(logloglogn+1/

))

41

Faster Approximation Scheme for DCLC

42

Faster Approximation Scheme for DCLC

43

Faster Approximation Scheme for DCLC










O(mn(loglogn + 1/

)) time

[Lorenz and Raz ORL’2001]


O(mn(logloglogn + 1/

)) time.



Conjecture:
O(mn/

) time both necessary and sufficient.

44

Dimension Reduction: OMCP


䑃䱃

45

(1+

⤨)
-

-
䅰灲潸 瑯⁏䵃倬癩v 䑃䱃

46

Faster Approximation Scheme for OMCP

47

Faster Approximation Scheme for OMCP










This is essentially O(m(n/

)
K
-
1
) time.

48

Faster Heuristic/Scheme for DMCP

49

Faster Heuristic/Scheme for DMCP










O(mn(n/

)
K
-
1
) time [Yuan TON’2002]


O(m(H/

)
K
-
1
) time.

50

Running Time

51

Running Time

52

Path Weight Ratios

53

Outline/Progress of the Talk


Problem Definitions



Related Works



Simple K
-
Approximation Algorithm



Faster Approximation Schemes



Conclusions

54

Conclusions


We know how to compute a
shortest path
. OSPF has been
proposed by IETF as an RFC.



We don’t know how to handle two or more QoS constraints
with guaranteed performance.



This is the first approach which is
both simple and
provably good
.


It is as simple as computing a shortest path.


The computed path is within a factor of K from optimal.



From the theoretical point of view, we have designed
faster FPTAS for several versions of the problem.

55





THANK YOU!