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Multiconstrained QoS Routing: Simple Multiconstrained QoS Routing: Simple Approximations to Hard ProblemsApproximations to Hard Problems

Guoliang (Larry) Xue Arizona State University

Research Supported by ARO and NSFCollaborators: W. Zhang, J. Tang, A. Sen,

and K. Thulasiraman

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Outline/Progress of the TalkOutline/Progress of the Talk

Problem DefinitionsProblem Definitions

Related WorksRelated Works

Simple K-Approximation AlgorithmsSimple K-Approximation Algorithms

Faster Approximation SchemesFaster Approximation Schemes

ConclusionsConclusions

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Multi-Constrained QoS RoutingMulti-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.

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Multi-Constrained QoS RoutingMulti-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 eE 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 W1, …, WK.

The multi-constrained QoS routing problem asks for an s—t path p such that k(p) ≤ Wk, 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.

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Illustration of the Problem (C=WIllustration of the Problem (C=W11, , D=WD=W22))

s x

y z

(2, 5)

(12, 5)

(12, 20)

(2, 2)

(10, 0)

K = 2

W1 = 16, W2 = 8

The shortest path with regard to the 1st edge weight is (s, z)

(14, 1)

The shortest path with regard to the 2nd edge weight is (s, y, z)

Neither of them is a feasible solution !

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

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Outline/Progress of the TalkOutline/Progress of the Talk

Problem DefinitionsProblem Definitions

Related WorksRelated Works

Simple K-Approximation AlgorithmsSimple K-Approximation Algorithms

Faster Approximation SchemesFaster Approximation Schemes

ConclusionsConclusions

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Related WorksRelated 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.

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Related WorksRelated 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.

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Outline/Progress of the TalkOutline/Progress of the Talk

Problem DefinitionsProblem Definitions

Related WorksRelated Works

Simple K-Approximation AlgorithmsSimple K-Approximation Algorithms

Faster Approximation SchemesFaster Approximation Schemes

ConclusionsConclusions

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A A SimpleSimple Idea 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.

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Illustration of the Concepts (C=WIllustration of the Concepts (C=W11, , D=WD=W22))

s x

y z

(2, 5)

(12, 5)

(12, 20)

(2, 2)

(10, 0)

K = 2

W1 = 16, W2 = 8

The shortest path with regard to the 1st edge weight is (s, z), l(p)=20/8.

(14, 1)

The shortest path with regard to the 2nd 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

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A Simple 2-Approximation AlgorithmA Simple 2-Approximation Algorithm

s x

y z

(2, 5)

(12, 5)

(12, 20)

(2, 2)

(10, 0)

K = 2

W1 = 16, W2 = 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

20/812/16

2/8

10/16

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A A BetterBetter Greedy 2-Approximation Greedy 2-Approximation AlgorithmAlgorithm

s x

y z

(2, 5)

(12, 5)

(12, 20)

(2, 2)

(10, 0)

K = 2

W1 = 16, W2 = 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

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Proof of CorrectnessProof 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.

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Numerical ResultsNumerical Results

Algorithms compared Greedy Previously best known K-approximation algorithm FPTAS for the OMCP problem

K = 3, W = W1 = W2 = W3 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)

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On ArpaNet TopologyOn ArpaNet Topology

The number of better paths: path p1 is better than path p2 if l(p1) < l(p2)For any path p, its relative error is calculated as (l(p) - l(pOMCP))/ l(pOMCP) , where pOMCP is the path found by OMCP for the source-destination pair.

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On Large Random Network TopologiesOn Large Random Network Topologies

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

Scalability of the algorithms, eps=0.5.80x314, 210x474, 140x560, 160x634.

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Outline/Progress of the TalkOutline/Progress of the Talk

Problem DefinitionsProblem Definitions

Related WorksRelated Works

Simple K-Approximation AlgorithmSimple K-Approximation Algorithm

Faster Approximation SchemesFaster Approximation Schemes

ConclusionsConclusions

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Approximation Scheme for SMCPApproximation Scheme for SMCP

We are given an undirected graph G(V, E) where each edge eE 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 W1, …, WK. We want to find an s-t path p s.t max{k(p)/ Wk, 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 (K2).

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Approximation Scheme for SMCPApproximation 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.

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Faster Approximation Schemes for Faster Approximation Schemes for OMCPOMCP

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.

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Basic Definitions AgainBasic Definitions Again

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Decision VersionDecision Version of the Basic Problem of the Basic Problem

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A A Restricted Decision VersionRestricted Decision Version (for (for comparison)comparison)

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Optimization VersionOptimization Version of the Problem of the Problem

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The DCLC Problem (used as a The DCLC Problem (used as a subproblem)subproblem)

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MCPN: MCPN: non-negative integersnon-negative integers (2 (2 weights)weights)

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MCPP: MCPP: positive integerspositive integers (K weights) (K weights)

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Solving MCPP in O(mCSolving MCPP in O(mCK-1K-1) Time) Time

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Solving MCPP in O(mCSolving MCPP in O(mCK-1K-1) Time) Time

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Illustration of the idea (acyclic graph)Illustration of the idea (acyclic graph)

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

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

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

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Non-negative Non-negative roundingrounding andand approximate approximate testingtesting

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Non-negative Non-negative roundingrounding andand approximate approximate testingtesting

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Positive Positive roundingrounding andand approximate approximate testingtesting

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Positive Positive roundingrounding andand approximate approximate testingtesting

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The Power of Approximate TestingThe Power of Approximate Testing

Assume UB2(1+)LB. Set C=sqrt(LBUB/(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.

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Faster Approximation Scheme for DCLCFaster Approximation Scheme for DCLC

Use the technique of Lorenz and Raz to compute LB and UB of DCLC so that LB ≤DCLC≤UB≤nLB. This takes O((m+nlogn)logn)) time: logn shortest path computations.

Set N to (logn)2, and apply TESTN to refine LB and UB so that LB≤DCLC≤UB≤2(1+(logn)2)LB. This takes O(mn) time: loglog(n) TESTN, each requires O((m+nlogn)n/N) time.

Set P to 1, and apply TESTP to refine LB and UB so that LB≤DCLC≤UB≤2(1+1)LB. This takes O(mnlogloglogn) time: loglog(logn) TESTP, 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/))

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Faster Approximation Scheme for DCLCFaster Approximation Scheme for DCLC

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Faster Approximation Scheme for DCLCFaster Approximation Scheme for DCLC

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Faster Approximation Scheme for DCLCFaster Approximation Scheme for DCLC

O(mn(loglogn + 1/O(mn(loglogn + 1/)) time)) time [Lorenz and Raz [Lorenz and Raz ORL’2001] ORL’2001] O(mn(logloglogn + 1/O(mn(logloglogn + 1/)) time.)) time.

Conjecture: Conjecture: O(mn/O(mn/) time both necessary and ) time both necessary and sufficient.sufficient.

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Dimension Reduction: OMCP Dimension Reduction: OMCP DCLC DCLC

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(1+(1+)(K-1)-Approx to OMCP, via DCLC)(K-1)-Approx to OMCP, via DCLC

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Faster Approximation Scheme for OMCPFaster Approximation Scheme for OMCP

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Faster Approximation Scheme for OMCPFaster Approximation Scheme for OMCP

This is essentially O(m(n/This is essentially O(m(n/))K-1K-1) time.) time.

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Faster Heuristic/Scheme for DMCPFaster Heuristic/Scheme for DMCP

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Faster Heuristic/Scheme for DMCPFaster Heuristic/Scheme for DMCP

O(mn(n/O(mn(n/))K-1K-1) time [Yuan TON’2002] ) time [Yuan TON’2002] O(m(H/ O(m(H/))K-1K-1) ) time.time.

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Running TimeRunning Time

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Running TimeRunning Time

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Path Weight RatiosPath Weight Ratios

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Outline/Progress of the TalkOutline/Progress of the Talk

Problem DefinitionsProblem Definitions

Related WorksRelated Works

Simple K-Approximation AlgorithmSimple K-Approximation Algorithm

Faster Approximation SchemesFaster Approximation Schemes

ConclusionsConclusions

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ConclusionsConclusions

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.

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THANK YOU!