Routing and Wavelength Assignment in

Multifiber WDM Networks with Non-Uniform

Fiber Cost ?

Christos Nomikos a, Aris Pagourtzis b, Katerina Potika b andStathis Zachos b

aDepartment of Computer Science, University of Ioannina, Greece,

e-mail: cnomikos@cs.uoi.gr

bComputer Science, ECE, National Technical University of Athens, Greece,

e-mail: {pagour,epotik,zachos}@cs.ece.ntua.gr

Abstract

Motivated by the increasing importance of multifiber WDM networks we study arouting and wavelength assignment problem in such networks. In this problem thenumber of wavelengths per fiber is given and the goal is to minimize the cost of fiberlinks that need to be reserved in order to satisfy a set of communication requests;we introduce a generalized setting where network pricing is non-uniform, that isthe cost of hiring a fiber may differ from link to link.

We consider two variations: undirected, which corresponds to full-duplex commu-nication, and directed, which corresponds to one-way communication. Moreover, forrings we also study the problem in the case of pre-determined routing. We presentexact or constant-ratio approximation algorithms for all the above variations inchain, ring and spider networks.

Key words: graph theory, wavelength assignment, multifiber optical networks

1 Introduction

In the past years we have been experiencing an enormous growth in band-width demand, due to the Internet and other data services. These services

? Some of the results presented in this paper have appeared in preliminary form in‘Fiber Cost Reduction and Wavelength Minimization in Multifiber WDM Networks’,Proc. Networking 2004, LNCS 3042, pp. 150-161.

Preprint submitted to Elsevier Science

require high bandwidth and thus extremely fast bit rates. All-optical networksemploying wavelength division multiplexing (WDM) satisfy efficiently this re-quirement. In these networks different optical signals can be transmitted viathe same fiber link using different wavelengths, thus increasing the aggregatebandwidth; however, it is desirable for each signal to remain on the same wave-length from transmitter to receiver, in order to avoid wavelength conversion.This is called wavelength continuity constraint . A second requirement, knownas distinct wavelength assignment constraint , is that signals carried throughthe same fiber should use different wavelengths. If we assume a single-fiberconnecting any two adjacent nodes of the network, each wavelength can beused for only one signal between these two nodes.

In networks with multiple fibers per link, signals that pass through the samelink may use the same wavelength provided that they traverse different parallelfibers. Such multi-fiber all-optical networks provide switching capabilities withthe help of optical crossconnects, where a signal that is on some fiber of anincoming link may continue on a different fiber of an outgoing link withoutchanging wavelength. Figure 1 shows an example where the use of multiplefibers can be beneficial in terms of network cost. In the single-fiber setting ofFigure 1(a) we need to hire three fiber units with three wavelengths each, whilein the multifiber setting of Figure 1(b) four fiber units with two wavelengthseach suffice. If we assume that the cost of a fiber is proportional to the numberof wavelengths it carries, then the latter setting is cheaper.

w

w2

1

w3

(a)

w2

w1 w1

(b)

Fig. 1. A comparison of coloring under (a) the single-fiber model and (b) the mul-tifiber model.

A multifiber network can be described by a graph G = (V, E) and a functionµ : E → N that defines the multiplicity of fibers on each link. The set ofrequests R is a set of pairs of nodes. A request can be satisfied by a pathconnecting these nodes (routing) and a wavelength (color) assignment alongthat path (coloring). Color collisions between paths that use the same edgeare allowed, so we use the term “path multicoloring”, as opposed to classical“path coloring” where paths that share an edge must receive different colors.

In this paper, we follow a general setting in which fiber costs are not thesame everywhere; we call such a situation non-uniform pricing, as opposed touniform pricing where the cost of a fiber on any link is the same. We consider

2

networks where each fiber has a limited bandwidth (number of wavelengths)w and each link has a cost, representing the cost of using a fiber on this linkfor a certain time period T . For a given a set of communication requests withduration at most T , we want to satisfy all requests, i.e. to route them andassign them a wavelength, minimizing the total cost of active fibers in thenetwork. Naturally, the number of active fibers needed between two adjacentnodes of the network is the maximum number of connections that use the samewavelength and pass through the link between the two nodes. An example withtwo different solutions is shown in Figure 2.

w=2

43

2

w

w

w

1

1

2

w=2

34

2

w1

w2

w2

Fig. 2. An instance of MinFibCost-RPMC with 2 colors (wavelengths) per fiber,a solution with cost 13 (left) and a solution with cost 11 (right).

We formalize this problem as the Minimum Fiber Cost Routing andPath MultiColoring (MinFibCost-RPMC). Given an undirected graphG = (V, E), a cost function c : E → R, a set of requests R and w colors(wavelengths), the goal is to assign simple paths to requests and colors to paths,so that the following objective function is minimized:

∑e∈E c(e) · µ(e), where

µ(e) is the maximum multiplicity of any color on edge e.

If instead of requests we are given paths (i.e. the routing is pre-determined)we refer to MinFibCost-RPMC with Pre-Routed Requests. Of course,this differs from MinFibCost-RPMC only in topologies that contain cycles.

We distinguish between two basic models: undirected and directed. The undi-rected model corresponds to the case where the communication for every re-quest is two-way and signals in both directions must use the same set of linksand the same wavelength (full-duplex communication). One-way communica-tion can be modeled by using directed requests and paths; the correspondingproblem variations have the same names, preceded by the word “Directed”.

MinFibCost-RPMC in rings is NP-hard for the undirected model, sincethe problem with uniform costs, which is a special case, is NP-hard [1]. Thesame holds for the directed case due to the NP-hardness of the single-fiberrouting and path coloring problem [2]. The special case of rings with pre-routed requests, for both models, can be also shown NP-hard in a similarmanner using the result in [3]. Finally, the problem is NP-hard for spiders

3

as well, but only under the undirected model, because so is the single-fiberproblem; this can be shown be reduction from edge coloring [4].

1.1 Technical Preliminaries

A chain is a graph that consists of a single path, while a ring is a graph thatconsists of a single cycle. A star is a tree with at most one internal node (i.e.of degree ≥ 2). A spider is a tree in which at most one internal node hasdegree ≥ 3, called central node and denoted by v0; that is, a spider is a graphresulting from a star whose edges have been replaced by chains (also calledlegs of the spider). Each leg starts at v0 and ends at a node of degree one (tipof the leg).

Given a network G = (V, E) and a set of requests over G we denote by n thenumber of nodes, and by m the number of requests. A routing of the requestsR is a set of paths P, each connecting the endpoints of a request. For a set ofpaths P and an edge e we denote by L(e,P) the load of edge e w.r.t. P, thatis the number of paths in P that use e.

Once a routing P is determined (either unique, or given in advance), each

edge e contributes at least cost dL(e,P)w

e · c(e), because at least dL(e,P)w

e fiberunits are needed for this edge. Summing up over all edges in E we get a lowerbound for the cost of an optimal solution for an instance I = (G, c,R, w) anda routing P:

OPT (I,P) ≥∑

e∈E

dL(e,P)

we · c(e) (1)

For the sake of brevity we will simply write OPT instead of OPT (I,P), when-ever I and P are clear from the context. An algorithm A for a minimizationproblem Π is a ρ-approximation (for ρ > 1) if for every instance I of Π, A runsin time polynomial in |I| and delivers a solution with cost SOL ≤ ρ · OPT ,where OPT denotes the cost of an optimal solution for I.

1.2 Related work

The problem of minimizing the number of active fibers in multifiber networkswith uniform fiber costs was introduced in [1], where polynomial-time solvabil-ity was shown for chains and 2-approximation algorithms were given for theundirected problem in ring and star networks. In [5] they also study the undi-rected problem with uniform fiber costs for chains and give a new polynomialtime algorithm for this class of graphs.

4

Very recently, in [6], a very similar problem has been examined where therouting is given and the goal is to minimize the maximum ratio between thenumber of active fibers on an edge and the corresponding lower bound. Theyshow that, in general topologies, no constant approximation ratio is possi-ble unless NP ⊆ ZPP and they present logarithmic factor approximationalgorithms. Moreover, they observe that their positive results apply to a gen-eralization defined exactly as our MinFibCost-RPMC with Pre-RoutedRequests.

Multifiber tree networks have been studied only recently. For the problemMinFibCost-RPMC with uniform fiber costs two approximation algorithms,with ratios 1+ 4|E| log |V |

OPTand 4 are presented in [7]. Another multifiber problem

where µ is given and the goal is to minimize the number of wavelengths isstudied in [8–10].

An interesting related problem for single-fiber networks is the MaximumRouting and Path Coloring Problem (MaxRPC), where the goal is tosatisfy as many requests as possible with a given number of wavelengths [11–13]. A generalization of MaxRPC to multifiber networks was introduced in [7]where a 2.54-approximation algorithm for trees was proposed.

Other related work includes traffic grooming. In this approach we can combinelow speed traffic components onto high speed channels in order to minimizethe network cost. Traffic grooming for path, star and tree networks is studiedin [14]; in [15] they consider the problem for ring networks.

1.3 Our results

For chains we observe that the algorithm for MinFibCost-RPMC with uni-form fiber costs [1] can be applied to MinFibCost-RPMC “as is”. Moreover,it can be adapted to give an exact algorithm for Directed MinFibCost-RPMC in chains.

In this paper we present two approximation algorithms for fiber minimizationproblems in rings: the first approximates MinFibCost-RPMC with Pre-Routed Requests within a factor of 2, and the second (which is based onlightest path routing) approximates MinFibCost-RPMC within a factor of2. Appropriate adaptations of these algorithms give a 2-approximation algo-rithm for Directed MinFibCost-RPMC with pre-routed requests and a3-approximation algorithm for Directed MinFibCost-RPMC.

We also propose a 2-approximation algorithm for MinFibCost-RPMC inspider networks; interestingly, this algorithm yields an exact algorithm for thedirected case.

5

2 Undirected Fiber Cost Minimization

In this section we deal with the undirected case of MinFibCost-RPMC inchains, rings (pre-routed requests or not) and spiders.

2.1 MinFibCost-RPMC in Chains

In chains MinFibCost-RPMC can be solved using the algorithm originallyproposed in [1] for chains with uniform fiber cost. We include this algorithmhere mainly for the sake of completeness, since it is used as a subroutine (fullyor partially) by some of the algorithms presented in this work.

Algorithm for MinFibCost-RPMC in chains

Input: I = (G, c,R, w), G = (V, E) is a chain network, c is the edge-cost function,

R is a set of requests and w is the number of colors.

Output: A multicoloring of paths in P, where P is the unique routing of R.

Consider the unique routing P of R.

1. For each edge e in E, add to P some unit length dummy paths, so that

the load of e becomes an integral multiple of w.

2. While there are paths p1, p2 ∈ P (as augmented with dummy paths), such

that p1 finishes and p2 starts at the same node, connect them to construct

a longer path. Let P ′ denote the resulting set of paths.

3. For every node v which is path endpoint (of some path ∈ P ′), P ′ contains

either paths starting at v or paths finishing at v. Partition them into

groups of exactly w paths in an arbitrary order. Let set S (F ) be the

set of all groups of starting (respectively finishing) paths.

4. Construct a bipartite multigraph H = (S ∪ F, A), where A contains

an edge for each path in P ′, connecting its starting and its finishing group.

5. Obtain an edge-coloring of H with w colors by using the

algorithm of Cole, Ost and Schirra [16].

6. Assign to each path in P ′ the color of the corresponding edge in H.

Assign to each path in P the color of the corresponding (long) path in P ′.

Theorem 1 The algorithm for MinFibCost-RPMC in chains computes amulticoloring with optimal cost.

6

Proof. It is proved in [1] that the algorithm obtains a solution that uses

exactly dL(e,P)w

e fiber units for each edge e. Thus, the total cost of this solution

equals∑

e∈EdL(e,P)

we · c(e), which is optimal by (1). 2

The complexity of the above algorithm is determined by the most costlystep, namely edge coloring of the multigraph H (Step 5), whose complex-ity is O(|A| logw) [16]. The number of edges in H (|A|) is at most m + n · w(the cardinality of path set P ′).

2.2 MinFibCost-RPMC with Pre-Routed Requests in Rings

Let G be a ring that consists of n nodes labeled clockwise v0,v1,. . ., vn−1. Wedenote the edge (vi, v(i+1) mod n), 0 ≤ i ≤ n− 1, by ei. We also denote the paththat connects vi and vj and contains v(i+1) mod n by 〈vi, vj〉 and we say that itbegins at vi and it ends at vj. Without loss of generality we assume that alledges of G are used by some path (otherwise we would eliminate an unusededge and obtain a chain instance, which can be solved optimally using thealgorithm for chains). Therefore, at least one fiber unit per edge is needed,thus the total cost of an optimal solution is at least OPT ≥

∑e∈E c(e) = C.

We denote by Pv the set of paths in P that contain v as an internal node. LetP ′

v be the set of paths that results from splitting paths in Pv at node v. Paths inP ′

v are called v-clockwise if they contain edge (v, u), where u is the neighbor of vin clockwise direction; the remaining paths in P ′

v are called v-counterclockwise.Consider the longest v-clockwise path and the longest v-counterclockwise pathin P ′

v; let p(v) be the one of the two using edges with minimum sum of costs.We define the tare t(v) of v to be the sum of edge-costs of p(v) and the spans(v) of v to be the length of p(v) (the number of its edges). If Pv is empty,then t(v) = 0 and s(v) = 0. We may reindex vertices so that v0 is the nodewith minimum tare and p(v0) is v0-clockwise. Let t = t(v0) and s = s(v0). Ouralgorithm for MinFibCost-RPMC with Pre-Routed Requests in ringsfirst selects node v0 as above (see Figure 3(b) for an example). The completealgorithm follows.

7

Algorithm for MinFibCost-RPMC with Pre-Routed Requests in rings

Input: I = (G, c,P, w); G = (V, E) is a ring network, c is the edge-cost function,

P is a set of paths and w is the number of colors.

Output: A multicoloring of paths in P.

1. Find node v0 with minimum tare and reindex nodes accordingly. Set s := s(v0).

2. Transform the given ring instance to a chain instance (G′, c′,P ′, w) as follows:

a. The chain graph G′ consists of n + s + 1 nodes, namely v ′0, . . . , v

′n+s.

Let e′i = (v′i, v

′i+1), 0 ≤ i ≤ n + s − 1.

for i := 0 to n − 1 do c′(e′i) := c(ei) and for i := 0 to s − 1 do c′(e′n+i) = c(ei).

b. For each path 〈vi, vj〉 ∈ P, add a path to P ′:

if i < j add 〈v′i, v

′j〉 to P ′, otherwise add 〈v′

i, v′n+j〉 to P ′.

3. Call Algorithm for MinFibCost-RPMC in chains on instance (G′, c′,P ′, w).

4. Color each path in P with the color of the corresponding path in P ′.

v

v

v

v 1

2

0

3

p

p

p

2

3

10 3

28

p4

1

(a)

v v v

1

p2

p3

1 42 3vv083 32 210

p

6v5v

p4

span 2 2 2 1tare 5 10 5 8

0 1 2 3 v v v v

(c)

(b)

Fig. 3. (a) An instance of MinFibCost-RPMC with Pre-Routed Requests inrings, (b) Step 1 of the algorithm; t = 5 and s = 2 and (c) Step 2 of the algorithm.

Theorem 2 The algorithm for MinFibCost-RPMC with Pre-RoutedRequests in rings computes a multicoloring with cost at most OPT + t.

Proof. It is easy to see that if a path in P uses edge ei and 0 ≤ i ≤ s − 1then the corresponding path of P ′ uses either edge e′i or edge e′n+i. Thus, for0 ≤ i ≤ s − 1, the load of ei in G is split into two parts in G′: L(ei,P) =L(e′i,P

′) + L(e′n+i,P′) (see an example in Figure 3). Notice that L(ei,P) =

L(e′i,P′) for s ≤ i ≤ n − 1. Due to the optimality of the chain algorithm, the

number of repetitions of any color on an edge e′i is at most µ(e′i) = dL(e′

i,P ′)

we.

Hence the cost of the solution computed by our algorithm is:

8

SOL ≤n+s−1∑

i=0

dL(e′i,P

′)

we · c′(e′i)

=s−1∑

i=0

(dL(e′i,P

′)

we + d

L(e′n+i,P′)

we) · c′(e′i) +

n−1∑

i=s

dL(e′i,P

′)

we · c′(e′i)

≤s−1∑

i=0

(dL(e′i,P

′) + L(e′n+i,P′)

we + 1) · c(ei) +

n−1∑

i=s

dL(ei,P)

we · c(ei)

=n−1∑

i=0

dL(ei,P)

we · c(ei) +

s−1∑

i=0

c(ei)

=n−1∑

i=0

dL(ei,P)

we · c(ei) + t (2)

The claim follows from (1) and (2). 2

Since t ≤ C ≤ OPT the following holds:

Corollary 3 The algorithm for MinFibCost-RPMC with Pre-RoutedRequests in rings achieves approximation ratio 1 + t

OPT≤ 2.

Remark 4 The above ratio gets close to 1 for instances with heavy commu-nication traffic.

The computation of each tare and span and the transformation can be per-formed in O(m+n) time. Therefore, the complexity of algorithm for MinFibCost-RPMC with Pre-Routed Requests in rings is determined by that of thechain algorithm, which is O((m + nw) log w).

2.3 MinFibCost-RPMC in Rings

We now propose an algorithm for MinFibCost-RPMC in rings, i.e. for thecase that the routing is also sought. Our algorithm is based on lightest-pathrouting: each request is routed along the path with minimum cost (sum ofedge-costs along path p) between the two alternative complementary paths(see an example in Figure 4). In order to guarantee an approximation ratioof 2 the algorithm considers some additional solutions.

9

Algorithm for MinFibCost-RPMC in rings

Input: I = (G, c,R, w); G = (V, E) is a ring network, c is the edge-cost function,

R is a set of requests and w is the number of colors.

Output: A routing P for R and a multicoloring of paths in P.

1. Let P be a lightest-path routing.

Call Algorithm for MinFibCost-RPMC with Pre-Routed Requests in rings

on (G, c,P, w). Let SOL1 be the cost of the solution.

2. For each edge e ∈ E:

route all requests in R avoiding e obtaining set of paths Pe and compute

the cost∑n−1

i=0 dL(ei,Pe)

we · c(ei) of an optimal solution for the chain

instance (G − e, c,Pe, w).

3. If the minimum cost of the solutions examined in Step 2 is smaller that SOL1

then call Algorithm for MinFibCost-RPMC in chains for the corresponding instance

and output the returned solution

else output the solution of Step 1.

5 2

3

21

2

2

v 5

v6

0v

v1

v2

v3

v4

(a)

5 2

3

21

2

2

v 5

v6

0v

v1

v2

v3

v4

(b)

Fig. 4. (a) A set of requests in a ring and (b) the corresponding lightest path routing

Theorem 5 The algorithm for MinFibCost-RPMC in rings computes amulticoloring with cost at most OPT + C/2 + t, where t is the tare of thelightest path routing.

Proof. It suffices to prove that OPT +C/2+ t is an upper bound for the costof the solution obtained by the lightest path routing.

For the sake of the analysis we consider a convenient transformation of thering, which possibly involves splitting of some edges into two or more parts.Note that splitting an edge of cost c into k edges of costs c1, . . . , ck, such that∑k

i=1 = c, does not affect the cost of the solution returned by the algorithm,since any path of the solution contains either all or none of these k edges.

10

Using this splitting we may assume that for every node vi there exists a nodevf(i) such that the total cost along both paths from vi to vf(i) in the ring isexactly C/2; if this is not the case in the original instance we can split an ap-propriate edge into two edges and set their cost so that the required conditionholds. Notice that this transformation is not necessary for the execution of thealgorithm, but for its analysis only. It is easy to observe that f(f(i)) = i andtherefore f is a permutation of indices (we assume a renumbering of nodes ifnew nodes have been added; of course the number of nodes n is updated too).

We claim that c(ei) = c(ef(i)). In order to prove this claim suppose for thesake of contradiction that c(ei) < c(ef(i)) (the other case is symmetric). Letj = (i + 1) mod n. Then the cost of path 〈vj, vf(i)〉 is less than C/2 and thecost of path 〈vj, v(f(i)+1) mod n〉 is greater than C/2. Since the length of thepath 〈vj, vf(j)〉 is C/2, vf(j) must fall between vf(i) and v(f(i)+1) mod n. This isa contradiction because vf(i) and v(f(i)+1) mod n are consecutive vertices in thering.

The above properties of f imply that for every set of paths P ′ the followingis true:

n−1∑

i=0

dL(ei,P ′)

we · c(ei) =

1

2

n−1∑

i=0

dL(ei,P ′)

we · c(ei) +

1

2

n−1∑

i=0

dL(ef(i),P ′)

we · c(ef(i))

=1

2

n−1∑

i=0

(dL(ei,P ′)

we + d

L(ef(i),P′)

we) · c(ei)

The first line holds simply because f is a permutation and the second becausec(ei) = c(ef(i)).

Since in Step 1 we solve this instance using the algorithm for MinFibCost-RPMC with Pre-Routed Requests in rings, inequality (2) holds for thelightest path routing P. Combining with the above equality, it turns out thatthe following holds for the cost of the obtained solution:

11

SOL≤n−1∑

i=0

(dL(ei,P)

we · c(ei) + t

=1

2

n−1∑

i=0

(dL(ei,P)

we + d

L(ef(i),P)

we) · c(ei) + t

≤1

2

n−1∑

i=0

(dL(ei,P)

w+

L(ef(i),P)

we + 1) · c(ei) + t

=1

2

n−1∑

i=0

(dL(ei,P)

w+

L(ef(i),P)

we) · c(ei) + C/2 + t (3)

Moreover, any path using both ei and ef(i) has total cost > C/2, hence it isnot a lightest path. This means that each path in P contains at most one ofthe two edges ei and ef(i), while its complementary path contains at least oneof them. Since an optimal routing P∗ results from P by replacing some pathswith their complementary ones, the following is true: L(ei,P) + L(ef(i),P) ≤L(ei,P∗) + L(ef(i),P∗). Using this inequality together with (3) we get

SOL≤1

2

n−1∑

i=0

(dL(ei,P∗)

w+

L(ef(i),P∗)

we) · c(ei) + C/2 + t

≤1

2

n−1∑

i=0

(dL(ei,P∗)

we + d

L(ef(i),P∗)

we) · c(ei) + C/2 + t

=n−1∑

i=0

dL(ei,P∗)

we · c(ei) + C/2 + t (4)

where again the third line holds because f is a permutation.

The theorem follows from (4) and (1). 2

Corollary 6 The algorithm for MinFibCost-RPMC in rings achieves ap-proximation ratio 1 + C

OPT≤ 2.

Proof.

Since P is a lightest path routing the tare t is smaller than t ≤ C/2. Hence,Theorem 5 also implies SOL ≤ OPT +C. Now, if OPT ≥ C the result followsimmediately.

If OPT < C, then the routing P∗ of the optimal solution does not use someedge, say e. In Step 2, the algorithm considers, among others, the (unique)routing in which all requests avoid e, i.e. routing P∗. The algorithm forMinFibCost-RPMC in rings then uses Algorithm for MinFibCost-RPMC

12

in chains, which returns an optimal solution for the corresponding chain in-stance. Hence, the solution returned is optimal. 2

The complexity of Step 1 is determined by the complexity of the algorithmfor MinFibCost-RPMC with Pre-Routed Requests in rings. In Step 2the algorithm computes the cost of optimal solutions for n different chaininstances. This does not require to construct the solution, but only to computethe edge loads, and can be performed in O(n+m) time per instance. A singlesolution is obtained by a call to algorithm for MinFibCost-RPMC in chains,if necessary. The overall cost is thus O((m + n)n + (m + nw) log w).

2.4 MinFibCost-RPMC in Spiders

The algorithm for MinFibCost-RPMC in spiders, starts by giving a direc-tion to every path in P. Then, it constructs an appropriate bipartite (multi-)graph and computes an edge coloring for this graph, which guarantees a uni-form distribution of colors among paths that contain the same edge. Thedetailed algorithm follows.

Fig. 5. Left: a spider (thick edges) and a path set on the spider. Right: directionassignment to paths (Step 1).

13

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����

ff

4

5

f1

s1

s2

s3

4s s5

f2

f3

Fig. 6. Adding dummy paths, joining and grouping of inward paths assuming 3 avail-able colors (w = 3) (Step 2.a). Starting groups are shown by rectangles, finishinggroups are shown by ovals.

Algorithm for MinFibCost-RPMC in spiders

Input: I = (G, c,R, w); G = (V, E) is a spider network, c is the edge-cost function,

R is a set of requests and w is the number of colors.

Output: A multicoloring of paths in P, where P is the unique routing of R.

Consider the unique routing P of R.

Let P0 be the set of paths that pass through the central node v0.

1. Assign an arbitrary direction to every path in P0. Direct all paths in P \ P0

toward the central node. Let Pin(r) (Pout(r)) be the subset of P consisting

of paths that use edges of leg r and are directed toward the center (respectively tip).

2. For each leg r of the spider:

a. Execute Steps 1,2 and 3 of Algorithm for MinFibCost-RPMC in chains on

instance (r, c,Pin(r), w) in order to group paths in Pin(r) into starting

and finishing groups. (Note that (i) paths finishing at the center may

be placed in groups that are not full, i.e. that contain fewer than w paths

and (ii) paths finishing in a leg other than r will be placed in a finishing

group of that leg.) Let P ′ be the resulting set of paths.

Add all starting groups to S and all finishing groups to F .

b. Partition P ′out(r) into groups of at most w paths as follows:

begin from the tip of r, select w paths that end closer to the tip and

group them together. Add this group to F and repeat the process

until there are no ungrouped paths in P ′out(r).

3. Execute Steps 4,5 and 6 of Algorithm for MinFibCost-RPMC in chains

on sets S, F and P ′ obtained in Step 2.

An illustration of the algorithm is presented in Figures 5, 6, 7, and 8.

14

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f

ff

6

7

8

11ff

10

f9

Fig. 7. Grouping of outward paths (Step 2.b).

c

c

a

a

ac

b

b

s

s

s

s

1

2

3

4

5

s

f

ff

9

10

11

b

a

7ff 1

f 6

8f

5ffff2

3

4b

a

c

b

c

b c

c

c

bb

c

a

b

a a

Fig. 8. Left: An edge coloring of the bipartite graph H (Step 3). Right: The coloringof P resulting from the bipartite edge coloring (after Step 3).

Lemma 7 For each edge e of a leg r, every color is assigned to at mostL(e,P ′

in(r))

w= dL(e,Pin(r))

we paths in P ′

in(r) that pass through e.

Proof. Let v1, v2, . . . , v`, v`+1 = v0 be the vertices of r, in consecutive orderstarting from the tip. Let also ei denote edge (vi, vi+1).

The set of paths P ′in(r) has the following properties:

(1) the load of every edge ei of leg r, is an integral multiple of w; that isL(ei,P ′

in(r)) = ki · w for some ki ∈ N,(2) none of the nodes v1, v2, . . . v`, is both starting node of a path and

finishing node of another.

Notice that the bipartite edge coloring (Step 3) guarantees that no color isused twice for paths in the same group. This implies that in any starting orfinishing group containing exactly w paths, every color is assigned to exactlyone path.

We claim that for every edge ei, each color is assigned to exactly ki =L(ei,P ′

in(r))

w

paths in P ′in(r) that pass through ei.

We will prove this claim by induction on i.

15

Consider edge e1. There are k1 · w paths passing through e1, which all startat v1. Hence, these paths are grouped in k1 starting groups, each containingw paths. Each color is used exactly once for each of the k1 groups, thus ourclaim is true for e1.

Suppose that our claim is true for ej. We consider three cases:

• If kj+1 > kj then there are some paths starting at vj+1. This implies (usingthe second property of P ′

in(r)) that there are no finishing paths at vj+1.Thus there there exist (kj+1 − kj) · w paths starting at vj+1, which formexactly (kj+1 − kj) starting groups, each containing w paths. This impliesthat each color appears exactly (kj+1 − kj) times in paths starting at vj+1.Since all paths passing through ej, also pass through ej+1, each color is used(kj+1 − kj) + kj = kj+1 times for paths passing through ej+1.

• If kj+1 < kj then there are exactly (kj − kj+1) · w finishing paths and nostarting paths at vj+1. The proof of the inductive step is similar to theprevious case.

• if kj+1 = kj, no path starts or finishes at vj+1 and the claim obviously holdsfor ej+1.

2

Lemma 8 For each edge e of a leg r, every color is assigned to at most

dL(e,P ′

out(r))

we = dL(e,Pout(r))

we paths in P ′

out(r) that pass through e.

Proof. Let v1, v2, . . . , v`, v`+1 = v0 be the vertices of r, in consecutive orderstarting from the tip. Let e be an edge of r and suppose that some coloris assigned to k paths in Pout(r) passing through e. We assign indices from{1, . . . , k} to these paths so that if j > i, then pi finishes closer to the tip thanpj (or they both finish at the same point). These paths belong to differentfinishing groups, as they have the same color.

Consider any path q in the same finishing group as pi, for i < k. Path pk doesnot end closer to the tip than q, because otherwise it would be placed in thesame group as pi instead of q. Moreover, q passes through the central node v0,since q ∈ P ′

out(r). Consequently q passes through edge e. This implies that allw paths in each of the k− 1 groups containing p1, p2, . . . , pk−1 pass through e.

Thus, L(e,P ′out(r)) ≥ w · (k − 1) + 1 which implies k ≤ dL(e,P ′

out(r))

we. 2

Theorem 9 The algorithm for MinFibCost-RPMC in spiders computes amulticoloring with cost at most OPT + C, where OPT is the optimal solutionand C =

∑e∈E c(e).

Proof. For each edge e, let re be the leg that contains e. It holds that L(e,P) =L(e,Pin(re)) + L(e,Pout(re)).

16

Therefore, by lemmata 7, 8 and inequality (1) the cost of the solution returnedby our algorithm is:

SOL≤∑

e∈E

(dL(e,Pin(re))

we + d

L(e,Pout(re))

we) · c(e)

≤∑

e∈E

(dL(e,Pin(re))

w+

L(e,Pout(re))

we + 1) · c(e)

=∑

e∈E

(dL(e,P)

we + 1) · c(e) ≤ OPT + C

2

Corollary 10 The algorithm for MinFibCost-RPMC in spiders achievesapproximation ratio 1 + C

OPT≤ 2.

Proof. This is because the cost of any solution is at least C, since we mayassume that all edges have a non-zero load (otherwise we can split the instanceinto a spider and several chains; each chain can then be solved independentlyand optimally using the algorithm for chains.) 2

Let m = |P| and m′ = |P ′| ≤ m+n(w−1). The complexity of Step 3 dependson the cost of bipartite edge coloring which is O(m′ log w) using the algorithmin [16]. The total time complexity is therefore O((m + nw) logw).

3 Directed Fiber Cost Minimization

In the directed version of the problem we are given directed requests and theunderlying graph is considered bidirected. We assume that the cost of an edgeis the same in both directions. In rings there are two versions of the problem,depending on whether the routing is known in advance or not.

3.1 Chains

For chain networks we can achieve optimal cost, by computing an optimalsolution for each direction independently; the cost of our solution is the sumof the costs of the two solutions. This gives exactly the lower bound of (1).

17

3.2 Rings with Pre-Routed Requests

For Directed MinFibCost-RPMC with Pre-Routed Requests in ringswe obtain the same approximation ratio as for the undirected case, i.e. 1 +

tOPT

≤ 2, by splitting the instance into one instance of clockwise direction andone of counterclockwise direction and solving the two instances as undirectedones. Note that now t is the maximum tare between the two sub-instances.

3.3 Rings without Pre-Routing

The algorithm for Directed MinFibCost-RPMC in rings, similarly to theundirected case, constructs n + 1 different instances: one instance is obtainedby lightest path routing; each one of the other is a chain instance, obtainedby omitting a pair of symmetric edges, i.e. edges that connect the same nodesbut have opposite directions. Then the algorithm computes a solution of costSOL1 for the lightest-routing instance by using the algorithm for DirectedMinFibCost-RPMC with Pre-Routed Requests in rings. For each ofthe n chain instances the algorithm computes the cost of an optimal solution;if the minimum among these costs is smaller than SOL1, then the algorithmfor Directed MinFibCost-RPMC in chains is called for the correspondinginstance, otherwise the solution of the lightest-routing instance is returned.

Theorem 11 The algorithm for Directed MinFibCost-RPMC in ringscomputes a multicoloring of cost at most OPT + 2C.

Proof. We denote the clockwise-directed edge (vi, v(i+1) mod n) by e+i , the

counterclockwise-directed edge (v(i+1) mod n, vi) by e−i and the common costof these edges by ci.

Let P be the set of lightest paths in the first instance constructed by ouralgorithm and P∗ be the path set of an optimal solution. Let t+ be the tare ofP in the clockwise direction and t− be the tare of P in the counterclockwisedirection.

As in the undirected case, w.l.o.g. we may assume that for every node vi thereexists a node vf(i) such that both paths from vi to vf(i) have cost C/2, andci = cf(i).

The lightest path routing guarantees that if a path in P passes through e+i then

its complementary path passes through e−f(i) and vice versa and no path in P

uses both e+i and e−f(i). This implies that L(e+

i ,P) + L(e−f(i),P) ≤ L(e+i ,P∗) +

L(e−f(i),P∗). Similarly L(e−i ,P) + L(e+

f(i),P) ≤ L(e−i ,P∗) + L(e+f(i),P

∗).

18

Using similar arguments as in section 2.3:

SOL≤1

2

n−1∑

i=0

(dL(e+

i ,P)

we + d

L(e−i ,P)

we +

dL(e+

f(i),P)

we + d

L(e−f(i),P)

we) · ci + t+ + t−

≤1

2

n−1∑

i=0

(dL(e+

i ,P)

w+

L(e−f(i),P)

we + 1) · ci +

1

2

n−1∑

i=0

(dL(e−i ,P)

w+

L(e+f(i),P)

we + 1) · ci + t+ + t−

≤1

2

n−1∑

i=0

(dL(e+

i ,P∗)

w+

L(e−f(i),P∗)

we) · ci + C/2 +

1

2

n−1∑

i=0

(dL(e−i ,P∗)

w+

L(e+f(i),P

∗)

we) · ci + C/2 + t+ + t−

≤n−1∑

i=0

(dL(e+

i ,P∗)

we + d

L(e−i ,P∗)

we) · ci + C + t+ + t−

≤OPT + 2 · C

because both tares t+ and t− are smaller than C/2. 2

The following corollary can be proved in a similar way as Corollary 6, byshowing that either OPT ≥ C or some pair of symmetric edges is completelyavoided by paths in P∗. In the latter case our algorithm achieves an optimalsolution.

Corollary 12 The algorithm for Directed MinFibCost-RPMC in ringsachieves approximation ratio 1 + 2C

OPT≤ 3.

3.4 Spiders

We now present an algorithm for Directed MinFibCost-RPMC in spidersby modifying the algorithm used for the undirected case. Intriguingly enoughthe new algorithm achieves optimal solutions.

Let us note that the algorithm for (undirected) MinFibCost-RPMC in spi-ders fails to find an optimal solution, due to the arbitrary direction that isassigned to each path, which may not be the best possible. In the directedcase the direction of each path is given in advance, which eliminates this dis-advantage.

19

The sets of paths Pin(r) and Pout(r) are defined in a similar way as in theundirected case. There is only one difference: since it is possible that a pathwith both endpoints in the same leg is directed toward the tip of the leg,Pout(r) may contain paths that do not pass through the central node v0.

In order to cope with this situation, Pout(r) must be treated in the sameway as Pin(r), that is, Step 2.b of the algorithm for MinFibCost-RPMCin spiders must be replaced by a step similar to Step 2.a, operating on in-stance (r, c,Pout(r), w). (Notice that this would also work for the undirectedcase; however, the original Step 2.b avoids the addition of dummy paths, thusdecreasing the running time.)

Using similar arguments as in Section 2.4 it is easy to prove that the modifiedalgorithm obtains a solution in which the number of fiber units used for eachpair of symmetric edges e+ and e− is exactly dL(e+,P)

we + dL(e−,P)

we, which is

optimal.

4 Summary of Results - Conclusions

We have studied an up-to-date optimization problem that we call Mini-mum Fiber Cost Routing and Path MultiColoring (MinFibCost-RPMC). This problem deals with the situation where the number of wave-lengths is limited and the cost of fiber usage varies from link to link (non-uniform pricing). The goal is to satisfy all requests with a minimum totalcost.

We have presented constant-ratio approximation algorithms for MinFibCost-RPMC in chains, spiders and rings. Additionally, we study the problem withpre-routed requests in rings. We also propose algorithms for the directed casefor all these topologies. We summarize our results in the following table, wherethe approximation ratio of each of them is shown (for algorithms that give op-timal solutions the “ratio” is ‘1’ and the term ‘pre-rings’ stands for “rings withpre-routed requests”). Recall that OPT denotes the cost of an optimal solu-tion, while C is the cost of using one fiber at each link (edge) of the networkand that OPT ≥ C (if not we can achieve optimal solutions).

Our algorithms are easy to implement and we have proven for all of them aguaranteed bound on their approximation ratio. However, their behavior iseven better in practice. In particular, it can be shown that for heavily loadedinstances the approximation ratio gets close to 1. This is due to the factthat for heavy communication traffic, OPT will be much greater than C. Forexample, if the load on each edge is at least k ·w, for any integer k, then from(1) we obtain OPT ≥ k · C.

20

Approximation ratios

for MinFibCost-RPMC

Network Undirected Directed

chains 1 1

pre-rings 1 + COPT

≤ 2 1 + COPT

≤ 2

rings 1 + COPT

≤ 2 1 + 2COPT

≤ 3

spiders 1 + COPT

≤ 2 1

References

[1] C. Nomikos, A. Pagourtzis, S. Zachos, Routing and Path Multicoloring,Information Processing Letters 80 (5) (2001) 249–256.

[2] P. Raghavan, E. Upfal, Efficient routing in all-optical networks, in: Proc. of thetwenty-sixth annual ACM symposium on Theory of computing, ACM Press,1994, pp. 134–143.

[3] M. R. Garey, D. S. Johnson, G. L. Miller, C. H. Papadimitriou, The complexityof coloring circular arcs and chords, SIAM J. Algebraic Discrete Methods 1 (2)(1980) 216–227.

[4] M. Golumbic, R. Jamison, Edge intersection graphs of paths in a tree, J.Combin. Theory B 38(1) (1985) 8–22.

[5] P. Winkler, L. Zhang, Wavelength assignment and generalized interval graphcoloring, in: Proc. of the 14th Annual ACM-SIAM Symposium on DiscreteAlgorithms, 2003, pp. 830–831.

[6] M. Andrews, L. Zhang, Wavelength assignment in optical networks withfixed fiber capacity, in: Proc. Automata, Languages and Programming, 31thInternational Colloquium, ICALP 2004, LNCS 3142, 2004, pp. 134–145.

[7] T. Erlebach, A. Pagourtzis, K. Potika, S. Stefanakos, Resource allocationproblems in multifiber WDM tree networks, in: Proc. of the 29th Workshop onGraph Theoretic Concepts in Computer Science, LNCS 2880, Springer-Verlag,2003, pp. 218–229.

[8] G. Li, R. Simha, On the wavelength assignment problem in multifiber WDMstar and ring networks, IEEE/ACM Transactions on Networking 9 (1) (2001)60–68.

[9] L. Margara, J. Simon, Wavelength assignment problem on all-optical networkswith k fibres per link, in: Proc. Automata, Languages and Programming, 27thInternational Colloquium, ICALP 2000, LNCS 1853, 2000, pp. 768–779.

[10] C. Chekuri, M. Mydlarz, F. B. Shepherd, Multicommodity demand flow ina tree, in: Proc. Automata, Languages and Programming, 30th InternationalColloquium, ICALP 2003, LNCS 2719, 2003, pp. 410–425.

[11] M. Carlislie, E. Lloyd, On the k-coloring of intervals, Discrete AppliedMathematics 59 (1995) 225–235.

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[12] C. Nomikos, A. Pagourtzis, S. Zachos, Minimizing request blocking in all-opticalrings, in: Proc. INFOCOM 2003, 2003.

[13] C. Nomikos, A. Pagourtzis, S. Zachos, Satisfying a maximum number of pre-routed requests in all-optical rings, Computer Networks 42 (2003) 55–63.

[14] R. Dutta, S. Huang, G. N. Rouskas, Traffic grooming in path, star, andtree networks: complexity, bounds, and algorithms, in: Proc. of the 2003ACM SIGMETRICS international conference on Measurement and modelingof computer systems, ACM Press, 2003, pp. 298–299.

[15] A. L. Chiu, E. Modiano, Traffic grooming algorithms for reducing electronicmultiplexing costs in WDM ring networks, Journal of Lightwave Technology18(1) (2000) 2–12.

[16] R. Cole, K. Ost, S. Schirra, Edge-coloring bipartite multigraphs in O(E log D)time, Combinatorica 21 (1) (2001) 5–12.

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