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Grid spanners with low forwarding index for energy

ecient networks

Frédéric Giroire, Stéphane Pérennes, Issam Tahiri

To cite this version:

Frédéric Giroire, Stéphane Pérennes, Issam Tahiri. Grid spanners with low forwarding index for

energy ecient networks. [Research Report] RR-8643, INRIA Sophia Antipolis; INRIA. 2014. <hal-

01095179>

ISSN 0249-6399 ISRN INRIA/RR--8643--FR+ENG

RESEARCH

REPORT

N° 8643

December 2014

Project-Team COATI

Grid spanners with low

forwarding index for

energy efﬁcient networks

Frédéric Giroire, Stéphane Pérennes , Issam Tahiri

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

Grid spanners with low forwarding index for energy

efﬁcient networks∗

Frédéric Giroire†, Stéphane Pérennes †, Issam Tahiri‡

Project-Team COATI

Research Report n° 8643 — December 2014 — 11 pages

Abstract: A routing Rof a connected graph Gis a collection that contains simple paths connecting every

ordered pair of vertices in G. The edge-forwarding index with respect to R(or simply the forwarding index

with respect to R)π(G, R)of Gis the maximum number of paths in Rpassing through any edge of G. The

forwarding index π(G)of Gis the minimum π(G, R)over all routings R’s of G. This parameter has been

studied for different graph classes [14], [1], [7], [5]. Motivated by energy efﬁciency, we look, for different

numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.

Key-words: spanning subgraphs, forwarding index, energy saving, routing, grid

∗This work has been partially supported by ANR project Stint under reference ANR-13-BS02- 0007, ANR program Invest-

ments for the Future under reference ANR-11-LABX-0031-01, ANR VISE, ANR Skyﬂow, CNRS-FUNCAP project GAIATO, the

associated Inria team AlDyNet, the project ECOS-Sud Chile and Paca Region.

†CNRS, Laboratoire I3S, UMR 7172, UNS, CNRS, Inria, COATI, 06900 Sophia Antipolis

‡Université de Bordeaux, Institut de Mathématiques, UMR 5251, CNRS, Inria, 33405 Talence

Graphes couvrants de la grille avec un indice de transmission faible

pour des réseaux efﬁcaces en énergie

Résumé : Un routage Rd’un graphe connexe Gest un ensemble de chemins simples connectant toutes

les paires ordonnées de sommets de G. L’indice de transmission arête relatif à R(ou simplement l’indice

de transmission relatif à R)π(G, R)de Gest le nombre maximum de chemins de Rqui utilisent une arête.

L’indice de transmission de G,π(G), est le π(G, R)minimum sur tous les routages Rde G. Ce paramètre

a été étudié pour différentes classes de graphes [14], [1], [7], [5]. Motivés par l’efﬁcacité énergétique des

réseaux, nous étudions, pour un nombre d’aêtes donné, les meilleurs sous-graphes couvrants de la grille

carré, c’est-à-dire ceux avec un indice de transmission faible.

Mots-clés : sous-graphes couvrants, indice de transmission, efﬁcacité énergétique, routage, grille

Grid spanners with low forwarding index 3

1 Introduction

A routing Rof a given connected graph Gof order Nis a collection of N(N−1) simple paths connecting

every ordered pair of vertices of G. The routing Rinduces on every edge eaload that is the number of

paths going through e. The edge-forwarding index (or simply the forwarding index) π(G, R)of Gwith

respect to Ris the maximum number of paths in Rpassing through any edge of G. It corresponds to the

maximum load over all edges of the graph when Ris used. Therefore, it is important to ﬁnd routings

minimizing this index. The forwarding index π(G)of Gis the minimum π(G, R)over all routings R’s

of G. This parameter has been studied for different graph classes (examples can be found in [1], [7], [5])

and this survey [14] gives a global view on the known results.

We call a connected spanning subgraph of a graph G, a spanner of G. More precisely, it is a connected

subgraph that has the same set of vertices as G. Our goal is to ﬁnd, for a given bound on the number

of edges, the best spanner of G, namely the one with the minimum forwarding index. The problem can

also be viewed as: for a given bound Uon the forwarding index, ﬁnd a spanner Fof Gwith minimum

number of edges such that π(F)≤U.

Knowing how to solve this problem is very interesting in practice for network operators willing to

reduce the energy consumed by their networks. In fact, most of the network links consume a constant

energy independently of the amount of trafﬁc they are ﬂowing [2], [13]. Therefore, it was proposed to

reduce the energy used by the network links by turning some of them off, or more conveniently, putting

them into an idle mode. Outside the rush hours, several studies [3], [4], [12], [9], [10], [11], show

that a good choice of the links to turn off can lead to signiﬁcant energy savings, while keeping the same

communication quality. In the case where the throughputs from every node to every other node are of

the same order, and where the capacities also lie in the same small range, a good choice of those links is

reduced to the problem of ﬁnding spanners of the network with low forwarding indices.

In this paper, we consider the case in which the initial graph is a square grid. We consider the

asymptotic case with nlarge. We have two main contributions.

On one side, we know that the forwarding index of the n×ngrid Gnis n3

2, see Proposition 1 [8]. Gn

has 2(n−1)2∼2n2edges. An important remark is that the load of the edges is lower in the corner than

in the middle of the grid. Using that, we show that we can build spanners of Gnwith much fewer edges

(only 13/18 ≈72% of the edges) and the same forwarding indices as Gn. We show that the proposed

spanners are close to optimum in the sense that we prove that it is impossible to build spanners with fewer

than 4/3n2edges (66% of the edges).

On the other side, the smallest possible spanner of the n×ngrid Gnis a spanning tree. The for-

warding index of the best spanning tree is asymptotically 3n4

8, see Proposition 2 [8]. When we add edges

and consider spanners with a larger number of edges, the load on the edges decreases, and so does the

forwarding index. In this paper, we study how the forwarding index decreases, when we increase the

number of edges.

The following table summarizes our results:

Spanning tree Spanners Grid

forwarding index 3

8n41

2an4(2 ≤a < n)1

2n31

2n3

lower bound on the number of edges n2−1'n2+4

9(0.1a)212

9n2

number of edges in the constructions n2−1n2+4

9a213

9n22n2

Proposition 1 [8] The forwarding index of Gnis asymptotically n3

2.

Proposition 2 [8] For n≥3, The spanning tree of Gnwith the minimum forwarding index is a tree with

centroid of degree 4 and 4 branches of almost equal sizes. its forwarding index is asymptotically 3n4

8.

RR n° 8643

4Giroire & Pérennes & Tahiri

2 Spanners with the forwarding index of the grid, n3

2, but much

fewer edges

In this section, we ﬁrst show that a spanner with the forwarding index of the grid has at least 4n2

3=12n2

9

edges. We then provide spanners with 13n2

9edges. But, before, we present some notations that will be

used throughout the paper.

Notations We note by Gn= (Vn, En)the n×nsquare grid, where Vnis the set of Vertices and Enis

the set of edges. A square grid can always be seen as nrows intersecting ncolumns. We name v(r, c)the

vertex at the intersection of row r∈[n]with column c∈[n], where [n]denotes the interval of the integer

numbers between 1and n. An edge joining v(r, c)to v(r, c + 1) is named eh(r, c)and an edge joining

v(r, c)to v(r+ 1, c)is named ev(r, c).

Proposition 3 For any Fspanner of Gnsuch that π(F)≤n3

2,Fmust have, asymptotically, at least 4n2

3

edges.

Proof: Consider Fa spanner of Gnand let Rbe a routing of Fsuch that π(F, R)≤n3

2. For an integer

l∈[n], we call load on line l, the sum of the load on the edges ev(l, j )∈E(F), for j∈[n]. The load

on line lis 2l(n−l)n2as there are ln vertices over line land (n−l)nvertices below. If Fhas n−xl

edges on line l, there exists at least one of these edges with load at least 2l(n−l)n2

n−xl. As π(F , R)≤n3

2, we

should have 2l(n−l)n2

n−xl≤n3

2.

That is

n−xl≥4l(n−l)

n.

Thus, Fshould have at least Pn

l=1

4l(n−l)

nvertical edges. The same argument independently holds for

the horizontal edges. Hence, a spanner of the grid, with load lower than n3

2on all edges, has at least

2

n

X

l=1

4l(n−l)

n≈4n2

3edges.

Theorem 1 There exists Fna spanner of Gnsuch that π(Fn)∼n3

2and its number of edges is asympot-

ically equal to 13n2

9.

Proof: Let us ﬁrst explain the intuition behind the construction of the spanner of the grid, Fn. We know

from the proof of Proposition 3 the ratio of edges needed in every row or column in order to satisfy the

lower bound. We cut the grid into small squares. Then, according to the position of the square, we use

only the number of needed horizontal edges and vertical edges in each square according to the lower

bound. It turns out that adding only few edges to ensure the connectivity is enough to get a spanner Fn

with a routing Rsuch that π(Fn, R)∼n3

2.

Construction of Fn.Let kbe an integer number such that 1≤k≤n. We divide Gninto small square

grids of size k×k. We do so by partitioning vertices of Gninto (n

k)2sets S(i,j)with i∈[n

k]and j∈[n

k]:

S(i,j)={v(r, c)∈Vn;i−1<r

k≤i, j −1<c

k≤j}. We call a vertex in S(i,j)that has a neighbour in

Gnoutside S(i,j)a border vertex.

Inria

Grid spanners with low forwarding index 5

Figure 1: Construction of the spanner Fnof Theorem 1, for n= 72, and an example of path of the routing

Rof Fn(from the yellow vertex to the pink vertex).

Let us now describe a spanner Fnthat veriﬁes our theorem. An example of it is shown in Figure 1

in the case of n=k2= 72. Let tbe the function deﬁned on integers by t(x) = d4xk(n−xk)k/n2e.

It represents the number of needed columns (respectively rows) for a square that is on the x-th position

horizontally (respectively vertically). We build Fnstarting from a subgraph that has all vertices of Gn

and no edges. For every S(i,j),i, j ∈[n

k], we choose edges to connect vertices in S(i,j)in the following

way:

- we add to Fnall edges ev(r, c)such that (rmod k)∈ {1, . . . , t(i)}(red edges in Figure 1) and

RR n° 8643

6Giroire & Pérennes & Tahiri

- all edges eh(r, c)such that (cmod k)∈ {1, . . . , t(j)}(blue edges in Figure 1);

- then we add to Fnsimple paths just to connect the remaining independent vertices (green edges in

Figure 1).

- We then add all edges that do not have both endpoints in the same set S(i,j)(black edges in 1). We

show in the following that adding all of them is not strictly necessary.

Description of the routing R.We now give a routing of the spanner Fn,R. For every ordered pair

of vertices (v(ra, ca), v(rb, cb)) of Vn, we describe the path connecting v(ra, ca)to v(rb, cb)in R. We

distinguish two types of ordered pairs of vertices:

• Type-1 pairs: dra/ke=drb/keor dca/ke=dcb/ke. Notice that this type includes ordered pairs

with vertices that belongs to the same set S(i,j).

• Type-2 pairs: All the ordered pairs that do not belong to the ﬁrst type.

For the Type-1 pairs, Ruses the shortest path routing. For Type-2 pairs, Ruses a three-segment path.

An example of such path is shown in Figure 1. We name ia=dra/ke,ib=drb/ke,ja=dca/keand

jb=dcb/ke:

• Step-1: Let im= min(ia, ib, n/k −ia, n/k −ib)and jm= min(ja, jb, n/k −ja, n/k −jb). The

ﬁrst segment is the shortest path from v(ra, ca)to one of the two border vertices of S(ia,ja)that are

on row k(ia−1) + t(jm). Among the two vertices, we choose v(rx, cx), which has the smallest

distance to S(ia,jb)(as the ﬁrst black vertex on the route in Figure 1).

• Step-2: Similarly, two border vertices of S(ib,jb)are on column k(ib−1) + t(im). Among these

two vertices, v(ry, cy)is the one that has the smallest distance to S(ia,jb)(as the third black vertex

on the route in Figure 1). The second segment will be linking v(rx, cx)to v(ry, cy)by using the

path [v(rx, cx)v(rx, cy)v(ry−cy)], which is the shortest path from v(rx, cx)to v(rx, cy)composed

of the two direct paths [v(rx, cx)v(rx, cy)], following row rx, and [v(rx, cy)v(ry−cy)], following

column cy.

• Step-3: The third and last segment will be the shortest path from v(ry, cy)to v(rb, cb).

Note that kmay be an arbitrary integer between 1and n. We choose a ksuch that 1k√n. For

instance, we may choose k=n1/3.

Number of edges of Fn. Let us compute the number of edges in the spanner, Fn. First the edges used

in the subgraph induced by S(i,j)are all the edges on a row from 1to t(i), all edges on a column from

1to t(j), to which we add the edges that connect the rest of vertices through a spanning tree. Hence the

number of edges in S(i,j)is:

≈t(i)·k+t(j)·k+ (k−t(i))(k−t(j))

≈k2+t(i)t(j)(1)

≈k2(1 + 16ijk2(n−ik)(n−jk)

n4)

≈k2(1 + 16ijk2

n2+16i2j2k4

n4−16i2jk3

n3−16ij2k3

n3)

Inria

Grid spanners with low forwarding index 7

The sum of those edges considering all the subsets S(i,j)(with i∈[n

k]and j∈[n

k]) is :

≈k2

n/k

X

i=1

n/k

X

j=1

(1 + 16ijk2

n2+16i2j2k4

n4−16i2jk3

n3−16ij2k3

n3)

≈k2[n2

k2+16k2

n2(X

i

i)2+16k4

n4(X

i

i2)2−2·16k3

n3(X

i

i2)(X

i

i)]

≈k2[n2

k2+16k2

n2·n4

4k4+16k4

n4·n6

9k6−2·16k3

n3·n5

6k5]

≈k2[n2

k2+4n2

k2+16n2

9k2−32n2

6k2]

=13

9n2+o(n2)

The number of the remaining edges is ≈2n2

k=o(n2), as k1. Therefore, as stated in the theorem,

the number edges of Fnis asymptotically equal to 13n2

9+o(n2).

Load of the edges of Fn.Lets now verify that every edge has an asymptotic load which is not greater

than n3

2+o(n3). Consider an edge eh(r, c)whose incident points are in S(i,j). The number of Type-1

pairs that may use eh(r, c)is bounded by the number of pairs having one endpoint in S(i,ja)and S(i,jb)

for some ja, jb∈[n

k]and those having one end point in S(ia,j)and S(ib,j)for some ia, ib∈[n

k]. The

number of these pairs is bounded by 2k2n2=o(n3)(as k2=o(n)).

Then, for Type-2 pairs, we can start by the load induced by the segments of paths described previously

in step-1 and step-3. This load is clearly bounded by the number of pairs having one endpoint inside S(i,j)

and another endpoint outside S(i,j). The number of these pairs is bounded by: 2k2(n−k)2=o(n3)(as

k2=o(n)). For Step-2, as the construction of the spanner Fnhas the needed density of edges, the load

is kept below n3

2+o(n3). The same argument holds for the black edges between two adjacent subsets

S(ia,ja)and S(ib,jb). This ends the proof.

3 Spanners with forwarding indices in the range ]n3

2,

3n4

8[and Lower

bounds

We ﬁrst provide spanners with forwarding indices in the range ]n3

2,3n4

8[in Proposition 4. We then prove

that these spanners have a number of edges of the optimum order, see Proposition 5

3.1 Spanners’ constructions

Proposition 4 Let abe an integer such that, 2≤a≤n. There exists a spanner Fn(a)of Gnwith

asymptotically n2+4

9a2edges and π(Fn(a)) ≤n4

2a.

Proof: We build a spanner of Gn,Fn(a), in the following way. We divide the grid into a2sectors.

A point is in Sector (i, j)if its coordonnates in the grid (x,y) are such that n

ai≤x < n

a(i+ 1) and

n

aj≤y < n

a(j+ 1). Each of these sectors has (n/a)2vertices. We call center of the sector (i, j)the

vertex ((i+ 1/2)n/a, (j+ 1/2)n/a). We consider the a×asubgrid linking all the sectors’ centers. We

then connect all the remaining vertices of a sector to its center with a spanning tree. This way, we get

Fn(a). Figure 2 provide a sketch of the construction of the spanner.

RR n° 8643

8Giroire & Pérennes & Tahiri

Figure 2: Spanner of Proposition 4 for n= 21 and a= 3. Edges of the a×agrid are in bold. Edges that

are not in a spanning tree of Gnare in red. Sectors with (n/a)2= 72vertices are separated by dashed

gray lines.

We now build a routing Rfor Fn(a). The demand between two vertices of the same sector are routed

on the tree spanning their sector using the unique shortest path between them. The demand between two

vertices of different sector is ﬁrst routed to their centers, and then is routed in the a×agrid.

Let us compute the load of the routing R. We ﬁrst consider the edges of the a×asubgrid. We know

that an a×agrid has a routing with load a3/2(Proposition 1). Thus, we know that it also has a w-routing

of load wa3/2. Each vertex of the a×agrid receives the load of the (n/a)2vertices connected to it.

Thus, we take w= (n/a)2and we obtain a w-routing of the a×agrid of load a3

2(n

a)4=n4

2a.

We then consider an edge that does not belong to the a×agrid. The only paths that can use this

edge are paths going from any vertex of the grid to a vertex of its sector. Thus, its load is smaller that

(n/a)2n2=n3

a2. This load is smaller than the maximum load on the a×agrid as soon as a2≥2awhich

means as soon as a≥2.

Therefore π(Fn(a), R) = n4

2a.

Let us now consider the number of edges of the spanner Fn(a). The number of edges necessary to

connect all the nodes is n2−1. If we choose well these edges, we just have to add a2edges to obtain the

a×agrid (see Figure 2, additional edges are in red). Fn(a)thus has n2+a2edges. We can improve the

Inria

Grid spanners with low forwarding index 9

spanner by using the results of Section 2. In Theorem 1, we show that we can ﬁnd a spanner of an a×a

subgrid with 13

9a2edges and a routing R0with the same load as a full grid with 2a2edges. By doing so,

we get a new spanner Fn(a), with n2+4

9a2edges and π(Fn, R0) = n4

2a.

We can rewrite the result of Proposition 4 to point out the impact of additional edges in general

(Corollary 1) and when we start from a spanning tree (Corollary 2).

Corollary 1 There exist:

- A spanner of Gnwith n2+p2edges, and an asymptotic forwarding index of n4

3p'0.33n4

p;

- A spanner of Gnwith n2+pedges, and an asymptotic forwarding index of n4

3√p'0.33 n4

√p.

Proof: Direct by Proposition 4 setting p2=4

9a2or p=4

9a2.

Corollary 2 There exists a spanner of forwarding index 1

α

3n4

8, that is a factor αless than the one of the

optimum spanning tree, while using 64

81 α2'0.79α2additional edges compared to a spanning tree.

Proof: Recall that an optimum spanning tree has forwarding index 3n4

8, see Proposition 2. Dividing it

by αmeans getting the forwarding index 3n4

8α=n4

2(4α/3) . This is achieved by the spanner Fn(a), with

a= 4α/3. The spanner has an additional number of edges compared to the spanning tree equal to

4

9(4α/3)2'0.79α2.

3.2 Lower bounds

Proposition 5 There exist no spanners of Gnwith n2+p2edges and a forwarding index less than

1

9√12

n4

p'0.032n4

p.

Proof: Let us consider a spanner of Gnthat has n2+p2edges. We build a multigraph in the following

way. We start by assigning to every node a weight of 1. Then, while there is still a vertex with degree 1

or 2, we delete this vertex and the edges connecting it to the graph and divide its weight evenly among

its neighbors; in case the removed vertex was of degree 2, we also connect the two neighbors afterwards.

At the end of this process, we get a multigraph Hsuch that the number of its vertices N0and the number

of its edges M0are related by the following equation: N0+p2=M0. Indeed, every time a vertex is

removed in the process leading to H, the number of edges is decreased by 1. Since all the vertices in H

have a degree strictly greater than 2, we have 3

2N0≤M0. This implies with the previous equation that

3

2N0≤N0+p2. Hence, we have N0≤2p2. Notice that the total weight is equal to n2. We now apply

the weighted version of the planar separator theorem [6] on H: there exists a partition of the vertices of

Hinto three subsets A,S, and B, such that each of Aand Bhas at most a weight 2n2/3,Shas less

than √6p2p2vertices (The original graph is of a bounded degree 4.) and there are no edges with one

endpoint in Aand another endpoint in B. This directly gives an edge cut of the original graph which has

less than 2√6p2p2edges and which partitions the original graph’s vertices into two subsets of size at

most 2n2/3. Therefore, any routing of this spanner will induce, at least on one edge of the cut, a load that

is greater than:

1

3n2·2

3n2·1

2√6p2p2=1

9√12

n4

p'0.032n4

p.

RR n° 8643

10 Giroire & Pérennes & Tahiri

4 Conclusion

We succeeded at providing spanners of the n×ngrid with a small number of edges for a given forwarding

index. Such spanners are important for energy efﬁcient networks in which the trafﬁc has to be routed in

the network while using a minimum number of equipments. The unused equipments are then turned off

to save energy. We leave as open two problems.

We propose spanners with a number of edges of optimum order for a forwarding index. More pre-

cisely, we have provided spanners of the grid with n2+4

9a2edges and forwarding indices 1

2an4(2 ≤a <

n). We proved that it is impossible to have spanners with the same FI and fewer than 'n2+4

9(0.1a)2

edges. It would be very nice to succeed in ﬁlling the gap between the lower bounds and the constructions.

Similarly, we describe spanners with 13/9n2edges and with the same forwarding index of the full

grid Gn. We proved that spanners with such a forwarding index should have at least 12/9n2edges.

Would it be possible to ﬁnd spanners with such a number of edges?

Last, we focused on a speciﬁc network in this work, the square grid. We are also interested by more

general graphs. In particular, the arguments to derive lower bounds can be used for more general planar

graphs with bounded degrees. It would be very interesting to ﬁnd results and constructions for other

families of planar graphs.

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RR n° 8643

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

2004 route des Lucioles - BP 93

06902 Sophia Antipolis Cedex

Publisher

Inria

Domaine de Voluceau - Rocquencourt

BP 105 - 78153 Le Chesnay Cedex

inria.fr

ISSN 0249-6399