Scheduling an Independent Set

We assume that we are given an independent set of links and we try to schedule it (in the SINR model) in a small number of subsets. The question of relationship between independent sets and SINR schedules is considered in deeper details in Chapter6. Here we use the ideas of [Hal09] for designing the algorithm (see Algorithm 5.2). In the pseudocode presented below, ∆ denotes the ratio of the longest link length and the shortest link length inS,l_min denotes the shortest link length in S and n=|S|.

Algorithm 5.2 Scheduling an Independent Set.

1. Input: a γ-independent set of linksS withγ ≥3.

2. SetW ← ∅. //A collection of sets.

3. LetS =Sdlog ∆e

t=1 Q_t, whereQ_t={i∈S : 2^t−1 ≤l_i/l_min <2^t}.

4. SetB_t←S∞

k=0Qt+kd2 log 2nefort= 1,2, . . . ,d2 log 2ne.

5. Fort= 1,2, . . . ,d2 log 2ne, do:

5.1. Take empty setsS_t¹, S_t², . . . in this fixed order.

5.2. Sort the links of Bt in an increasing order of length: i1, i2, . . . , i_|Bt|. 5.3. Fork= 1,2, . . . ,|B_t|, do:

5.3.1. Let f be the smallest index such that S_t^f does not contain links j such thatl_j ≥n²l_ik and j and i_k aren^1/α-conflicting.

5.3.2. Set S_t^f ←S_t^f ∪ {i_k}.

5.4. Set W ←W ∪ {S_t¹, S_t², . . .}.

6. If any set S_t^f from W is not SINR feasible, split it into SINR feasible subsets T₁, T₂, . . . and setW ←(W \ {S_t^f})∪ {T₁, T₂, . . .}.

7. Output: the schedule W.

The first four steps of the algorithm constitute the preprocessing stage. The purpose of this stage is to splitS into a number of subsetsB_tthat have certain desirable properties (see Proposition5.8). After preprocessing, each setB_t is passed to the loop at step5.3., which is a first-fit procedure. This is the core of the algorithm and produces an “almost feasible” schedule W, which is then refined to a feasible schedule. Below we present a detailed analysis of the algorithm.

Let us start with the properties of the sets Bt. Proposition 5.8. Sd2 log 2ne

t=1 B_t = S, and for each set B_t and for each pair of links i, j∈B_t with l_i≥l_j, eitherl_i <2l_j or l_i≥n²l_j.

Proof. The first property follows from the fact that Sd2 log 2ne

t=1 B_t contains all the sets Q_k. Note also that B_t is a union of sets Q_k, such that if k 6= k⁰ and Q_k, Q_k⁰ ⊆ B_t,

|k−k⁰| ≥ d2 log 2ne, which implies the second property.

Definition 5.9. For each linki∈S, letφ_γ(S, i) denote the number of linksj ∈S such thatl_j ≥γ²l_i andi andj are γ^1/α-conflicting, whereγ ≥1.

Lemma 5.10. Suppose that the mean power scheme is used for a γ-independent set of linksT with γ >1 and (γ−1)⁻¹ ∈O(1). Then for all i∈T, φ_n(T, i)∈O(1).

Proof. Let us fix a linki∈S and letS⁰⊆Sbe the set of links such that for eachj ∈S⁰, l_j ≥ n²l_i and j and i are n^1/α-conflicting. The conflicting condition can be stated as follows:

min{d_ji, d_ij} ≤n^1/αp l_il_j.

Thus, we can apply Lemma5.3 with the following values of parameters:

T =S⁰, i0 =i, γ, ρ=n², δ=n^1/α, which will give

φ_n(S, i) =|S⁰| ∈O





n^1/α (γ−1)n

!2m

+ 1



=O(1), taking into account that α >1.

We will also need the following theorem from [Hal09] for our analysis.

Definition 5.11. A set of links S is called a set of similar length links if for each pair i, j∈S,li ≤2lj and lj ≤2li.

Theorem 5.12. [Hal09] Suppose that α > m. Let S be a set of similar length links that is γ-independent with respect to some power scheme, where γ > 2. Then S is a Ω(γ^α)-feasible set w.r.t. U.

Remark 1. Note that ifS is a set of similar length links that is SINR-feasible w.r.t. one of power schemesM,L,U, it can be split into anO(1) number of SINR-feasible subsets w.r.t. anyone of those power schemes, using Algorithm 2.4. Thus, we can replaceU by L orMin the formulation of the theorem above.

Remark 2. If one replaces the requirement about γ-independence w.r.t. any power scheme by a requirement ofγ-independence w.r.t. M, then we can weaken the assump-tion on γ: γ >1 and (γ−1)⁻¹ ∈O(1) instead ofγ >2 (can be proven along the same lines as the original proof of Theorem5.12).

Below is the main lemma of this section.

Lemma 5.13. Suppose that α > mand let S be a γ-independent set of links w.r.t. M, where γ ≥3. Then Algorithm 5.2schedules S into O(logn) subsets, wheren=|S|.

Proof. The algorithm schedules each set B_t separately, and the number of setsB_t is in O(logn). It remains to show that each such set is scheduled into O(1) subsets. Let us consider, w.l.o.g., B1. We need to bound the number of subsets S₁¹, S₁², . . . in the end of the main loop. Note that at the time of consideration of a link i_k ∈ B₁, only the links from B1[i⁺_k] (i.e. links longer than i_k) have been assigned to subsets. The algorithm finds the smallest index f such that S₁^f does not contain links j ∈ B₁ such that lj ≥ n²lik and j and i_k are n^1/α-conflicting. Note that the number of such links is φ_n(B1, i_k). Recall that B1 is γ-independent, so by Lemma 5.10, φ_n(B1, i_k) ∈ O(1).

Thus, the indexf will be inO(1) at each step, which means that the number of subsets S₁^f is inO(1). Now let us examine the feasibility of the setsS₁^f. Consider, w.l.o.g.,S₁¹and leti∈S₁¹. By construction ofB₁ and Proposition5.8,S₁¹ can be split into the following two subsets: T1 = Qt∩S₁¹ where t is the unique index such that i ∈ Qt ⊆ B1, and T₂ ={j ∈S₁¹ :l_j ≥n²l_i orl_i ≥n²l_j}. By Theorem 5.12,ISRM(T₁, i)∈ O(1). On the other hand, by construction of S₁¹ (see step 5.(3.)1.), for eachj ∈T₂, ISRM(j, i) < _n¹; hence, ISRM(T2, i) < |T₂|/n ≤ 1. Thus, we have that each of the sets S_t^f is O(1)-feasible, which means that at the refinement step it can be split into at most O(1) feasible subsets. It follows that the number of feasible subsets on the output of the algorithm is inO(logn).

Theorem 5.14. Algorithm 5.1 is an O(logn)-approximation algorithm for scheduling w.r.t. mean power scheme.

Proof. By Theorem5.4, the application of Algorithm2.2yields a coloring ofG^γ_M(Γ) with O(χ(G^γ_M(Γ))) colors. We know from Lemma 2.16 that χ(G^γ_M(Γ)) ∈ O(OP T SM(Γ)).

Applying also Lemma 5.13, we get that the length of the schedule on the output of Algorithm 5.1is in O(χ(G^γ_M(Γ)) logn)⊆O(OP T SM(Γ) logn).

5.2.2 Comparing the Mean Power Scheme with Optimal Power As-signments

Recall that Theorem5.7shows that in the Conflict Graph model, the mean power scheme approximates the schedules of the best possible power assignments within a factor in O(log log ∆). Combining this result with Algorithm5.2allows to obtain a similar result for the SINR model.

The theorem presented below was first proven in [Hal12] using a slightly different method.

It can also be obtained from the results of [HHMW13]. We will present a stronger result in Chapter 7, similar to the one obtained in [HHMW13], using a stronger result on the relationship between the SINR model and the Conflict Graph model.

Theorem 5.15. [Hal12] Let Γ be a set of links in a doubling metric space with α > m.

Let P be any power assignment forΓ and let n=|Γ|. Then, OP T SM(Γ)∈O(OP T S_P(Γ) lognlog log ∆), and this approximation can be obtained in polynomial time.

Proof. First we construct the graphG²_M(Γ) and apply Algorithm2.2, getting a constant factor approximation of the best coloring. Recall that Theorem 5.7 implies that the number of colors used is as follows:

χ(G²_M(Γ))∈O(χ(G²_P(Γ)) log log ∆)⊆O(OP T SP(Γ) log log ∆).

Let S be any of the color classes obtained. Since S is a 2-independent set w.r.t. mean power scheme, it can be split into anO(1) number of 3-independent subsets using The-orem 5.5. Then we can apply Algorithm 5.2 and obtain a schedule of length O(logn) for each of those subsets which gives a schedule of length O(logn) forS. Thus, we can find a schedule of length in O(OP T S_P(Γ) lognlog log ∆) for Γ, using the mean power scheme.

Chapter 6

The Relationship Between the

SINR and Conflict Graph Models

In this chapter we address the following question: how well does the conflict graph model approximate the SINR model, and what are the circumstances that decide the approximation level. We study the uniform, linear and mean power schemes in this context. The study is mainly concentrated around the capacity problem, from which results about the scheduling problem follow as well (see Chapter7).

First we consider the relationship between the two models in the case when the links are located in a doubling metric space with dimension m < α. We show that if the uniform or linear power schemes are used, the difference factor between the capacities obtained in the two models is in O(log ∆), and this bound is tight. In terms of n, the difference can be up to a factor Θ(n). On the other hand, when the mean power scheme is used, the capacity in the two models differs by only a factor in O(1). This result brings the two models close in the case of the mean power scheme, which means that we can use bounds and algorithms developed for the Conflict Graph model in order to obtain bounds and algorithms for the SINR model. As an example, we apply the maximum weighted independent set algorithm for graphs to the Conflict Graph model in order to get an O(logn)-approximation algorithm for the weighted capacity problem with respect to the mean power scheme in the SINR model.

We also consider the relationship between the two models for the case when the links are located in a general metric space. In this case it is easily shown that there are instances of the capacity problem with links havingequal lengths, for which the capacity in the Conflict Graph model is in Θ(n/logn) when using e.g. the uniform power scheme, whereas in the SINR model the capacity is in O(1) even when using the best possible

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power assignment. This means that the doubling property has a decisive role in the relationship between the two models.

6.1 Sets of Similar Length Links

Recall Theorem 5.12 which shows that, given a γ-independent set of links of similar length (see Definition 5.11), where γ ∈ O(1), it can be partitioned into O(1) SINR-feasible subsets. Here γ-independence and SINR-feasibility is meant with respect to any of the three power schemes, U,M,L, considered in this chapter. In fact, the claim holds for any power scheme P such that changing the length of a link iby a factor of O(1) results in a change of P(i) by a factor of O(1), independent of i. It follows that γ-independence is a good approximation for SINR-feasibility, i.e.

OP T C_P(Γ)∈Θ(τ(G^γ_P(Γ))) and OP T S_P(Γ)∈Θ(χ(G^γ_P(Γ)))

when Γ is a set of similar length links (see Definition5.11) andP is as above. A natural question is: how do these relations scale when the link lengths are not close to each other? The following simple corollary of Theorem5.12provides an upper bound for the gap between the two models, when the network is located in a doubling metric space withα > m.

Theorem 6.1. Let Γ be a set of arbitrary links located in a doubling metric space with m < α, γ >2 and P ∈ {U,M,L}. Then

OP T CP(Γ) = Ω(τ(G^γ_P(Γ))/log ∆) and OP T SP(Γ) =O(log ∆χ(G^γ_P(Γ))), where ∆ = max_i,j∈Γ^l_lⁱ

j denotes the ratio of the lengths of the longest and the shortest links.

Proof. Let S⊆Γ be aγ-independent set. S can be split intoO(log ∆) subsets, each of which is a set of similar length links. Indeed, if l₁ is the length of the shortest link, we can splitS as follows:

S=

dlog ∆e+1

[

t=1

{i∈L: 2^t−1 ≤ l_i

l₁ <2^t}.

According to Theorem 5.12, each of these subsets can be scheduled into O(1) number of feasible subsets, which gives O(log ∆) subsets in total. It follows that each coloring of G^γ_P(Γ) can be transformed into a SINR-feasible schedule with O(χ(G^γ_P(Γ)) log ∆) feasible subsets.

Note that, in general, the parameter ∆ does not depend on the number of links, so it can theoretically be arbitrarily large with respect to n=|Γ|. In the following two sections we discuss if this upper bound can be improved for the power schemes that we consider.