Limitations of current wireless link scheduling algorithms☆
Introduction
The task of the MAC layer in TDMA-based (time-division multiple access) wireless networks is to determine which nodes can communicate in which time-frequency slot. A scheduler aims to optimize criteria involving throughput and fairness. This requires obtaining effective spatial reuse while satisfying the interference constraints. We treat the fundamental scheduling problem of partitioning a given set of communication links into the fewest possible feasible sets or slots.
We adopt the physical or SINR model of communication, where signal decays (loses power) as it travels and a transmission is successful if its strength at the receiver exceeds the accumulated signal strength of interfering transmission by a sufficient (technology determined) factor. Considerable progress has been made in recent years in elucidating essential algorithmic properties of the SINR model, involving various aspects of networks such as connectivity scheduling [1], [2], maximum feasible set of links [3], [4], [5], SINR diagrams [6], distributed scheduling [7], [8], distributed broadcast [9], [10], and others. Early work on the scheduling problem includes [11], [12], [13]. Gupta and Kumar [14] proposed the geometric version of SINR and initiated average-case analysis of network capacity known as scaling laws. NP-completeness has been shown for scheduling with different forms of power control: none [15], limited [16], and unbounded [17]. Moscibroda and Wattenhofer [1] initiated worst-case analysis in the SINR model.
Although the standard analytic assumption that signal decays polynomially with the distance traveled is far from realistic [18], [19], it has been shown that results obtained with that assumption can be translated to the setting of arbitrary measured signal decay [20], [21], as well as to Rayleigh fading model – a stochastic fading model [22].
The setting of transmission powers of the links – power control – is an integral part of scheduling, as the same set of links can be feasible with one power assignment and infeasible with another. The scheduling problem has been considered both for arbitrary power control and for fixed oblivious power assignments, where the power chosen for a link is local and depends only on the link itself. This paper addresses algorithms dealing with both scenarios.
Finding constant-factor approximation to the scheduling problem has proved challenging. However, there are several approaches giving logarithmic approximation.
I. Greedy first-fit algorithms. The links are processed in a non-decreasing order of length (the transmitter-receiver distance) and are assigned to the first slot where they have low mutual interference with already scheduled links [23], [4], [3], [24]. An approximation ratio of can be proved for such algorithms, by first arguing that a similar first-fit algorithm provides constant factor approximation to the capacity problem, where the goal is to select maximal size feasible subset, then extending this result to the scheduling problem via a generic argument. It has also been shown that such algorithms perform well for some randomly generated network instances [25].
II. Randomized Contention Resolution. The links act in rounds , where in each round r, every link transmits independently with probability or stays silent, for a sequence of probability values , , and each link is assigned to the time slot r if its transmission succeeded in round r [8], [26]. This is another approach giving a -approximation (for fixed power assignments) with high probability, under certain choices of probabilities.
III. Partitioning into Length Classes. The strategy is to divide the links into groups of nearly equal length and schedule each group separately. Following this approach, numerous -approximation results have been argued [15], [27], [28], where Δ is the ratio between the longest and shortest link length.
The only known algorithms to achieve non-trivial approximation for scheduling in terms of the number of links n are from the first two families. The only known constant-factor approximation algorithms for scheduling are obtained in the case of the linear power scheme [29], [30]. In recent papers, we have developed a conflict-graph based approach that gives and -approximation algorithms for scheduling with power control and fixed power assignments, respectively, but here too, as in the third family of algorithms above, the approximation factor is only in terms of n [31], [32]. The algorithms from [31], [32] are somewhat similar to first-fit algorithms considered here, but they process the links in the opposite order. This rather unintuitive approach gives better approximation in terms of Δ but no non-trivial approximation in terms of n.
The optimum number of slots for scheduling has also been approximated through interference measures [8], [26], [33]. However, no measure is known to give better than approximation. It is also not evident how to efficiently compute such measures. Many variants of the scheduling problem are known to be NP-hard but we are not aware of any significant inapproximability result.
While for the third family of algorithms it is easy to find examples attaining the approximation ratio , we are not aware of constructions showing that the approximation ratio of of the first and second families cannot be improved. For randomized contention resolution algorithms (the second family), an additive approximation bound has been proven in [26], and it was asked whether another analysis of their algorithm could give approximation ratio smaller than . In fact, for one particular power assignment (the linear power scheme, defined below), it is known that such an algorithm provides additive approximation.
Our results We show that first-fit algorithms (including the algorithms for both fixed power assignments and optimized powers) achieve no better than (in terms of only n) or (in terms of only Δ) approximation, and randomized contention resolution algorithms achieve no better than or approximation, for any sequence of probabilities , These results hold even in one-dimensional networks, where the nodes are embedded in the real line. Our constructions are obtained by modeling sets of links after certain graphs that are hard instances for analogous algorithms for graph coloring.
Note, however, that our constructions do not work for the uniform power assignment (where all links use equal power) and the linear power assignment (defined below). For the latter, constant factor approximation algorithms are known [29], [30]. The case of the uniform power assignment is still open.
These results suggest that new methods are needed for better than logarithmic approximation of scheduling.
Section snippets
Model and definitions
Communication links Consider a set L of n links, labeled by numbers from 1 to n. Each link i represents a unit-demand communication request from a sender to a receiver which are point-size wireless transmitter/receivers located in a metric space with distance function d. We denote by the distance from the sender of link i to the receiver of link j, the length of link i and the minimum distance between two sets and . We also adopt for
Scheduling with fixed power schemes
The first-fit algorithm considered in [23] was originally used for the uniform power scheme, but applies also to other oblivious power schemes [4]. The algorithm is a simple greedy procedure, where one starts with empty slots in a fixed order, then the links are processed in non-decreasing order by length and a link i is assigned to the first slot S such that , for a given constant γ. One may also generalize the acceptance condition with , where is
Lower bounds for randomized contention resolution algorithms
Next, we consider a generalization of the distributed algorithm presented in [8]. In this algorithm, the sender nodes of the links act in synchronous rounds, and each sender node transmits with probability or waits with probability in round i independently from others, where is the same for all links (but may change across the rounds). Once the transmission succeeds in round i, the node is silent in subsequent rounds.
Our construction in this case is modeled after a complete
Conclusions
We demonstrated the limitations of current wireless links scheduling algorithms, by showing that in many cases, they do not give better than logarithmic approximation. A notable exception is the case of uniform power assignment.
Open Problem. Consider the NDFirstFit algorithm for scheduling with the uniform power assignment : Start with empty slots in a fixed order and process the links in non-decreasing order by length, assigning each link i to the first slot S, s.t. . Does
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This work is supported by Icelandic Research Fund grants 120032011, 152679-051, and 174484-051.