In wireless networks, a great challenge is the management of simultaneous transmissions among nodes in an environment which is characterized by real conditions. We are concentrated on the scheduling problem, where the nodes are located in an arbitrary decay space. In this space, the transmission signal may be reduced by the interference of other communication links, the obstacles, the reflections and the shadowing. Thus, we consider two conditions: Firstly, the distances between nodes are not symmetrical. Secondly, each of the nodes has not the same transmission power. These two conditions add a greater degree of difficulty in our study. Therefore, our aim is to seek the fewest number of different time slots needed to schedule all the communication links in such a network. The key point of our study is to ensure the successful transmission of messages in a decay space. Then, we use acknowledgement messages and determine guards to protect the transmissions providing quality of services.
Also in this paper, we are focused on the broadcast problem, where a sender node transmits messages to all the nodes in the network when they follow the symmetry property. In particular, we present an online mechanism in a metric space that each receiver node is activated to get a message from its sender. We consider that each mobile user has a limited battery capacity or equivalently battery-feasibility. In  and , the authors study online mechanism and mainly under budget of an user (agent) in an online procurement market.
We adopt the Signal-to-Interferences-and-Noise-Ratio (SINR) physical model which is based on the physical assumptions that the strength of signals reduces gradually because of the cumulative interference of other communication links. The SINR model recently acquires the attracted study of algorithmic community. Moscibroda and Wattenhofer  initiated a scheduling algorithm in the SINR model in which a set of links is successfully scheduled into polylogarithmic number of slots. In 
, randomized distributed algorithms were proposed for the scheduling problem, where a transmission probability is used, as a parameter which works for short schedules. Jurdzinski et al. presented a randomized algorithm, in which all nodes start the algorithm at the same time, and a randomized algorithm, in which the source node is only actived during the initiation phase of the algorithm. They studied this problem in an uniform network using a communication graph in a metric space with a distance function at most 1. Bodlaender and Halldórsson  used an abstract SINR model in order to solve the capacity problem with uniform power in a decay space. In , the authors presented a randomized multiple-message broadcast protocol.
1.1 Additional Related Work
The study of scheduling (and capacity) problem in the SINR model through algorithmic analysis using oblivious power schemes presented recently in the literature. In these schemes, the power chosen for a link depends only on the link length itself and these can be categorized into three cases: 1) the uniform power; 2) the linear power; and 3) the mean power scheme. The first -approximation algorithm for oblivious power schemes is presented in [7, 6] for the wireless scheduling problem and the weighted capacity problem, where is the ratio of the maximum and minimum link lengths. The result is achieved by the representing of interference by a conflict graph. However, the unweighted Capacity problem admits constant-factor approximation according to . The WCapacity problem admits -approximation according to [6, 7]. The result for the scheduling problem is -approximation. In the case of the grouping of link lengths, the -approximation, is according to [24, 25, 11]. A -approximation for the scheduling problem is presented at [7, 6], which is the best bound.
Note that in , Halldórsson and Wattenhofer prove that the wireless scheduling problem is in APX. More, in , the author present an approximation algorithm for the wireless scheduling problem with ratio . These results hold also for the weighted capacity problem. In , an algorithm for the capacity problem that achieves is proposed. In addition, T. Tonoyan  prove that a maximum feasible subset under mean power scheme is always within a constant factor of subsets feasible under linear or uniform power scheme for the capacity problem.
Following , we study the scheduling problem in a more general space, in an arbitrary decay space. This means that the strength of a transmitted signal of any sender node is vulnerable because of interference of other nodes, obstacles, reflections and shadowing. Therefore, there is a reduction of the strength signal and the receiver may not get the message from its sender. In this paper, we achieve the ideal solution of the scheduling problem using power control through two different efficient algorithms. The main contributions are summarized:
Firstly, we propose the first randomized distributed algorithm in order to control the power of each node and to solve the minimum scheduling problem in a non-uniform network. The algorithm is based on the coloring method in , which assigns probability/color to each node taking part in an implementation. We propose a scheduling and power selection algorithm in a decay space, which is called as SPAIDS. Therefore, we propose an randomized algorithm, where is the number of nodes and is the ratio between the maximum and the minimum power assignment. More details:
We determine a set of probability transmissions to each node in order to achieve transmissions of the messages in the network and we separate it into subsets. The SPAIDS algorithm needs a time in order to assign colors in the nodes because of the separation into feasible subsets.
We consider that the message is successfully received when the sender receives an acknowledgement message from its receiver because the nodes are located in a decay space.
We use guards in order to protect the receiver from interference of other links and to boost the signal. Also, we protect the sending of an acknowledgement message from a receiver. Then, we guarantee the successful transmission and the quality of service.
Secondly, we propose online algorithm in a metric space, which is called as OAMS, in order to control the power of each node and to achieve the deliver of messages to all nodes in the space. We focus on the online broadcast problem that each receiver node is activated at each time step. Each mobile user has a battery, who can store power at most . Also, we assume that there is unknown distribution of nodes in our network. The algorithm assigns probability/color to each node taking part in an implementation. The proposed algorithm is constant-competitive.
1.3 Paper Organization
The rest of this paper is organized as follows: Section 2 describes the system model used in this work and gives some useful definitions. Section 3 presents a conflict graph and its properties in a decay space as well as upper bounds. Section 4 presents the scheduling and power selection algorithm (SPAIDS). Section 5 presents the online algorithm in a metric space (OAMS).
2 System Model and Definitions
In this section, we describe the proposed model of wireless ad-hoc networks, which consists of pairs of nodes. A pair of nodes is denoted as a quasi-link , where is the sender and is the receiver of quasi-link . We consider that quasi-links are the communication links in decay spaces (Section 2.2). The model is characterized by the following components: SINR formula, decay signal among nodes using quasi-metrics and bounded growth properties. We study the case that the power transmission is non-uniform in all the nodes as well as the case that the distances among nodes are not symmetrical. Thus, our proposed model is characterized as a realistic model. Moreover, we introduce a new notion of affectance, the Weighted Average Affectance (WAFF).
2.1 System Model
We consider that a wireless network can be represented as a graph , where is the set of nodes and is the set of edges (or quasi-links). Each directed edge is denoted as a communication request from a sender to a receiver in decay spaces. We consider is a set of quasi-links. Each sender transmits packets to its receiver at power multiplied by the gain . The gain represents the distance between sender and receiver, which is denoted as , where is the path-loss exponent and is the quasi-distance (in Section 2.2) among two nodes and .
In our model, we use the SINR interference model and assume that is the noise (constant) at the receiver and is a threshold of SINR. The signal transmission can be successful if and only if for all the senders , where .
2.2 Metric and Decay Spaces
, the nodes of network are embedded in a general metric space. A metric space consists of an ordered pair of, where is a set of nodes and is a distance function. is defined as a metric such that for any , the following holds: (i) symmetry property, (ii) triangle property and (iii) non-negativity property .
On the other hand, a real network has not the symmetry property. In , the authors study their network in a metric space when there is not the symmetry property. This metric space is defined as a decay space or else quasi-metric. A quasi-metric on a set is defined as a function such that for all : (i) , (ii) and (iii) . We denote the quasi-distance of two nodes : , where . Each quasi-link is .
Moreover, we bound the arbitrary growth of space. The bounded growth decay space consists of two properties: (i) Doubling Dimension. This property is the infimum of all numbers such that every ball of radius has at most points of mutual distance at least where is an absolute constant and . Metrics with finite doubling dimensions are said to be doubling. (ii) Independent Dimension. In the decay spaces, the concept of independence-dimension is applied in [2, 3] and is defined as follows: Let be a metric space and . A set is called independent with respect to if for all . The size of the largest independent point set is called the independent-dimension of and denoted by .
2.3 Power Conditions
In addition, we give two conditions for the power assignments: (i) for some constant . (ii) If then and , that large quasi-link in a decay space has small power assignment. While small quasi-link has better power condition in order to transmit a message to the receiver in a decay space.
In this part, we introduce a new notion of affectance. It is defined as a ”Weighted Average Affectance (WAFF)” and depends on the quasi-link lengths, the power assignments and the density bounding properties. Note that the measure of affectance is introduced by  and recently reused by . In [6, 7, 9], the authors study only the quasi-link lengths.
We study the model in decay spaces when the distances between of nodes are not symmetrical. Thus, we have . Also, the nodes have not the same power assignment. The next definition means that if each quasi-link has weighted affectance on a quasi-link then we have the weighted average affectance of a set of quasi-links on a quasi-link , where the nodes are located in a decay metric space. In this paper, the weight of a node is the transmitted probability of node . For simplicity we redefined the as . Then, we have:
Let be a set of quasi-links. We consider a quasi-link . The ”Weighted Average Affectance” of on in a decay space is defined as follows:
where is the affectance of quasi-link on quasi-link using power assignments and in a decay space, respectively. The decay distance is and .
3 Conflict Graph in Decay Space
In this section, we study conflict graphs and their properties in non-uniform wireless networks. Note that conflict graphs are graphs defined over a set of links (quasi-links in decay spaces). Our interest is situated in the case of the non-symmetry property and the non-uniform power assignment. Useful definition is the independence of quasi-links, as it determines the less distance of quasi-links when they are not in conflict. Let be the quasi-link in which sends a message to . Our goal is to seek an upper bound of the WAFF of a set of quasi-links on the given quasi-link .
In this paper, our study is based on non-unit balls (of radius ) because of quasi-links. We divide the set in annuli disks (or n-spheres in distance ) centered at the (or ) of quasi-link . In Figure 1, the concentric disks surrounded around the endpoint of quasi-link are represented. The set consists of equilength subsets . Let be the annulus disk with center and radius for each disk. Each has a number of active nodes, which can cause an affectance on the quasi-link . Each disk has a number of annuli disks and , where is the radius at the small disk and at the large disk. The large disk is at most a factor of the radius .
In the following part, we introduce the concepts of DP-feasible and acknowledgment messages. Also, we seek lower and upper bounds.
A set of quasi-links is called as DP-feasible if the holds for each quasi-link in case that we use a power assignment . The set is feasible if there exists a power assignment for which is DP-feasible. Thus, a set S of quasi-links is feasible if and only if the average weighted affectance satisfies: . A set of quasi-links is called -DP-feasible if it is DP-feasible. The Proposition 1 gives the feasibility of WAFF.
We assume and are the upper and the lower bound of the sum of probabilities of the transmitted nodes, correspondingly. A set of quasi-links S is DP-feasible if and only if .
Initially, we use the WAFF and the fact that the affectance of a set of links on the quasi-link is upper bounded by the value according to . In , the authors use unit balls in uniform networks. In decay spaces, we consider the following: Let disks . For each node , there is a color such that the sum of probabilities of this color, in ball , is at least whp: , where is the ratio of power assignment , is a constant; and is the decay distance.
Then, . ∎
Two quasi-links , in a decay space with are -independent iff and are -independent iff or with probability of transmitting .
Let S be a set of links. We consider power for each link i. A set S is DP-feasible iff . From the feasibility of S holds the next inequality: and therefore the quasi-distance links , where we denote , is defined by or . Then, . ∎
In general decay space, we introduce a lower bound in the WAFF for each node , as follows: A set of quasi-links S is GDP-feasible if and only if , where and is an upper bound of the sum of probabilities of the transmitted nodes in independent dimension D.
In this part, we study the reverse case that a receiver transmits an acknowledgement message to its sender in order to be known that successfully received the message. The transmission of an acknowledgement message needs a proper power assignment. We take into account that the nodes are located in a decay space. We need to study the affectance of quasi-link by quasi-link in the case of acknowledgement transmissions, where and are links in uniform networks. Also, we compare the affectance of acknowledgments with the standard affectance.
Let a quasi-link and its reverse quasi-link for the acknowledgement transmission. We consider that a quasi-link has a power assignment . However, the acknowledgement transmission needs a power in order to arrive at the sender. We consider that the power is defined by .
For all quasi-links of a set , it holds that when the symmetry property is not satisfied and each node has its own power assignment .
The affectance of acknowledgement of quasi-link by quasi-link in a decay space is: . By the Definition 2,
By the triangle inequality, it holds the last inequality: .
However, we want: . ∎
For all quasi-links of a set , it holds that when the symmetry property is not satisfied.
3.3 Upper Bound Graphs
A conflict graph for a set of quasi-links is an upper bound graph, if each independent set in this conflict graph is -feasible using a power assignment. A conflict graph for a set of quasi-links is a lower bound graph, if each -feasible set is an independent set in this conflict graph. Therefore, upper and lower bounds for the scheduling problem are sought. In [6, 7], the authors introduce the initial idea to seek bounds and show that there are -approximation algorithms for Scheduling and WCapacity using oblivious power schemes in a metric space.
In the case of doubling dimension, we seek an upper bound of the WAFF of an set of quasi-links on a given quasi-link . The set consists of equilength subsets . Each has a number of nodes. The set is divided into two subsets and . The subset contains the quasi-links that are closer to sender of quasi-link : , where and . The subset contains the quasi-links that are closer to receiver of quasi-link : , where and .
Let , be a set of 1-independent quasi-links. The quasi-links are -independent, . Then, the WAFF in a decay space is given as follows:
where the transmission probability of nodes in each disk is bounded by and .
The affectance of quasi-link by quasi-link in a decay space is:
Then, the weighted average affectance for each disks is calculated as follows:
Let L be a 1-independent set of links. The quasi-links s.t. and are -independent, . There exists and the path-loss , the power of link i is greater than the power of link j, . Then, the weighted average affectance in a decay space is given by ,where the transmission probability of nodes in each disk is bounded by the parameters and .
4 Scheduling and Power Selection Algorithm in a Decay Space
In this section, we propose a randomized distributed algorithm in order to control the power of each node and to solve the minimum scheduling problem in a non-uniform network.
The algorithm is based on the coloring method in , which assigns probability/color to each node taking part in an implementation. Also, we consider that the message is successfully received when the sender receives an acknowledgement message to inform it.
In Section 4.2, we prove the existence of a dense ball that the message is successfully received with high probability. In Section 4.3, we prove that there is a set of guards that is guarding the receiver of quasi-link . We use guards in order to protect the receiver from interference of other quasi-links and to boost the signal. Also, we prove that there is a set of guards that is guarding the sender of quasi-link . In the last case, we want to protect the sending of an acknowledgement message from its receiver. Then, we guarantee the successful transmission.
4.1 Overview of the Algorithm
First of all, we consider that Algorithm 1 determines a probability/color from the set of probability transmissions to each node in order to achieve transmissions of the messages in the network. Then, the number of colors is , where is the ratio between the maximum and the minimum power assignment as well using the Theorem of  and the Theorem of . Then, it holds the following: any - feasible set can be partitioned into subsets, each of which is - feasible.
In the next step of Algorithm 1, we use a restriction in order to schedule quasi-links whose the quasi-distance is at most , where is the ratio of the power assignments of quasi-links and , and is a constant which indicates the disk . Then, we control three states: 1) The restriction of affectance on quasi-link by other quasi-links , 2) The verification if the messages are received with success (Algorithm 3) and 3) The verification if the network is density (Algorithm 3).
Algorithm 2 controls the sum of probabilities in a ball and waits acknowledgement transmissions. This algorithm returns true if the sum of probabilities in a ball is at most , where . It means that the expected affectance from a set of quasi-links on a quasi-link is not large when all nodes wants to transmit concurrently using a probability transmission each of them.
Algorithm 3 controls the density. It means that the probability transmission is constant. More, applying the technique of , Algorithm 3 blocks the sum of probabilities in a ball to overcome the constant . This happens because of the positive results of both Algorithm 2 and Algorithm 3. After each successful transmission of nodes , they disable and the sum of probabilities is reduced.
More, there is the case that a probability transmission can overcome , it happens in sparse area. Then, Algorithm 1 is executed (the last line of algorithm) and the sum of probabilities is at least .
4.2 Without Guards
Let . For each node , we assume a ball , whenever , there exists , which is the center of and is included in . Let , and . Then,
Let be a receiver node and . We consider that . Also, we assume that ball with center and radius is the largest mass of probability that . Then, . If there is a node in a ball and satisfies the inequalities (5a), (5b) then is also satisfied the (5c) inequality (as it is proved in ).
In addition, we consider a ball . The average probability mass of is at least . The average probability mass of is at least , where and is the ball of radius with the highest probability mass. If the (5b) inequality is satisfied for a node then (5c) is also satisfied. If the (5b) inequality is not satisfied for a node , there is a ball of radius and the probability mass is at least in distance at most from . Then, there is a ball with radius and probability mass at least , which are guaranteed by bounded growth property. Thus, a sequence of balls is created with probability mass at least that a ball has radius for . The distance between and according to their centers is , which means that (5b) inequality is not satisfied. If (5b) is satisfied for some node and radius then the center of the ball is in distance at most . According to , we have an event success. ∎
Let be a ball satisfying the previous Lemma 4, where . Then, for every , the probability of receiving a message .
Seek the probability that the receiver of a quasi-link successfully gets the message that can be affected by other nodes . We use the acknowledgement transmission to inform its sender in odrer to guarantee the successful reception of message. Then, the probability of successfully receiving a message from a node is computed from the joint of the following probabilities:
Event . The message is transmitted from only one node, which belongs to the disk . Using Fact 4 of : , as . Thus, .
Event . There are not sender-nodes in the ball . Using Fact 5 of : There are nodes with probabilities for each node and then . Also, we observe that the ball consists of balls with radius and be the dimension. Then, . Thus, .
Event . The sender receives the acknowledgement message from its receiver, given that a message is transmitted from the sender-node . Let
be the random variable, that is equal to 1 if the message is successfully received, otherwise 0. Thus, we compute. Using Markov inequality in the second inequality: . The transmission probability is
, for . Therefore, .
Event . The WAFF by nodes from the union of disks on quasi-link is at most . Let be a set of nodes. By Markov inequality, . The expected value is given by .
Therefore, we choose in order to bound the expected value by . Thus, .
Event . The WAFF by nodes from outside of the ball is at most . Then, . The proof is based on the division of the space that nodes are located outside the ball . The idea of the proof is same as the previous event.
Therefore, the probability of receiving a message is . ∎
Let be a ball satisfying the previous Lemma 4, where . Then, for every , the probability of receiving a message .
4.3 With Guards
In this part, we seek a set of nodes that is guarding a receiver/sender node from the affectance of other quasi-links in a metric decay space, which is bounded-growth. This space has bounded independence dimension and bounded doubling dimension, which is defined in . Specifically, we use guards in order to protect the receiver from the interference of other quasi-links and to boost the signal. Then, we show that there is a set of guards which is guarding a sender of a quasi-link . Thus, we protect the sending of an acknowledgement message of its receiver from various interference types, such as the simultaneous transmissions of quasi-links.
First of all, we need the following definition in order to prove the Lemma 8. We define a -density-dominant node in order to provide an efficient measure with regard to the transmission probability of nodes and the number of active nodes. In particular, we assume two disjoint subsets of nodes, the set of receiver