Mobile Information Systems

Mobile Information Systems / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 8836517 | https://doi.org/10.1155/2020/8836517

Qiufen Ni, Chuanhe Huang, Panos M. Pardalos, Jia Ye, Bin Fu, "Different Approximation Algorithms for Channel Scheduling in Wireless Networks", Mobile Information Systems, vol. 2020, Article ID 8836517, 13 pages, 2020. https://doi.org/10.1155/2020/8836517

Different Approximation Algorithms for Channel Scheduling in Wireless Networks

Academic Editor: Alessandro Bazzi
Received09 Jun 2020
Revised22 Sep 2020
Accepted29 Oct 2020
Published16 Nov 2020

Abstract

We introduce a new two-side approximation method for the channel scheduling problem, which controls the accuracy of approximation in two sides by a pair of parameters . We present a series of simple and practical-for-implementation greedy algorithms which give constant factor approximation in both sides. First, we propose four approximation algorithms for the weighted channel allocation problem: 1. a greedy algorithm for the multichannel with fixed interference radius scheduling problem is proposed and an one side -IS-approximation is obtained; 2. a greedy -approximation algorithm for single channel with fixed interference radius scheduling problem is presented; 3. we improve the existing algorithm for the multichannel scheduling and show an time -approximation algorithm; 4. we speed up the polynomial time approximation scheme for single-channel scheduling through merging two algorithms and show a -approximation algorithm. Next, we study two polynomial time constant factor greedy approximation algorithms for the unweighted channel allocation with variate interference radius. A greedy -approximation algorithm for the multichannel scheduling problem and an -approximation algorithm for single-channel scheduling problem are developed. At last, we do some experiments to verify the effectiveness of our proposed methods.

1. Introduction

An important problem in wireless network is to develop efficient algorithms for maximum throughput by scheduling channels among many nodes. Based on the Shannon capacity formula, we can know that there is a minimum SINR required for each user in the network of wireless communication, as the existence of signal propagation loss during the process of data transmission; when the distance of a communication user pair is larger than the threshold, the effect of the received power in the receiver can be neglected because the receiver’s SINR is less than the threshold. There is an interference between two users in the same channel when their distance is less than a certain distance. Given a set of users, we find a way which schedules a maximal number of them without introducing interference between any two of them. The weighted version of this problem is to achieve the sum of maximal weights for the users assigned channels for communication. The channel scheduling problem is NP-hard even in the single-channel model [1] and has been studied in a series of papers (for example, [24]). According to IEEE standard 802.11a, there are 13 orthogonal channels provided for wireless network [5]. The multichannel or multi-radio scheduling problem is essential to guarantee the performance of wireless networks.

A channel scheduling problem for a given graph is to select an edge subset and assign a channel to each edge in under the restriction so that all edges in are interference-free. The node exclusive interference model has been studied in many articles (for example, [612]). The algorithms for the single-channel or single-radio scheduling are reported in [13, 14]. For the model not allowing two communication edges to share the same node, the optimal scheduling problem is converted into matching problem [1517] which has a polynomial time solution. Some approximation algorithms for single channel were reported in [18] which show the existence of a theoretical polynomial time approximation scheme for the single-channel assignment. A simple greedy algorithm which is easy to implement is shown to have a constant factor approximation [18] for the single-channel assignment.

The theoretical research about the multichannel scheduling has been reported in [1923]. In [4], they formulate the channel allocation problem as an NP-hard non-linear integer programming problem and then propose a probabilistic polynomial time -approximation algorithm based on linear programming. A joint approach between routing and channel scheduling is shown in [24]. Some constant factor approximation algorithms are shown in [25] for assigning a minimum number of channels for a conflict-free communication, which are also NP-hard [26, 27].

A polynomial time approximation scheme was developed in [28] for multi-radio and multichannel wireless network with very high computational complexity for -approximation, where is the number of channels. This theoretical result is not practical for implementation.

A maximal independent set problem based on joint scheduling and routing optimization is studied in [2], which is a mixed integer non-linear programming NP-hard problem. It developed a column generation based -approximation-bounded approximation algorithm, which can find tight -bounded approximate solutions and the optimal solutions.

We introduce the notion of two-side approximation for the channel scheduling problem. A pair of parameters controls the accuracy of approximation in this paper. A -approximation satisfies and , where is the set of edges assigned channels in an optimal solution and is the weight of edge . A -approximation satisfies . Our two-side approximation approach combines two complementary problems. The first one is to maximize the total weights of communication edges that have been assigned channels, and the second one is to minimize the total weight of edges that do not receive channel for communication. It brings a more accurate tool for the approximation algorithms for the channel scheduling problem.

As the approximation scheme developed by Cheng et al. [28] has very high computational complexity, it is impossible to implement with software. Finding simple and efficient algorithm for the multichannel scheduling problem is still a challenging research topic. We show that a simple greedy algorithm can obtain an -approximation for the single-channel scheduling problem. In many cases, the greedy algorithms give much more accurate results than the worst ratio. Furthermore, we develop an time -approximation algorithm for the single-channel scheduling problem. We also show that a simple greedy algorithm satisfies , which can obtain an -approximation for the single-channel scheduling problem. We also develop a time -approximation algorithm for the multi-channel scheduling problem.

Our proposed multi-channel scheduling algorithm can be used in many wireless systems in which the number of users is larger than the number of subchannels; this case is considered in [29] where they study the channel assignment problem in uplink wireless communication system. It also can be used in the distributed data transmission system with limited communication resources, which is shown in [30]. The proposed methods are also suitable for channel assignment problem in the cellular-VANET heterogeneous wireless networks, which is studied in [25]. Our proposed channel scheduling approach has a wide range of applications and can be applied to different wireless resource allocation scenarios.

We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate interference radius among those nodes. The paper is organized as follows: in Section 2, we present approximation algorithms for the weighted scheduling problems. The interference range is determined by a uniform parameter , but the edges of communication have different weights. In Section 2.2, we show that a greedy algorithm for multi-channel assignment has a constant factor approximation. In Section 2.3, we show a two-side approximation algorithm for the single-channel assignment problem and show that there does not exist such an approximation for the multi-channel assignment problem. In Section 2.4, we give improved approximation schemes for both single-channel and multi-channel assignment problems. In Section 3, we develop the approximation for the unweighted channel assignment problem. This is the case that all the edges have weight 1, but the interference radius is not fixed. Some simulation results for the greedy approximation algorithm for the multi-channel assignment are given in Section 4. We draw conclusion in Section 5.

2. Weighted Channel Assignment

We develop approximation algorithms for the weighted channel assignment. The interference range is controlled by a fixed parameter . The polynomial time approximation scheme for multi-channel scheduling problem is based on this model [28].

2.1. Definitions and Models

The network is modeled as an edge-weighted graph , where is the set of nodes and is the set of edges for traffic flow. The weight of each edge is often represented by the rate of network traffic. For each , let be the weight of .

Definition 1. Given an edge-weighted network graph , let denote the distance between nodes and . The function can be either the geometric distance between and in Euclidean space if is a set of points in Euclidean space or the hop distance between and in the graph . The edge distance between two edges and is defined by . Note that the graph only considers those edges that need to do communication over a wireless network. No silent edge is included in .
A set of edges is -interference-free matching if for any two edges and in , where is distance threshold for interference.
A (multiple) channel assignment problem has a demand graph that requests communication for . A wireless network may not have the resource to satisfy the communications for all edges. A channel assignment algorithm selects a subset and assigns a channel to each edge in so that the edges in the same channel form a -interference-free matching, where is distance threshold for interference. When there is only one channel available for the entire demand graph, the channel assignment problem is called single-channel assignment problem. Otherwise, it is called multi-channel assignment problem.
A channel assignment for an edge is represented by , where is a channel. Assume that is a set of channel assignments. A channel assignment is interfered by if there is a channel assignment such that . Define to be the set of all edges with for some channel . If is the weight function for the edges in , define . An optimal solution for a channel assignment problem with weight function is a set of channel assignments for edges such that is the maximum.

Definition 2. Assume that is a set of channel assignments for .Let .Define .Define ,where is the interference degree of an edge , is the interference degree of the graph , and is distance threshold for interference.
If the distance is the Euclidean distance and all nodes are on the plane, then , which is shown in Lemma 1. If the distance is the hop distance and all nodes are on the plane, then , which was proved in [18]. Therefore, does not depend on the threshold.

Definition 3. Assume that is a channel scheduling problem. We define some measures for approximations.(i)An -IS-approximation for the channel scheduling problem satisfies the condition .(ii) A -VC-approximation for the channel scheduling problem satisfies the condition .(iii)A -approximation for the channel scheduling problem satisfies the conditions and .

Lemma 1. Assume that all nodes in the demand graph are points on a plane. Then, for the Euclidean distance as the threshold for the interference.

Proof. Let be an edge in the graph . The distance between and is at most (otherwise, they cannot communicate). Edge and another edge have interference if one of and and one of and have distance at most . Let each node be a center of a circle of diameter on a plane. Let represent a circle with center at and radius . Let be a channel assignment for , which gives a list of disjoint circles such that each node in those edges with channel assigned is a circle center. Circle touches at most 6 circles among , and so does . If both and touch 6 circles, there must exist at least one of them touched by both and because the distance between and is at most .

2.2. Multiple Channel with Fixed Interference Radius Scheduling

In this section, we present a greedy approximation algorithm for the multi-channel scheduling problem shown in Algorithm 1. We can only show a one-side -approximation for the multi-channel scheduling problem.

 Input: a weighted graph , and a distance parameter for the largest distance of interference.
 Output:
(1)Sort all edges by the increasing order of their weights and put those edges in a list ;
(2)Let ;
(3)Repeat;
(4) Select an edge with the largest weight from ;
(5) If there is a channel for not being interfered by
(6) Then, assign the channel to and put into ;
(7) Remove from ;
(8)Until is empty;
(9)Return ;

In the multi-channel scheduling problem, we assume that each node has channels available for allocation. Two nodes and can communicate if their corresponding edge is assigned a channel (the distance between and is at most the threshold ) and has no interference with other edges in the same channel.

We note that when there is a set of edges that have been assigned channels, it is straightforward to check if a new edge will be interfered at certain channel . This can be done by checking for all , where is the subset of edges in assigned channel .

For the case that all nodes are on a plane, and the distance is either Euclidean distance or hop distance, the following theorem gives a constant factor approximation for the multiple channel assignment problem as is bounded by constants in both distances.

Theorem 1. The Algorithm 1 greedy-multi-channel (.) is a -IS-approximation algorithm for the multi-radio multi-channel scheduling problem and has the computational complexity , where and .

Proof. Assume that is an optimal solution for the channel scheduling problem. Let be an approximate solution derived by greedy-multi-channel (.).
For each edge in , assign an edge , denoted by , in such that is the least among all edges with interference with . When the algorithm greedy-multi-channel (.) processes an edge , the available channels for are those which are not allocated to the edges with end points at either or .

Claim 1. For each edge in , for each that is already processed before .

Proof. It follows from the greedy algorithm which processes the edges according the decreasing order of their weights.
Let be the total number of channels. For each edge , we consider two cases. The constant used in the two cases will be assigned later.

Case 1. There are at least channels available for the edge when is processed. Such an edge is of type 1.
In this case, since is not assigned for a channel, there must be at least edges that are already assigned channels and have . For such an edge with , let . Since is processed before in the algorithm, we have . Thus, . Let for all the other edges . For each edge of type 1, we have inequalityFor each edge in , we havewhere .

Case 2. There are fewer than channels available for the edge when is processed in greedy-multi-channel (.). Such an edge is of type 2.
For at least one of and , say , there are at least channels that are already assigned. For each edge with an end point in already assigned channels, let . Let for all the other edges . For each edge , we havewhere . For each edge of type 2, Each edge in is of either type 1 or type 2. We haveWe haveSelect the constant so that is minimal. Let . Let . We have . Thus, the ratio of approximation is .

2.3. Single Channel with Fixed Interference Radius Scheduling

In this section, we present a greedy approximation algorithm for the channel scheduling problem shown in Algorithm 2. Our two-side approximation bound for the greedy algorithm improves the one-side approximation bound in [18].

 Require: a weighted graph , and a distance parameter for the largest distance of interference.
 Ensure: ;
(1)Sort all edges by the decreasing order of their weights and put those edges in a list ;
(2)Let ;
(3)Repeat;
(4) Select an edge with the largest weight from ;
(5) If is not being interfered by ;
(6) Then, assign the channel to and put into ;
(7) Remove from ;
(8)Until is empty;
(9)Return ;

We note that when there is a set of edges that have been assigned channels, it is straightforward to check if a new edge will be interfered at certain channel . This can be done by checking for all , where is the subset of edges in in channel .

Theorem 2. The Algorithm 2 greedy-single-channel (.) is a -approximation algorithm for the single-channel scheduling problem and has the computational complexity .

Proof. Assume that is an optimal solution for the channel scheduling problem. Let is an approximate solution derived by greedy-single-channel (.).
For each edge in , assign a edge , denoted by , in such that has the largest weight among all edges with interference with .
Consider an edge selected in . Let be the list of edges with .

Claim 2. For each edge in , .

Proof. If is in , we consider an edge has interference with itself. Thus, it is trivial. By the definition of , has the largest of weight among all edges in with interference with .
Assume that is the first edge in and has the same channel with . Before selecting for assigning a channel, there is no interference between and other edges in . Therefore, (otherwise, should not be selected for channel assignment). Since has the largest weight among all edges with interference with in , we have .
Assume that contains channel assignments for edges . Partition into . By Claim 2, we have thatTherefore, we have . Thus, is a -IS approximation for .
On the other hand, for each , we always have (by Claim 2). For each edge in , there is an edge such that and have the interference at the same channel (otherwise, would be assigned some channel by Greed1 (.)). Furthermore, since has interference with . Therefore, there is an edge such that . By inequality (1), we haveWe have the inequalities: This gives . Therefore, greedy-single-channel (.) gives a -approximation.

Theorem 3. Let be the Euclidean distance and the input has the geometric position of all nodes in a Euclidean plane. The Algorithm 2 greedy-single-channel (.) gives a -approximation for the single-channel scheduling problem and runs in time.

Proof. For the implementation, we partition the plane into grid of size . When an edge is assigned a channel, put the assignment into the corresponding grid for one of its two nodes in the edge. When assigning a channel a new edge, check the assigned edges in the nearby grids. Thus, each edge only costs an time.
If is defined to be the hop distance, then is at most 49, which is shown in [18]. If is the Euclidean distance, we show that is at most 11.

Theorem 4. Let be the hop distance in network graph . The Algorithm 2 greedy-single-channel (.) gives a -approximation for the single-channel scheduling problem and runs in time.

Proof. It follows from Theorem 2 and the fact , which is shown in [18]. A brute force implementation takes time.
For the multi-channel scheduling problem, we show that it does not have two-side approximation unless P=NP.

Theorem 5. Assume that is a function from to with . Then, there is no polynomial time -VC-approximation for the multiple channel scheduling problem unless P=NP.

Proof. Let be the input graph of the multi-channel assignment problem. Assume that there exists a polynomial time -VC-approximation algorithm, the multiple channel scheduling problem. Let be the least number of channels that can support the communications of all edges in . When the number of total available channels is equal to , an optimal solution assigns channels to all edges in . This makes . When the approximate solution satisfies , it becomes an optimal solution. Thus, we can search the least number of channels from 1 to to support all edges in . This brings a polynomial time solution for the conflict-free channel assignment problem, which was proved to be NP-hard [26]. Therefore, P=NP.
We have the following corollary that shows we cannot have a two-side approximation for the multi-channel scheduling problem.

Corollary 1. Assume that is a function from to with . Then, there is no polynomial time -approximation for the multiple channel scheduling problem unless P=NP.
In Section 2.4, we show a faster approximation for -IS-approximation than that in [28]. Part of the algorithm is based on the shifting technology which has originated from [31] and has been widely used in developing approximation for networking problem (for example, [18, 28]).
In Section 3.1, we present a greedy -approximation algorithm for the unweighted multi-channel scheduling problem with variate interference radii.
In Section 3.2, we present a greedy approximation algorithm for the unweighted single channel with variate interference radii scheduling problem.
In Section 4, we did some simulation for the multi-channel assignment problems for the greedy algorithm. Greedy algorithm is easy to implement and fast to output the result. Its experimental results show much better performance than the theoretical approximation ratio, which is derived under the worst case analysis.

2.4. Improving the Existing Algorithm for the Multichannel Scheduling

We show a faster approximation for -IS-approximation than that in [28]. Let PTAS (.) (polynomial-time approximation scheme) represent the algorithm described below. Part of the algorithm is based on the shifting technology which has originated from [31] and has been widely used in developing approximation for networking problem (for example, [18, 28]).

Assume that the maximal distance of two nodes for communication is one. Let be the distance of interference. The grid size is . The plane is partitioned into grids of size .

Let be a constant in (0, 1). Define m = .

For two integers , define to be a partition such that the plane is partitioned into the disjoint union of squares of size , and the left bottom point of each square of size in has coordinates for some integers and .

Lemma 2. There is a time algorithm to find an optimal solution for the multi-channel scheduling problem in a square with at most channels in each node.

Proof. We apply a division and conquest method to find an optimal solution. Partition a square into four squares by one strip in the vertical middle and one strip in the horizontal middle. The width of the two stripes is equal to the width of the grid . We can only select at most edges in the two strips. The number of cases of choices is for a single channel. The number of cases of choices is for channels. The four sub-problems in the four sub-squares can be solved independently.
Let be the computational time for solving the channel scheduling problem in a square. We have the recursive equation: . Select a constant so that . Expanding the recursion, we have

Theorem 6. Assume that is an arbitrary constant in . Then, there is an time algorithm to give a -IS-approximation for the multi-channel scheduling with channels.

Proof. A point is in the boundary of if it is in a grid in the boundary of a big grid of . Let be the set of all edges that connect the points in the boundary of in an optimal solution.
Assume is the disjoint union of grids . Let be the optimal solution for the set of edges connecting at least one node not in any boundary grid of . Let . We haveEach boundary grid is in at most . We also have . Therefore, there are and such thatBy inequalities (11) and (12), we haveThe computational time is reduced to , where is the time for finding the optimal solution in a area. By Lemma 2, the algorithm runs in time and gives an -approximation.

2.5. Merging Two Algorithms in Single-Channel Scheduling

In this section, we show that merging Algorithm 1 and PTAS (.) can speed up the polynomial time approximation scheme in many cases for the single-channel scheduling. We propose Algorithm 3 which shows a polynomial time -approximation scheme for the single-channel scheduling problem.

 Require: constant and graph .
 Ensure:
(1)Let ;
(2)Run greedy-single-channel (.) to find an -approximation ;
(3)Let ;
(4)If ;
(5)Then, run PTAS (.) to find an -IS-approximation ;
(6)Return with ;

Lemma 3. Assume that a channel scheduling problem satisfies the condition . Then, a -VC-approximation for the channel scheduling implies a -IS-approximation.

Proof. Assume an approximation satisfies .

Theorem 7. Assume that is an arbitrary constant in . Then, there is an time algorithm to give a -approximation for the single-channel scheduling. Furthermore, Algorithm 3 runs in time if , where is the graph of wireless network.

Proof. By Theorem 2, we have that is a -approximation for the single-channel scheduling problem with and . By Lemma 3, if , then is a -IS-approximation, where .
Since is a -IS-approximation for the channel scheduling problem and is also an approximation for the channeling problem, we have with being at most and at least . Therefore, is a -IS-approximation for the channel scheduling problem.
Since , we have that is a -approximation for the channel scheduling problem.

3. Unweighted Channel Assignment

In this section, we develop the approximations for the unweighted channel assignment problem. This is the case that all the edges have weight 1. The radii of interference are not fixed.

In the unweighted model, each node has a radius for the range that has the interference. Any node with distance at most to is interfered by when is active. To an edge in an active communication, we assume .

For edges and , there is an interference between them if for some and in the nodes of the two edges.

Definition 4. Assume that is a channel scheduling problem. We define some measures for approximations.An optimal solution for is a set of edges without interference with largest .A -IS-approximation for the channel scheduling problem satisfies the condition .A -VC-approximation for the channel scheduling problem satisfies the condition .A -approximation for the channel scheduling problem satisfies the conditions and .By Definition 3, a -approximation for the channel scheduling problem is both -IS-approximation and -VC-approximation for it. Many existing papers used the -IS-approximation to measure the accuracy.

Definition 5. Assume that is a set of channel assignments for .For each edge , define , where is the distance of interference from .Let ; there is an interference between and .Define .

Lemma 4. Assume that all nodes in the demand graph are points on a plane. Then, for the Euclidean distance for the interference.

Proof. Let be an edge in the graph . We only consider a fixed channel . Let represent a circle with center at and radius . Define to be the set of edges assigned with channel such that has interference with or (in other words, or . Define : is a node in with . Define : is a node in with .
Assume that is large. Consider the case that node has interference with other edges. For the case of , we have similar conclusions.

Case 1. . Let . Using as the center, evenly partition the plane into the fan areas with angle each. By the pigeon hole principle, there is a fan area that has at least . In this case, the node with the largest radius interferes all other nodes of . This is a contradiction since all nodes are from the solution and have no interference each other. .

Case 2. . We have more than 25 nodes within the distance at most to , and . The number of circles of radius at least without overlap inside a big circle (with center at ) of radius is at most . This gives a contradiction. Thus, we have .
Similarly, we also have and . Therefore, .

3.1. Multiple Channels with Variate Interference Radii

In this section, we present a greedy -approximation algorithm for the unweighted multi-channel scheduling problem with variate interference radii, which is shown in Algorithm 4.

 Require: a weighted graph , and a distance parameter for the largest distance of interference.
 Ensure:
(1) Sort all edges by the increasing order of their sum of radii and put those edges in a list .
(2) Let ;
(3) Repeat;
(4) Select an edge with the least from ;
(5) If there is a channel for not being interfered by
(6) Then, assign the channel to and put into ;
(7) Remove from ;
(8) Until is empty;
(9) Return ;

We note that when there is a set of edges that have been assigned channels, it is straightforward to check if a new edge will be interfered at certain channel . This can be done by checking for all , where is the subset of edges in in channel .

Theorem 8. The Algorithm 4 greedy (.) is a -IS-approximation algorithm for the multi-radio multi-channel scheduling problem and has the computational complexity , where and .

Proof. Assume that is an optimal solution for the channel scheduling problem. Let be an approximate solution derived by greedy (.).
For each edge in , assign an edge , denoted by , in such that is the least among all edges with interference with .

Claim 3. For each edge in , .

Proof. If is in , we consider an edge has interference with itself. Thus, it is trivial. By the definition of , is the least among all edges in with interference with .
Assume that is the first edge in and has the same channel with and has interference with . Before selecting for assigning a channel, there is no interference between and other edges in . Therefore, (otherwise, should not be selected for channel assignment). Since has the least among all edges with interference with in , we have .
Let be the total number of channels. For each edge , we consider two cases.

Case 1. There are at least channels available for the edge when is processed. Each of such an edge is called type A.
In this case, since is not assigned for a channel, must have interference with at least edges already assigned channels. For each edge with interference with , let , and let otherwise. For each of type , we haveLet . For each edge in , let . We have

Case 2. There are less than channels available for the edge when is processed. Each of such an edge is called type B.
For at least one of and , say , there are at least channels that are already assigned. For each edge with an end point in already assigned channels, let . For each edge in , let . For each of type , we haveLet