Topology Control in Heterogeneous Wireless

Topology Control in Heterogeneous Wireless Networks:Problems and Solutions

Ning Li and Jennifer C.Hou

Department of Computer Science

University of Illinois at Urbana-Champaign

Urbana,IL61801

{nli,jhou}@https://www.wendangku.net/doc/767867144.html,

Abstract—Previous work on topology control usually assumes homogeneous wireless nodes with uniform transmission ranges. In this paper,we propose two localized topology control algo-rithms for heterogeneous wireless multi-hop networks with non-uniform transmission ranges:Directed Relative Neighborhood Graph(DRNG)and Directed Local Spanning Subgraph(DLSS). In both algorithms,each node selects a set of neighbors based on the locally collected information.We prove that(1)the topologies derived under DRNG and DLSS preserve the network connectivity;(2)the out degree of any node in the resulting topology by DLSS is bounded,while the out degree cannot be bounded in DRNG;and(3)the topologies generated by DRNG and DLSS preserve the network bi-directionality.

I.I NTRODUCTION

Energy ef?ciency[1]and network capacity are perhaps two of the most important issues in wireless ad hoc networks and sensor networks.Topology control algorithms have been proposed to maintain network connectivity while reducing energy consumption and improving network capacity.The key idea to topology control is that,instead of transmitting using the maximal power,nodes in a wireless multi-hop network collaboratively determine their transmission power and de?ne the network topology by forming the proper neighbor relation under certain criteria.

By enabling wireless nodes to use adequate transmission power(which is usually much smaller than the maximal trans-mission power),topology control can not only save energy and prolong network lifetime,but also improve spatial reuse(and hence the network capacity)[2]and mitigate the MAC-level medium contention[3].Several topology control algorithms [3]–[10]have been proposed to create power-ef?cient network topology in wireless multi-hop networks with limited mobility (a summary is given in Section III).However,most of them as-sume homogeneous wireless nodes with uniform transmission ranges(except[4]).

The assumption of homogeneous nodes does not always hold in practice,since even devices of the same type may have slightly different maximal transmission power.There also exist heterogeneous wireless networks in which devices have dramatically different capabilities,for instance,the communi-cation network in the Future Combat System which involves wireless devices on soldiers,vehicles and UA Vs.As will be exempli?ed in Section III,most existing algorithms cannot be directly applied to heterogeneous wireless multi-hop networks in which the transmission range of each node may be different. To the best of our knowledge,this paper is the?rst effort to address the connectivity and bi-directionality issue in the heterogeneous wireless networks.

In this paper,we propose two localized topology control al-gorithms for heterogeneous wireless multi-hop networks with non-uniform transmission ranges:Directed Relative Neighbor-hood Graph(DRNG)and Directed Local Spanning Subgraph (DLSS).In both algorithms,the topology is constructed by having each node build its neighbor set and adjust its trans-mission power based on the locally collected information. We are able to prove that(1)the topology derived under both DRNG and DLSS preserves network connectivity,i.e., if the original topology generated by having every node use its maximal transmission power is strongly connected,then the topologies generated by both DRNG and DLSS are also strongly connected;(2)the out degree of any node in the topology by DLSS is bounded,while the out degree of nodes in the topology by DRNG may be unbounded;and(3)the topology generated by DRNG and DLSS preserves network bi-directionality,i.e.,if the original topology by having every node use its maximal transmission power is bi-directional,then the topology generated by either DRNG or DLSS is also bi-directional after some simple operations.

Simulation results indicate that,compared with the other known topology control algorithms that can be applied to het-erogeneous networks,DRNG and DLSS have smaller average node degree(both logical and physical)and smaller average link length.The former reduces the MAC-level contention, while the latter implies a small transmission power needed to maintain connectivity.

The rest of the paper is organized as follows.In Section II, we give the network model.In Section III,we summarize previous work on topology control,and give examples to show why existing algorithms cannot be directly applied to heterogeneous networks.Following that,we present both the DRNG and DLSS algorithms in Section IV,and prove several of their useful properties in Section V.Finally,we evaluate the performance of the proposed algorithms in Section VI, and conclude the paper in Section VII.

使用DRNG和DLSS进行拓扑控制丗连通性和双向性

II.N ETWORK M ODEL

Consider a set of nodes(vertices),V={v1,v2,...,v n}, which are randomly distributed in the2-D plane.Assume the area that a transmission can cover is a disk.We de?ne the range of a node v i as the radius of the disk that v i

can cover using its maximal transmission power,denoted r v

i .

In a heterogeneous network,the transmission ranges of all nodes may not be the same.Let r min=min v∈V{r v}and r max=max v∈V{r v}.

We denote the network topology generated by having each node use its own maximal transmission power as a simple directed graph G=(V(G),E(G)),where E(G)={(u,v): d(u,v)≤r u,u,v∈V(G)}is the edge(link)set of G,and d(u,v)is the Euclidean distance between node u and node v.Note that(u,v)is an ordered pair representing an edge from node u to node v,i.e.,(u,v)and(v,u)are two different edges.A unique id(such as an IP/MAC address)is assigned to each node.Here we let id(v i)=i for simplicity.

We assume that the wireless channel is symmetric and obstacle-free,and each node is equipped with the capability to gather its location information via,for example,GPS for outdoor applications and pseudolite[11]for indoor applica-tions,and many other lightweight localization techniques for wireless networks(see[12]for a summary).

Before delving into the technical discussion and algorithm description,we give the de?nition of several terms that will be used throughout the paper.

De?nition1(Reachable Neighborhood):The reachable neighborhood N R u is the set of nodes that node u can reach using its maximal transmission power,i.e., N R u={v∈V(G):d(u,v)≤r u}.For each node u∈V(G), let G R u=(V(G R u),E(G R u))be an induced subgraph of G such that V(G R u)=N R u.

De?nition2(Weight Function):Given two edges (u1,v1),(u2,v2)∈E and the Euclidean distance function d(·,·),weight function w:E→R satis?es:

w(u1,v1)>w(u2,v2)

?d(u1,v1)>d(u2,v2)

or(d(u1,v1)=d(u2,v2)

&&max{id(u1),id(v1)}>max{id(u2),id(v2)}) or(d(u1,v1)=d(u2,v2)

&&max{id(u1),id(v1)}=max{id(u2),id(v2)}

&&min{id(u1),id(v1)}>min{id(u2),id(v2)}). This weight function ensures that two edges with different end-vertices have different weights.Note,however,that w(u,v)= w(v,u).

De?nition3(Neighbor Set):Node v is a neighbor of node u under an algorithm A,denoted u A?→v,if and only if there exists an edge(u,v)in the topology generated by the algorithm.In particular,we use u→v to denote the neighbor relation in G.u A←→v if and only if u A?→v and v A?→u.The Neighbor Set of node u is N A(u)={v∈V(G):u A?→v}.

De?nition4(Topology):The topology generated by an al-

gorithm A is a directed graph G A=(E(G A),V(G A)),where V(G A)=V(G),E(G A)={(u,v)∈E(G):u A?→}.

De?nition5(Radius):The radius,R u,of node u is de?ned as the distance between node u and its farthest neighbor(in terms of Euclidean distance),i.e,R u=max v∈N

A

(u)

{d(u,v)}. De?nition6(Connectivity):For any topology generated by an algorithm A,node u is said to be connected to node v(denoted u?v)if there exists a path(p0= u,p1,...,p m?1,p m=v)such that p i A?→p i+1,i= 0,1,...,m?1,where p k∈V(G A),k=0,1,...,m.It follows that u?v if u?p and p?v for some p∈V(G A). De?nition7(Bi-Directionality):A topology generated by an algorithm A is bi-directional,if for any two nodes u,v∈V(G A),u∈N A(v)implies v∈N A(u).In other words,the topology generated by A is bi-directional if all edges in the topology are bi-directional.

De?nition8(Bi-Directional Connectivity):For any topol-ogy generated by an algorithm A,node u is said to be bi-directionally connected to node v(denoted u?v)if there exists a path(p0=u,p1,...,p m?1,p m=v)such that p i A←→p i+1,i=0,1,...,m?1,where p k∈V(G A),k= 0,1,...,m.It follows that u?v if u?p and p?v for some p∈V(G A).

Deriving network topology consisting of only bi-directional links facilitates link level acknowledgment,which is a critical operation for packet transmissions and retransmissions over unreliable wireless media.Bi-directionality is also important in ?oor acquisition mechanisms such as the RTS/CTS mechanism in IEEE802.11.

De?nition9(Addition and Removal):The operation Addi-tion is to add an extra edge(v,u)into G A if(u,v)∈E(G A), (v,u)/∈E(G A),and d(u,v)≤r v.The operation Removal is to delete any edge(u,v)∈E(G A)if(v,u)/∈E(G A).

Let G+

A

and G?

A

denote the resulting topologies after applying Addition and Removal to G A,respectively.

Both the Addition and Removal operations attempt to create a bi-directional topology by removing uni-directional edges or converting uni-directional edges into bi-directional.The result-ing topology after Removal is alway bi-directional,although it may not be strongly connected.The resulting topology after Addition is not necessarily bi-directional,as it essentially tries to increases the transmission power of a node v to a level that may be beyond its capability.

III.R ELATED W ORK AND W HY T HEY C ANNOT BE

D IRECTLY A PPLIED TO H ETEROGENEOUS N ETWORKS Several topology control algorithms[3]–[10]have been proposed.In this section,we?rst summarize these algorithm and then give examples on why they cannot be directly applied to heterogeneous networks.

A.Related Work

Rodoplu et al.[4](denoted R&M)introduced the notion of relay region and enclosure for the purpose of power control.Instead of transmitting directly,a node chooses to

be de?ned in Section

III-B).

be de?ned in Section IV).

Fig.1.The de?nition of the Directed Relative Neighborhood Graph. relay through other nodes if less power will be consumed.It is shown in the paper that the network is strongly connected if every node maintains links with the nodes in its enclosure and the resulting topology is a minimum power topology. The major drawback is that it requires an explicit propagation channel model to compute the relay region(in the simulation study presented in Section VI,we assume that the free-space model is used),hence the resulting topology is sensitive to the model used in the computation.Also,it assumes there is only one data sink(destination)in the network.

Ramanathan et al.[5]presented two centralized algorithms to minimize the maximal power used per node while maintain-ing the(bi)connectivity of the network.They introduced two distributed heuristics for mobile networks.Both centralized algorithms require global information,and thus cannot be directly deployed in the case of mobility.On the other hand, the proposed heuristics cannot guarantee the preservation of the network connectivity.

COMPOW[3]and CLUSTERPOW[7]are approaches im-plemented in the network layer.Both hinge on the idea that if each node uses the smallest common power required to main-tain network connectivity,the traf?c carrying capacity of the entire network is maximized,the battery life is extended,and the MAC-level contention is mitigated.The major drawback is its signi?cant message overhead,since each node has to run multiple daemons,each of which has to exchange link state information with their counterparts at other nodes.

CBTC(α)[6]is a two-phase algorithm in which each node ?nds the minimum power p such that some node can be reached in every cone of degreeα.The algorithm has been proved to preserve network connectivity ifα<5π/6.Several optimization methods(that are applied after the topology is derived under the base algorithm)are also discussed to further reduce the transmitting power.

To facilitate the following discussion,the de?nition of the Relative Neighborhood Graph(RNG)is given below.

De?nition10(Neighbor Relation in RNG):For RNG[13], [14],u RNG

←??→v if and only if there does not exist a third node p such that w(u,p)

Borbash and Jennings[8]proposed to use RNG for the topology initialization of wireless networks.Based on the local knowledge,each node makes decisions to derive the network topology based on RNG.The network topology thus derived has been reported to exhibit good overall performance in terms of power usage,low interference,and reliability.

Li et al.[9]presented the Localized Delaunay Triangula-tiona,a localized protocol that constructs a planar spanner of the Unit Disk Graph(UDG).The topology contains all edges that are both in the unit-disk graph and the Delaunay triangulation of all nodes.It is proved that the shortest path in this topology between any two nodes u and v is at most a constant factor of the shortest path connecting u and v in UDG. However,the notion of UDG and Delaunay triangulation cannot be directly extended to heterogeneous networks.

In[10],we proposed LMST(Local Minimum Spanning Tree)for topology control in homogeneous wireless multi-hop networks.In this algorithm,each node builds its local minimum spanning tree independently and only keeps on-tree nodes that are one-hop away as its neighbors in the ?nal topology.It is proved that(1)the topology derived under LMST preserves the network connectivity;(2)the node degree of any node in the resulting topology is bounded by 6;and(3)the topology can be transformed into one with bi-directional links(without impairing the network connectivity) after removal of all uni-directional links.Simulation results show that LMST can increase the network capacity as well as reduce the energy consumption.

Instead of adjusting the transmission power of individual devices,there also exist other approaches to generate power-ef?cient topology.By following a probabilistic approach,Santi et al.derived the suitable common transmission range which preserves network connectivity,and established the lower and upper bounds on the probability of connectedness[15].In[16], a“backbone protocol”is proposed to manage large wireless ad hoc networks,in which a small subset of nodes is selected to construct the backbone.In[17],a method of calculating the power-aware connected dominating sets was proposed to

trol)is strongly connected.

23

mization is not strongly connected:there is no path from v 1to v 3.

Fig.2.An example that shows CBT C (23

π)may render disconnectivity in heterogeneous networks.There is no path from v 1to v 3due to the loss of edge (v 2,v 3),which is discarded by v 2since v 1and v 4have already provided the necessary

coverage.

trol)is strongly

connected.nected:there is no path from v 5to v 2

.

Fig.3.An example that shows RNG may render disconnectivity in heterogeneous networks.There is no path from v 5to v 2due to the loss of edge (v 4,v 2),which is discarded since |(v 4,v 5)|<|(v 4,v 2)|,and |(v 2,v 5)|<|(v 4,v 2)|

.

trol)is strongly

connected.nected:there is no path from v 3to v 5

.

Fig.4.An example that shows MRNG may render disconnectivity in heterogeneous networks.There is no path from v 3to v 5due to the loss of edge (v 2,v 5),which is discarded since |(v 2,v 3)|<|(v 2,v 5)|,and |(v 5,v 3)|<|(v 2,v 5)|

.

topology control)is strongly

connected.

at v 8.

at v 7.

strongly connected:there is no path from v 7to v 4.

Fig.5.An example that shows the algorithm in which each node builds a local directed minimum spanning tree and only keeps the one-hop neighbors may result in disconnectivity.

establish an underlying topology for the network.

B.Why Existing Algorithms Cannot be Directly Applied to Heterogeneous Networks

Most existing topology control algorithms(except[4])as-sume homogeneous wireless nodes with uniform transmission ranges.When directly applied to heterogeneous networks, these algorithms may render disconnectivity.In this subsec-tion,we give several examples to motivate the need for new topology control algorithms for heterogeneous networks.

As shown in Fig.2(a)-(b)(note that in Figs.2–5we use arrows to indicate the direction of the links to represent a link

from u to v),the network topology derived under CBT C(2

3π)

(without optimization)may not preserver the connectivity, when the algorithm is directly applied to a heterogeneous

network.CBT C(5

6π)also has the same problem.

Similarly we show in Fig.3(a)-(b)that the network topology derived under RNG may be disconnected when the algorithm is directly applied to a heterogeneous network.As RNG is de?ned for undirected graphs,one may tailor the de?nition of RNG for directed graphs.

De?nition11(Neighbor Relation in MRNG):For Modi?ed Relative Neighborhood Graph(MRNG),u MRNG

?????→v if and only if there does not exist a third node p such that w(u,p)< w(u,v),d(u,p)≤r u and w(p,v)

As shown in Fig.4(a)-(b),the topology derived under MRNG may still be disconnected(we will give another variation of RNG for directed graphs in the next section). One possible extension of LMST[10]is for each node to build a local directed minimum spanning tree[18]–[20] and keep only neighbors within one hop.Unfortunately,the resulting topology does not preserve the strong connectivity, as shown in Fig.5.In the next section,we will improve on this approach to preserve the connectivity.

IV.DRNG AND DLSS

In this section,we propose two localized topology con-trol algorithms for heterogeneous wireless multi-hop net-works with non-uniform transmission ranges:Directed Rela-tive Neighborhood Graph(DRNG)and Directed Local Span-ning Subgraph(DLSS).In both algorithms,the topology is derived by having each node build its neighbor set and adjust its transmission power based on locally collected information. Several nice properties of both algorithms will be discussed in Section V.

Both algorithms are composed of three phases:

1)Information Collection:each node collects the local

information of neighbors such as position and id,and identi?es the Reachable Neighborhood N R.

2)Topology Construction:each node de?nes(in compli-

ance with the algorithm)the proper list of neighbors for the?nal topology using the information in N R.

3)Construction of Topology with Only Bi-Directional Links

(Optional):each node adjusts its list of neighbors to make sure that all the edges are https://www.wendangku.net/doc/767867144.html,rmation collection

The information needed by each node u for topology control is the information of its reachable neighborhood N R.This can be obtained locally,in the case of homogeneous networks, by having each node broadcast periodically a Hello message using its maximal transmission power.The information con-tained in a Hello message should at least include the node id and the position of the node.These periodic messages can be sent either in the data channel or in a separate control channel. In heterogeneous networks,having each node broadcast a Hello message using its maximal transmission power may be insuf?cient.For example,as shown in Fig.6,v1is unable to know the position of v4since v4cannot reach v1.We will treat this issue rigorously in Section V-D.For the time being, we assume that by the end of the?rst phase every node u obtains its N R u.

Fig.6.An example that shows having each node broadcast a Hello message using its maximal transmission power may be insuf?cient for some nodes (e.g.,node v1)to know their reachable neighborhood.This?gure also serves to show that given an arbitrary direct graph,it may be impossible to derive a bi-directional topology.

B.Topology construction

First we de?ne the neighbor relation used in both algo-rithms.

De?nition12(Neighbor Relation in DRNG):For Directed Relative Neighborhood Graph(DRNG),u DRNG

?????→v if and only if d(u,v)≤r u and there does not exist a third node p such that w(u,p)

De?nition13(Neighbor Relation in DLSS):For Directed Local Spanning SubGraph(DLSS),u DLSS

????→v if and only if (u,v)∈E(T u),where T u is obtained by applying Algorithm1 to G R u.T u is a directed local spanning subgraph that spans N R u.Hence node v is a neighbor of node u if and only if node v is on node u’s directed local MST T u,and is one-hop away from node u.

DLSS is a natural extension of LMST[10]for hetero-geneous networks.Instead of computing a directed local MST(which minimizes the total cost of the all edges in the subgraph,and is shown to be wrong in Section III-B),

Algorithm1DLSS(u)

INPUT:G R u,the induced subgraph of G that spans the reachable neighborhood of u;

OUTPUT:T u=(V T

u ,E T

u

),a local spanning subgraph of

G R u;

1:V T u:=V,E T u:=?;

2:sort all edges in E(G R u)in the ascending order of weight (as de?ned in De?nition2);

3:for each edge(u,v)in the order do

4:if u is not connected to v in T u then

5:E T u:=E T u∪{(u,v)};

6:end if

7:if u is connected to all nodes in N R u then

8:exit;

9:end if

10:end for

each node u computes a directed local subgraph according to Algorithm1(which minimizes the maximum cost of all edges in the subgraph)and takes on-tree nodes that are one-hop away as its neighbors.

Each node can broadcast its own maximal transmission power in the Hello message.By measuring the receiving power of Hello messages,each node u can determine the speci?c power level required to reach each of its neighbors [10].Node u then uses the power level that can reach its farthest neighbor as its transmission power.This approach can be applied without knowing the actual propagation model. C.Construction of topology with only bi-directional edges As illustrated in the previous section,some links in G DLSS may be uni-directional.There also exist uni-directional links in G DRNG.We can apply either Addition or Removal to G DLSS and G DRNG to obtain bi-directional topologies.We will discuss some properties of these solutions in Section V-B.

V.P ROPERTIES OF DRNG AND DLSS

In this section,we discuss the connectivity,bi-directionality and degree bound of DLSS and DRNG.We always assume G is strongly connected,i.e.,u?v in G for any u,v∈V(G).

A.Connectivity

Lemma1:For any edge(u,v)∈E(G),we have u?v in G DLSS.

Proof:Let all the edges(u,v)∈E(G)be sorted in the increasing order of weight,i.e.,w(u1,v1)

1)Basis:The?rst edge(u1,v1)satis?es w(u1,v1)=

min(u,v)∈E(G){w(u,v)}.According to Algorithm1, (u1,v1)and(v1,u1)will be inserted into G DLSS,i.e., u1DLSS

←??→v1.

2)Induction:Assume the hypothesis holds for all edges

(u i,v i),1≤i

If u k DLSS

????→v k,then u k?v k.Otherwise in the local

topology construction of u,before edge(u k,v k)was inserted into T u

k

,there must already exist a path p= (w0=u k,w1,w2,···,w m?1,w m=v k)from u k to v k, where(w i,w i+1)∈E(T u

k

),i=0,1,···,m?1.Since edges are inserted in a ascending order of weight,we have w(w i,w i+1)

Theorem1:G DLSS preserves the connectivity of G,i.e., G DLSS is strongly connected if G is strongly connected.

Proof:Suppose G is strongly connected.For any two nodes u,v∈V(G),there exists at least one path p= (w0=u,w1,w2,···,w m?1,w m=v)from u to v,where (w i,w i+1)∈E(G),i=0,1,···,m?1.Since w i?w i+1 by Lemma1,we have u?v.

Lemma2:Given three nodes u,v,w∈V(G DLSS)satisfy-ing w(u,v)>w(u,w)and w(u,v)>w(w,v),d(w,v)≤r w, then u v in G DLSS.

Proof:We only need to consider the case where d(u,v)≤r u since d(u,v)>r u would imply u v.Consider the local topology construction of u.Before we insert(u,v) into T u,the two edges(u,p)and(p,v)have already been processed since w(u,p)

Theorem2:The edge set of G DLSS is a subset of the edge set of G DRNG,i.e.,E(G DLSS)?E(G DRNG).

Proof:We prove by contradiction.Given any edge (u,v)∈E(G DLSS),assume(u,v)/∈E(G DRNG).According to the de?nition of DRNG,there must exist a third node p such that w(u,p)

Theorem3(Connectivity of DRNG):If G is strongly con-nected,then G DRNG is also strongly connected.

Proof:This is a direct result of Theorem1and Theo-rem2.

B.Bi-directionality

Now we discuss the bi-directionality property of DLSS and DRNG.Since Addition may not always result in bi-directional topologies,we?rst apply Removal to topologies by DLSS and DRNG.It turns out the simple Removal operation may lead to disconnectivity.Examples are given in Figs.7–8to show, respectively,that DLSS and DRNG with Removal may result in disconnectivity.

In general,G may not be bi-directional if the transmission ranges are non-uniform.Since the maximal transmission range can not be increased,it may be impossible to?nd a bi-directional connected subgraph of G for some cases.An

trol)is strongly connected.

strongly connected:there are 2components.

Fig.7.

An example that shows DLSS with Removal may result in

disconnectivity.

trol)is strongly

connected.

nected.

strongly connected:there are 2components.

Fig.8.

An example that shows DRNG with Removal may result in disconnectivity.

example is given in Fig.6:v 1can reach v 2and v 4,v 2can reach v 1and v 3,v 3can reach v 2and v 4,and v 4can reach v 2only.Addition does not lead to bi-directionality since all edges entering or leaving v 4are uni-directional with all nodes already transmitting with their maximal power.On the other hand,Removal will partition the network.In this example,although the graph G is strongly connected,its subgraph with the same vertex set cannot be both connected and bi-directional.

Now we show that bi-directionality can be ensured if the original topology is both strongly connected and bi-directional.Theorem 4:If the original topology G is strongly connected and bi-directional,then G DLSS and G DRNG are also strongly connected and bi-directional after Addition or Removal .Proof:Since E (G DLSS )?E (G DRNG ),we have E (G ?DLSS )?E (G +DLSS )and E (G ?DLSS )?E (G ?DRNG )?E (G +DRNG ).Therefore,we only need to prove that G ?DLSS preserves the strong connectivity.

In the Induction step in Lemma 1,the only reason we

cannot prove that u k DLSS

←??→v k is that edge (v k ,u k )may not exist.Given that G is bi-directional,we are able to prove

that u k DLSS

←??→v k .Hence for any edge (u,v )∈E (G ),we have u ?v in G DLSS .The removal of asymmetric edges in G DLSS does not affect this property.Therefore,G ?DLSS is still strongly

connected.

u

Fig.9.The de?nition of Cone (u,α,v ).

C.Degree Bound

It has been observed that any minimum spanning tree of a simple undirected graph in the plane has a maximum node degree of 6[21].However,this bound does not hold for directed graphs.An example is shown in Fig.10,where node u has 18neighbors.In this section,we will discuss the node degree in the topology by DLSS and DRNG.

De?nition 14(Disk):Disk (u,r )is the disk centered at node u with a radius of r .

De?nition 15(Cone):Cone (u,α,v )is the unbounded shaded region shown in Fig.9.

The following corollary is a direct result of Lemma 1.Corollary 1:If v is a neighbor of u ’s in G DLSS ,and d (u,v )≥r min ,then u can not have any other neighbor inside Disk (v,r min ).

Theorem 5:For any node u ∈V (G DLSS ),the number of

Fig.10.An example that shows the out degree in a heterogenous network can be very large.The transmission range of u is r max and the transmission range for all other nodes is r min ,where r max =2(r min + ), >0.All nodes are so arranged that the distance between any node and its closest neighbor is r min + .Therefore,the only links in the network are those from u to all the other nodes.Since relaying packets is impossible,u has to use its maximal transmission power and keeps all 18neighbors.

neighbors in G DLSS that are inside Disk (u,r min )is at most 6.

Proof:Let N (u )be the set of neighbors of u in G DLSS that are inside Disk (u,r min ).Let the nodes in N (u )be ordered such that for the i th node w i and the j th node w j (j >i ),w (u,w j )>w (u,w i ).By Lemma 2,we have w (u,w j )≤w (w i ,w j )(otherwise u w j ).Thus ∠w i uw j ≥π/3,i.e.,node w j cannot reside inside Cone (u,2π/3,w i ).Therefore,node u cannot have neighbors other than node w i inside Cone (u,2π/3,w i ).By induction on the rank of nodes in N (u ),the maximal number of neighbors that u can have is at most 6.

Theorem 6:The out degree of node in G DLSS is bounded by a constant that depends only on r max and r min .

Proof:For any node u in G DLSS ,there are at most 6neighbors inside Disk (u,r min )from Theorem 5.Also from Corollary 1,the set of disks {Disk (v,r min

2):v ∈

N DLSS (u ),v /∈Disk (u,r min )}are disjoint.Therefore,the total number of neighbors of u is bounded by:

c 1=6+ π[(r max +r min

2)2?(r min 2)2]π(r min 2

)2 =4 β(β+1) +6,where β=r max

r min .Actually we can observe that Fig.10shows the scenario where the maximum out degree of u is achieved if →0.Therefore,we can further tighten the bound.Since the hexagonal area (as shown in Fig.10)centered at every neighbor of u is disjoint with each other,the total number of neighbors of u is bounded by:

c 2=

π(r max +r min √3)2√32r 2min

?1= 2π

√3(β+1√3)2 ?1.Fig.11.The out degree may be unbounded in G DRNG .

The bound given in Theorem 5is actually quite large.We

will show in Section VI that the average maximum degree is much smaller for networks with random distributed nodes.Also note that what has been discussed so far is actually the logical node degree,i.e.,the number of logical neighbors.In practice,it is more important to consider the physical node degree,i.e.,the number of nodes within the transmission radius.If omni-directional antennas are used,the physical degree cannot be bounded for an arbitrary topology.However,with the help of directional antennas,we will be able to bound the physical degree given that the logical degree is bounded under DLSS (except in some extreme cases, e.g.,a large number of nodes are of the same distance from one node).The idea is that,when transmitting to a speci?c neighbor,node u should adjust the direction and limit the transmission power so that no other nodes will be affected.

Notice that the out degree is not bounded in G DRNG .An example is given in Fig.11.For all p i that lies inside the shaded area,as long as r p i

As mentioned in Section IV,in the case that nodes may have different maximal transmission powers,the operation of having each node u broadcast its own position information to all the other nodes within r u is not suf?cient to ensure each

node u obtains the information of reachable neighborhood N R

u (Fig.(6)).Fortunately with the desirable properties of DRNG and DLSS proved in Sections V-A and V-B,we show that it is suf?cient for node u to collect neighborhood information only from nodes whose maximal transmission range covers node u .That is,the original information exchange algorithm that requires only “one-hop”information suf?ces.

Consider a directed simple graph with less edges:G =(V (G ),E (G )),where E (G )={(u,v ):d (u,v )≤min(r u ,r v ),u,v ∈V (G )}.For any edge (u,v )∈E (G ),since d (u,v )≤min(r u ,r v ),we have (v,u )∈E (G ),which

means G is bi-directional.De?ne N R u

={v ∈V (G ):

d (u,v )≤min(r u ,r v )},r u

=max v ∈N R u {d (u,v )},where

Fig.12.Topologies derived by R&M,DRNG,and DLSS.

r u ≤r u since for any v∈N R u ,d(u,v)≤r u.Let r min = min v∈V{r v }and r max =max v∈V{r v }.By requiring each node u to broadcast its position and id to all other nodes within r u,we are able to determine N R u and r u .We can then apply DRNG and DLSS on top of G and prove that Theorems1-5 still hold even if the original topology is G .

Theorem7:Theorems1–6still holds if the original topol-ogy is G .

Proof:We replace G,r u,N R u,r min,and r max with G ,r u ,N R u ,r min and r max in the proof of Lemma1–2and Theorem1–6.Then following the same line of arguments,we can prove that they still hold if the original topology is G . Theorem8:If the original topology is G (which is a subgraph of G),G DLSS and G DRNG are bi-directional after Addition or Removal.

Proof:We apply Theorem4to G ,for G is bi-directional.

VI.S IMULATION S TUDY

In this section,we evaluate the performance of R&M,DRNG,and DLSS by simulations.All three algorithms are known to preserve network connectivity in heterogeneous networks.

In the ?rst simulation,50nodes are uniformly distributed in a 1000m ×1000m region.The transmission ranges of nodes are uniformly distributed in [200m,250m ].Fig.12gives the topologies derived using the maximal transmission power (labeled as NONE),R&M (under the two-ray ground model),DRNG,and DLSS for one simulation instance.As shown in Fig.12,R&M,DRNG and LMST all signi?cantly reduce the average node degree,while maintaining network connectivity.Moreover,both DRNG and DLSS outperforms R&M in the sense that fewer edges are formed in the topology.

# Nodes

A v e r a g e R a d i u s (m )

(a)Average

radius.

# Nodes

A v e r a g e L i n k L e n g t h (m )

(b)Average link length

https://www.wendangku.net/doc/767867144.html,parison of DLSS,DRNG and R&M with respect to average radius and average edge length.

In the second simulation,we vary the number of nodes in the region from 100to 300,and each data point is an average

100

120140160180200220240260280300

22.5

3

3.5

(a)Average logical out degree.

# Nodes

A v e r a g e P h y s i c a l D e g r e e

(b)Average physical out degree.

https://www.wendangku.net/doc/767867144.html,parison of R&M,DRNG and DLSS with respect to average out degree.

of 50simulation runs.The transmission ranges of nodes are uniformly distributed in [200m,250m ].Fig.13shows the average radius and the average link length for the topologies derived under NONE(no topology control),R&M,DRNG,and DLSS.DLSS outperforms the others,which implies that DLSS can provide a better spatial reuse and use less energy to communicate.

We also compare the out degree of the topologies by different algorithms.The result of NONE is not shown because its out degrees increase almost linearly with the number of nodes and are signi?cantly larger than those under R&M,DRNG,and DLSS.Fig.14shows the average logical/physical out degree for the topologies derived by R&M,DRNG,and DLSS.The average out degrees under R&M and DRNG increase with the increase in the number of nodes,while those under DLSS actually decrease.Fig .15shows the average maximum logical degree and the largest maximum logical

# Nodes

A v e r a g e M a x L o g i c a l D e g r e e

(a)Average maximum logical out degree.

# Nodes

L a r g e s t M a x L o g i c a l D e g r e e

(b)Largest maximum logical out degree.

https://www.wendangku.net/doc/767867144.html,parison of R&M,DRNG and DLSS with respect to the maximum logical degree.

out degree for each number of nodes.The largest maximum logical degree under DLSS is at most 4,and is well below the theoretical upper bound obtained in Theorem 6.Also DLSS has much smaller degrees than the other topologies.Similar results can be observed in Fig.16for physical degrees.The only difference is that the physical degrees are in general larger than the logical degrees for the same network.

VII.C ONCLUSIONS

In this paper,we have proposed two local topology control algorithms,Directed Relative Neighborhood Graph (DRNG)and Directed Local Spanning Subgraph (DLSS),for heteroge-neous wireless multi-hop networks in which each node may have different maximal transmission ranges.We show that as most existing topology control algorithms (except R&M [4])do not consider the fact that nodes may have different maximal transmission ranges,they render disconnected network

topol-

# Nodes

A v e r a g e M a x P h y s i c a l D e g r e e

(a)Average maximum physical out

degree.

# Nodes

L a r g e s t M a x P h y s i c a l D e g r e e

(b)Largest maximum physical out degree.

https://www.wendangku.net/doc/767867144.html,parison of R&M,DRNG and DLSS with respect to the maximum physical degree.

ogy when directly applied to heterogeneous networks.Then we devise DRNG and DLSS and prove that (i)both DRNG and DLSS preserve network connectivity;(ii)both DRNG and DLSS preserve network bi-directionality if Addition and Remove operations are applied to the topologies derived under these algorithms;and (iii)the out degree of any node is bounded in the topology derived under DLSS,while that may be unbounded under DRNG.The simulation study validates the superiority of DRNG and DLSS over R&M.

As part of our future research,we will pursue the following open problems:(1)given a topology in which each node transmits with different maximal transmission power,what is the probability that the topology is bi-directional with respect to the distribution and the density of nodes,and the distribution of the transmission ranges?and (2)How will MAC-level interference affect network connectivity and bi-directionality?

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系泊系统的设计和探究

赛区评阅编号（由赛区组委会填写）： 2016年高教社杯全国大学生数学建模竞赛 承诺书 我们仔细阅读了《全国大学生数学建模竞赛章程》和《全国大学生数学建模竞赛参赛规则》（以下简称为“竞赛章程和参赛规则”，可从全国大学生数学建模竞赛网站下载）。 料 我们的报名参赛队号（12位数字全国统一编号）： 参赛学校（完整的学校全称，不含院系名）： 参赛队员 (打印并签名) ：1. 2.

3. 指导教师或指导教师组负责人 (打印并签名)： （指导教师签名意味着对参赛队的行为和论文的真实性负责） 日期：年月日 送全国评阅统一编号（赛区组委会填写）： 全国评阅随机编号（全国组委会填写）： （请勿改动此页内容和格式。此编号专用页仅供赛区和全国评阅使用，参赛队打印后装订到纸质论文的第二页上。注意电子版论文中不得出现此页。）

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usb驱动程序教程

编写Windows https://www.wendangku.net/doc/767867144.html,的usb驱动程序教程 Windows https://www.wendangku.net/doc/767867144.html, 是微软推出的功能强大的嵌入式操作系统，国内采用此操作系统的厂商已经很多了，本文就以windows https://www.wendangku.net/doc/767867144.html,为例，简单介绍一下如何开发windows https://www.wendangku.net/doc/767867144.html, 下的USB驱动程序。 Windows https://www.wendangku.net/doc/767867144.html, 的USB系统软件分为两层： USB Client设备驱动程序和底层的Windows CE实现的函数层。USB设备驱动程序主要负责利用系统提供的底层接口配置设备，和设备进行通讯。底层的函数提本身又由两部分组成，通用串行总线驱动程序（USBD）模块和较低的主控制器驱动程序（HCD）模块。HCD负责最最底层的处理，USBD模块实现较高的USBD函数接口。USB设备驱动主要利用 USBD接口函数和他们的外围设备打交道。 USB设备驱动程序主要和USBD打交道，所以我们必须详细的了解USBD提供的函数。 主要的传输函数有： abourttransfer issuecontroltransfer closetransfer issuein te rruptransfer getisochresult issueisochtransfer gettransferstatus istransfercomplete issuebulktransfer issuevendortransfer 主要的用于打开和关闭usbd和usb设备之间的通信通道的函数有： abortpipetransfers closepipe isdefaultpipehalted ispipehalted openpipe resetdefaultpipe resetpipe 相应的打包函数接口有： getframelength getframenumber releaseframelengthcontrol setframelength takeframelengthcontrol 取得设置设备配置函数： clearfeature setdescriptor getdescriptor setfeature

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数学建模A题系泊系统设计完整版

数学建模A题系泊系统 设计 HEN system office room 【HEN16H-HENS2AHENS8Q8-HENH1688】

系泊系统的设计 摘要 本题要求观测近海观测网的组成，建立模型对其中系泊系统进行设计，在不同风速和水流的情况下确定锚链，重物球，钢管及浮标等的状态，从而使通讯设备的工作效果最佳。求解的具体流程如下： 针对问题一，分别对系统中的受力物体在水平方向和竖直方向上的力进行分析，找出锚链对锚无拉力时的临界风速，运用力矩平衡求出钢管与钢桶的倾斜角度。对于锚链，将其等效为悬链线模型，根据风速不同判断锚链的状态，从而求出结果。 ?时能够正常工作。为针对问题二，需要调节重物球的质量，使通讯设备在36m m 了确定重物球的质量，首先将实际风速与临界风速进行比较，判断此时系统中各物体的状态，与题目中已知数据进行比较。在钢桶倾斜角度达到临界角度时，计算锚链与海床的夹角并于题中数据进行比较，计算重物球的质量。在浮标完全没入海面时，计算相应条件下重物球的质量，从而确定满足条件的重物球的质量范围。 针对问题三，要求在不同条件下，求出系泊系统中各物体的状态。以型号I锚链为例，当水流方向与风速方向相同时，系统条件最差，分析在不同水深条件下的系泊系统设计。由题中已知条件确定系统设计的限制条件，对系统各物体进行受力分析，以使整体结果最小，即可得出最优的系泊系统设计。 关键词：悬链线多目标非线性规划 一、问题重述 近浅海观测网的传输节点由浮标系统、系泊系统和水声通讯系统组成（如图1所示）。某型传输节点的浮标系统可简化为底面直径2m、高2m的圆柱体，浮标的质量为1000kg。系泊系统由钢管、钢桶、重物球、电焊锚链和特制的抗拖移锚组成。锚的质量为600kg，锚链选用无档普通链环，近浅海观测网的常用型号及其参数在附表中列出。钢管共4节，每节长度1m，直径为50mm，每节钢管的质量为10kg。要求锚链末端与锚的链接处的切线方向与海床的夹角不超过16度，否则锚会被拖行，致使节点移位丢失。水声通讯系统安装在一个长1m、外径30cm的密封圆柱形钢桶内，设备和钢桶总质量为100kg。钢桶上接第4节钢管，下接电焊锚链。钢桶竖直时，水声通讯设备的工作效果最佳。若钢桶倾斜，则影响设备的工作效果。钢桶的倾斜角度（钢桶与竖直线的夹角）超过5度时，设备的工作效果较差。为了控制钢桶的倾斜角度，钢桶与电焊锚链链接处可悬挂重物球。 系泊系统的设计问题就是确定锚链的型号、长度和重物球的质量，使得浮标的吃水深度和游动区域及钢桶的倾斜角度尽可能小。