Theorem 2 The induced graph GЈ = G[VЈ] is a connected graph. Proof: We prove this theorem by contradiction. Assume that GЈ is disconnected and v and u are two disconnected vertices in GЈ. Assume dis G (v, u) = k + 1 > 1 and (v, v 1 , v 2 , , v k , u) is a shortest path between vertices v and u in G. Clearly, all v 1 , v 2 , , v k are distinct; and among them there is at least one v i such that m(v i ) = F (otherwise, v and u are con- nected in GЈ). On the other hand, the two adjacent vertices of v i , v i–1 and v i+1 , are not con- nected in G; otherwise, (v, v 1 , v 2 , , v k , u) is not a shortest path. Therefore, m(v i ) = T based on the marking process. This brings a contradiction. २ The next theorem shows that, except for source and destination vertices, all intermedi- ate vertices in a shortest path are contained in the dominating set derived from the mark- ing process. Theorem 3 The shortest path between any two nodes does not include any nongateway node as an intermediate node. Proof: We prove this theorem also by contradiction. Assume that a shortest path between two vertices v and u includes a nongateway node v i as an intermediate node; in other words, this path can be represented as (v, , v i–1 , v i , v i+1 , , u). We label the vertex that precedes v i on the path as v i–1 ; similarly, the vertex that follows v i on the path is la- beled as v i+1 . Because vertex v i is a nongateway node, i.e., m(v i ) = F, there must be a con- nection between v i–1 and v i+1 . Therefore, a shorter path between v and u can be found as (v, , v i–1 , v i+1 , , u). This contradicts the original assumption. २ Since the problem of determining a minimum connected dominating set of a given con- nected graph is NP-complete, the connected dominating set derived from the marking process is normally nonminimum. In some cases, the resultant dominating set is trivial, i.e., VЈ = V or VЈ = { }. For example, any vertex-symmetric graph will generate a trivial dominating set using the proposed marking process. However, the marking process is effi- cient for an ad hoc wireless network where the corresponding graph tends to form a set of localized clusters (or cliques). The simulation results shown in [36] confirm this observa- tion. When the transmission radius of a mobile host is not too large, the proposed algo- rithm generates a small connected dominating set. 20.3.3 Dominating Set Reduction In the following, we propose two rules to reduce the size of a connected dominating set generated from the marking process. We first assign a distinct ID, id(v), to each vertex v in GЈ. N[v] = N(v) ʜ {v} is the closed neighbor set of v, as opposed to the open one, N(v). Rule 1: Consider two vertices v and u in GЈ. If N[v] ʕ N[u] in G and id(v) < id(u), change the marker of v to F if node v is marked, i.e., GЈ is changed to GЈ – {v}. The above rule indicates that the closed neighbor set of v is covered by that of u and 20.3 FORMATION OF A CONNECTED DOMINATING SET 433 vertex v can be removed from GЈ if the ID of v is smaller than that of u. Note that if v is marked and its closed neighbor set is covered by the one of u, it implies vertex u is also marked. When v and u have the same closed neighbor set, the vertex with the smaller ID is removed. It is easy to prove that GЈ – {v} is still a connected dominating set of G. The condition N[v] ʕ N[u] implies that v and u are connected in GЈ. In Figure 20.4 (a), since N[v] Ͻ N[u], vertex v is removed from GЈ if id(v) < id(u) and vertex u is the only dominating node in the graph. In Figure 20.4 (b), since N[v] = N[u], either v or u can be removed from GЈ. To ensure one and only one is removed, we pick the one with the smaller ID. Rule 2: Assume that u and w are two marked neighbors of marked vertex v in GЈ. If N(v) ʕ N(u) ʜ N(w) in G and id(v) = min{id(v), id(u), id(w)}, then change the marker of v to F. The above rule indicates that when the open neighbor set of v is covered by the open neighbor sets of two of its marked neighbors, u and w, if v has the smallest ID of the three, it can be removed from GЈ. The condition N(v) ʕ N(u) ʜ N(w) in Rule 2 implies that u and w are connected. The subtle difference between Rule 1 and Rule 2 is the use of open and closed neighbor sets. Again, it is easy to prove that GЈ – {v} is still a connected domi- nating set. Both u and w are marked, because the facts that v is marked and N(v) ʕ N(u) ʜ N(w) in G usually do not imply that u and w are marked. Therefore, if one set of u and w is not marked, v cannot be unmarked (change the marker to F). To apply Rule 2, an addition- al step last step needs to be included in the marking process: If v is marked [m(v) = T], send its status to all its neighbors. Consider the example in Figure 20.4 (c). Clearly, N(v) ʕ N(u) ʜ N(w). If id(v) = min{id(v), id(u), id(w)}, vertex v can be removed from GЈ based on Rule 2. If id(u) = min{id(v), id(u), id(w)}, then vertex u can be removed based on Rule 1, since N[u] ʕ N[v]. If id(w) = min{id(v), id(u), id(w)}, no vertex can be removed. Therefore, the ID as- signment also decides the final outcome of the dominating set. Note that Rule 2 can be easily extended to a more general case where the open neighbor set of vertex v is covered by the union of open neighbor sets of more than two neighbors of v in GЈ. However, the connectivity requirement for these neighbors is more difficult to specify at vertex v. The role of ID is very important for avoiding “illegal simultaneous” removal of vertices 434 DOMINATING-SET-BASED ROUTING IN AD HOC WIRELESS NETWORKS uvw vu vu (a) (b) (c) Figure 20.4 Three examples of dominating set reduction. in GЈ. In general, vertex v cannot be removed even if N[v] Ͻ N[u], unless id(v) < id(u). Consider the example of Figure 20.4 (c) with id(v) = min{id(v), id(u), id(w)}. If the above rule were not followed, vertex u would be unmarked to F (because N[u] ʚ N[v] even though id(v) < id(u)); and based on Rule 2, vertex v would be unmarked to F. Clearly, the only vertex w in VЈ does not form a dominating set any more. 20.3.4 Example Figure 20.5 shows an example of using the proposed marking process and its extensions to identify a set of connected dominating nodes. Each node keeps a list of its neighbors and sends the list to all its neighbors. By doing so, each node has distance-2 neighborhood in- formation, i.e., information about its neighbors and the neighbors of all its neighbors. Node 1 does not mark itself as a gateway node because its only neighbors, 2 and 3, are connected. Node 3 marks itself as a gateway node because there is no connection between neighbors 1 and 4 (2 and 4). After node 3 marks itself, it sends its status to its neighbors 1, 20.3 FORMATION OF A CONNECTED DOMINATING SET 435 (a) 2 3 4 5 6 7 12 9 10 11 16 15 13 20 19 8 14 18 17 1 Figure 20.5 (a) Marked gateways without applying rules. (b) Marked gateways by applying Rules 1 and 2. 2 3 4 5 6 7 12 9 10 11 16 15 13 20 19 8 14 18 17 1 (b) 2, and 4. This gateway status is used to apply Rule 2 to unmark some gateway nodes to nongateway nodes. Figure 20.5 (a) shows the gateway nodes (nodes with double cycles) derived by the marking process without applying two rules. After applying Rule 1, node 17 is unmarked to the nongateway status. The closed neighbor set of node 17 is N[17] = {17, 18, 19, 20}, and the closed neighbor set of node 18 is N[18] = {16, 17, 18, 19, 20}. Apparently, N[17]ʕ N[18]. Also the ID of node 17 is less than the ID of node 18, thus node 17 can unmark itself by applying Rule 1. After applying Rule 2, node 8 is unmarked to nongateway status, as shown in Figure 20.5 (b). Node 8 knows that its two neighbors 14 and 16 are all marked. This invokes node 8 to apply Rule 2 to check whether condition N(8) ʕ N(14) ʜ N(16) holds or not. The neighbor set of node 14 is N(14) = {7, 8, 9, 10, 11, 12, 13, 16}, the neighbor set of node 8 is N(8) = {12, 13, 14, 15, 16}, the neighbor set of node 16 is N(16) = {8, 14, 15, 18}, and therefore, N(14) ʜ N(16) = {7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18. Apparently, N(8) ʕ N(14) ʜ N(16). The ID of node 8 is the smallest among nodes 8, 14, and 16. Thus node 8 can unmark itself by applying Rule 2. 20.3.5 Mobility Management In an ad hoc wireless network, each host can move around without speed and distance lim- itations. Also, in order to reduce power consumption, mobile hosts may switch off at any time and then switch on later. We can categorize topological changes of an ad hoc wireless network into three different types: mobile host switching on, mobile host switching off, and mobile host movement. The challenge here is to find when and how each vertex should update/recalculate gate- way information. The gateway update means that only individual mobile hosts update their gateway/nongateway status. The gateway recalculation means that the entire network re- calculates gateway/nongateway status. If many mobile hosts in the network are in move- ment, gateway recalculation may be a better approach, i.e., the connected dominating set is recalculated from scratch. On the other hand, if only a few mobile hosts are in move- ment, then gateway information can be updated locally. It is still an open problem as to when to update gateways and when to recalculate gateways from scratch. In the following, we will focus only on the gateway update for the three types of topol- ogy changes mentioned above. Without lost of generality, we assume that the underlying graph of an ad hoc wireless network always remains connected. We show that for both switching on and switching off operations, the update of node status (gateway/nongate- way) can be limited to neighbors of the node that switches on or off. When a mobile host v switches on, only its nongateway neighbors, along with host v, need to update their status, because any gateway neighbor will still remain as gateway after a new vertex v is added. For example, in Figure 20.6 (a), when host v switches on, the sta- tus of gateway neighbor host u is not affected, because at least two of u’s three neighbors, u 1 , u 2 , and u 3 , are not connected originally and these connections will not be affected by host v’s switching on. On the other hand, in Figure 20.6 (b), host v’s switching on may lead to non-gateway neighbor w to mark itself as gateway, depending on the connection between host w’s neighbors w 1 , w 2 , and w 3 . The corresponding update process can be as follows: 436 DOMINATING-SET-BASED ROUTING IN AD HOC WIRELESS NETWORKS Switching On 1. Mobile host v broadcasts to its neighbors about its switching on. 2. Each host w ʦ v ʜ N(v) exchanges its open neighbor set N(w) with its neighbors. 3. Host v assigns its marker m(v) to T if there are two unconnected neighbors. 4. Each nongateway host w ʦ N(v) assigns its marker m(w) to T if it has two uncon- nected neighbors. 5. Whenever there is a newly marked gateway, host v and all its gateway neighbors ap- ply Rule 1 and Rule 2 to reduce the number of gateway hosts. When a mobile host v switches off, only gateway neighbors of that host need to update their status, because any nongateway neighbor will still remain as nongateway after vertex v is deleted. The corresponding update process can be as follows. Switching Off 1. Mobile host v broadcasts to its neighbors about its switching off. 2. Each gateway neighbor w ʦ N(v) exchanges its open neighbor set N(w) with its neighbors. 3. Each gateway neighbor w changes its marker m(w) to F if all neighbors are pairwise connected. Note that since the underlying graph is connected, we can easily prove by contradiction that the resultant dominating set (derived from the above marking process) is still connect- ed when a host (gateway or nongateway) switches off. A mobile host v’s movement can be viewed as several simultaneous or nonsimultane- ous link connections and disconnections. For example, when a mobile host moves, it may lead to several link disconnections from its neighbor hosts and, at the same time, it may have new link connections to the hosts within its wireless transmission range. These new 20.3 FORMATION OF A CONNECTED DOMINATING SET 437 v u1 u3 u2 u (a) g atewa y nei g hbor u w1 w2 w3 wv (b) non- g atewa y nei g hbor w new li n k Figure 20.6 Mobile host v switching on. links may be disconnected again depending on the way host v moves. Other details of mo- bility management can be found in [38]. 20.4 EXTENSIONS 20.4.1 Networks with Unidirectional Links In this subsection, we extend the dominating-set-based routing to ad hoc wireless net- works with unidirectional links. In an ad hoc wireless network, some links may be unidi- rectional due to different transmission ranges of hosts or the hidden terminal problem [34], in which several packets intended for the same host collide and, as a result, they are lost. With few exceptions, such as the dynamic source routing protocol (DSR) [3], most existing protocols assume bidirectional links. Prakash [27] studied the impact of unidirec- tional links on some of the existing distance vector routing protocols such as destination- sequenced distance vector (DSDV) [26], and found that unidirectional links prove costly. It is shown that hosts need to exchange O(|V| 2 ) amount of information with each other, where |V| is the number of hosts in the network. In a network with directed links, the domination concept has to be redefined. Specifi- cally, an ad hoc wireless network is represented as a directed graph D = (V, A) consisting of a finite set V of vertices and a set A of directed edges, where A ʚ V × V. D is a simple graph without self-loop and multiple edges. A directed (also called unidirectional) edge from u to v is denoted by an ordered pair uv. If uv is an edge in D, we say that u dominates v and v is an absorbant of u. A set VЈ ʚ V is a dominating set of D if every vertex v ʦ V – VЈ is dominated by at least one vertex u ʦ VЈ. Also, a set VЈ ʚ V is called an absorbant set if for each vertex u ʦ V – VЈ, there exists a vertex v ʦ VЈ which is an absorbant of u. The dominating neighbor set of vertex u is defined as {w|wu ʦ A}. The absorbant neighbor set of vertex u is defined as {v|uv ʦ A}. A directed graph D is strongly connected if for any two vertices u and v, a uv path (i.e., a path connecting u to v) exists. It is assumed that D is strongly connected. If it is not strongly connected, the network management subsystem will partition the network into a set of independent subnetworks, each of which is strongly connected. Other concepts related to graph theory and, in particular, directed graphs can be found in [2]. The objective here is to quickly find a small set that is both dominating and absorbant in a given directed graph. Note that the absorbant subset may overlap with the dominating subset. In an undirected graph, these two concepts are the same and, hence, a dominating set is an absorbant set. To determine a set that is both dominating and absorbant, we propose an extended marking process. m(u) is a marker for vertex u ʦ V, which is either T (marked) or F (un- marked). We will show later that the marked set is both dominating and absorbant. Extended Marking Process 1. Initially assign F to each u ʦ V. 2. u changes its marker m(u) to T if there exist vertices v and w such that wu ʦ A and uv ʦ A, but wv  A. 438 DOMINATING-SET-BASED ROUTING IN AD HOC WIRELESS NETWORKS Figure 20.7 (a) shows four gateway hosts, 4, 7, 8, and 9, derived from the extended marking process. Figure 20.7 (b) and (c) show gateway domain number at host 8 and gate- way routing table at host 8, respectively. Node ids appended with subscripts a and d corre- spond to absorbant neighbors and dominating neighbors, respectively. A bidirectional edge (v, u) can be considered as two unidirectional edges vu and uv. Arrow dashed lines correspond to unidirectional edges and solid lines represent bidirectional edges. Note that the above extended marking process requires each vertex u to know only its absorbant neighbor set. Figure 20.8 shows three assignments of u, with one dominating neighbor w and one absorbant neighbor v. The only case in Figure 20.8 with m(u) = F is when wv ʦ A, for each dominating neighbor w and each absorbant neighbor v of u. The fourth case, where v and w are bidirectionally connected [a combination of Figures 20.8 (a) and (b)], is not shown. Assume that VЈ is the set of vertices that are marked T in V, i.e., VЈ = {u|u ʦ V, m(u) = T}. The induced graph DЈ is the subgraph of D induced by VЈ, i.e., DЈ = D[VЈ]. Most of results for undirected graphs (Theorems 1 to 4) also hold for directed graphs, as shown in the following propositions. The proofs of these results can be found in [37]. 20.4 EXTENSIONS 439 Figure 20.7 (a) A sample ad hoc wireless network with unidirectional links. (b) Gateway domain member list at host 8. (c) Gateway routing table at host 8. ( a ) gateway domain member list gateway routing table destination member list next hop distance (c)(b) 3 10 11a 9 (1,2,3,11) 4 (5,6) 7 (6d) 9 7 7 1 2 1 2 1 11 10 7 8 5 9 4 6 3 () Proposition 1: Given a D = (V, A) that is strongly connected, the vertex subset VЈ, derived from the extended marking process, has the following properties: ț VЈ is empty if and only if D is completely connected, i.e., for every pair of vertices u and v, there are two edges uv and vu. ț If D is not completely connected, VЈ forms a dominating and absorbant set. When the given D is completely connected, all vertices are marked F. This make sense, because if all vertices are directly connected, there is no need to use a dominating and ab- sorbant set to reduce D. Proposition 2: VЈ includes all the intermediate vertices of any shortest path. Proposition 3: The induced graph DЈ = D[VЈ] is a strongly connected graph. Propositions 1, 2, and 3 serve as bases of the dominating-set-based routing. The domi- nating and absorbant set derived from the extended marking process has the desirable properties of routing optimality (Proposition 2) and connectivity (Proposition 3). Howev- er, in general, the derived dominating and absorbant set is not minimum. In the following, we propose two rules to reduce the size of a connected dominating and absorbant set generated from the extended marking process. We first randomly assign a distinct label, id(v), to each vertex v in V. In a directed graph, N d (u)[N a (u)] represents the dominating (absorbant) neighbor set of vertex u. In general, the neighbor set is the union of the corresponding dominating neighbor and absorbant neighbor sets, i.e., N(u) = N a (u) ʜ N d (u). Vertex u is called neighbor of vertex v if u is a dominating, absorbant, or dominating and absorbant neighbor of v. Rule 1a: Consider two vertices u and v in induced graph DЈ. Unmark u, i.e., DЈ is changed to DЈ u = DЈ – {u}, if the following conditions hold. 440 DOMINATING-SET-BASED ROUTING IN AD HOC WIRELESS NETWORKS m(u)=F wv u (a) m(u)=T wv u (b) m(u)=T wv u (c) Figure 20.8 Marker of u for three different situations. 1. N d (u) – {v ʕ N d (v) and N a (u) – {v ʕ N a (v) in D. 2. id(u) < id(v). The above rule indicates that when the dominating (absorbant) neighbor set of u (ex- cluding v) is covered by the dominating (absorbant) of v, vertex u can be removed from DЈ if the ID of u is smaller than that of v. Note that u and v may or may not be connected (they are bidirectional or unidirectional). The role of ID is very important in avoiding “il- legal simultaneous” removal of vertices in VЈ when Rule 1a is applied “simultaneously” to each vertex. In general, vertex u cannot be removed even if N d (u) – {v} ʕ N d (v) and N a (u) – {v} ʕ N a (v) in D, unless id(u) < id(v). Consider a graph of four vertices, u, v, s, and t, with four undirected edges (u, s), (s, v), (v, t), and (t, u). All four vertices will be marked using the extended marking process. Also, N d (u) = N d (v) = N a (u) = N a (v) = (s, t)[N d (s) = N d (t) = N a (s) = N a (t) = (u, v)]. Without using ID, both u and v (also s and t) will be un- marked, leaving no marked vertex. With ID, one of u and v (also s and t) will be unmarked, leaving two marked vertices. Rule 2a: Assume that v and w are two marked vertices in DЈ. Unmark u if the following conditions hold. 1. N d (u) – {v, w} ʕ N d (v) ʜ N d (w) and N a (u) – {v, w} ʕ N a (v) ʜ N a (w) in D. 2. id(u) = min{id(u), id(v), id(w)}. 3. v and w are bidirectionally connected. The above rule indicates that when u’s dominating (absorbant) neighbor set (excluding v and w) is covered by the union of dominating (absorbant) sets of v and w, vertex u can be removed from DЈ if the ID of u is smaller than those of v and w. Again, u and v(w) may or may not be connected. Figure 20.9 shows an example of using the extended marking process and its exten- sions (two rules) to identify a set of connected dominating and absorbant nodes. Figure 20.9 (a) shows the gateway nodes (nodes with double cycles) derived by the extended marking process without applying two rules. Figure 20.9 (b) shows the remaining gateway nodes after applying two rules. Assume that VЈ * is the resultant dominating and absorbant set when Rule 1a and Rule 2a are simultaneously applied to all vertices in VЈ. The following result shows that VЈ * (its in- duced graph is DЈ * ) is still a connected dominating and absorbant set of V. The shortest path property of Proposition 3 still holds in DЈ * for Rule 1a, but not for Rule 2a. Proposition 4: If VЈ is a strongly connected dominating and absorbant set of D derived by using the extended marking process, then VЈ * derived by using Rule 1a and Rule 2a on all vertices in VЈ is still a strongly connected dominating and absorbant set of V. In addition, if VЈ * is derived by applying Rule 1 alone, then VЈ * still includes all intermediate vertices of at least one shortest path for any pair of vertices in V. 20.4 EXTENSIONS 441 Actually, for each application of Rule 2a, the length of a shortest path (that includes u as an intermediate node) increases by at most one. 20.4.2 Hierarchical Dominating Sets Hierarchical routing aggregates hosts into clusters, clusters into superclusters, and so on. If addresses of the destination host and the host that is forwarding the packet belong to dif- ferent superclusters, then forwarding will be done via an intersupercluster route; if they belong to the same supercluster but to different clusters, forwarding will be done via inter- cluster routes; if they belong to the same cluster, forwarding will be done via intracluster routes. The extended marking process can be applied to the induced graph to generate a domi- nating set of a given dominating set (here interpreted as dominating and absorbant set). 442 DOMINATING-SET-BASED ROUTING IN AD HOC WIRELESS NETWORKS 2 3 4 5 6 9 10 11 16 15 13 20 19 8 14 18 17 1 7 12 (a) 2 3 4 5 6 9 10 11 16 15 13 20 19 8 14 18 17 7 12 1 (b) Figure 20.9 (a) Marked gateways using the extended marking process. (b) Marked gateways ob- tained by applying Rules 1a and 2a. [...]... 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