Topology control for fault-tolerant communication in wireless ad hoc networks

Abstract

Fault-tolerant communication and energy efficiency are important requirements for future-generation wireless ad hoc networks, which are increasingly being considered also for critical application domains like embedded systems in automotive and aerospace. Topology control, which enables multi-hop communication between any two network nodes via a suitably constructed overlay network, is the primary target for increasing connectivity and saving energy here. In this paper, we present a fault-tolerant distributed topology control algorithm that constructs and continuously maintains a k-regular and k-node-connected overlay for energy-efficient multi-hop communication. As a by-product, it also builds a hierarchy of clusters that reflects the node density in the network, with guaranteed and localized fault-tolerant communication between any pair of cluster members. The construction algorithm automatically adapts to a dynamically changing environment, is guaranteed to converge, and exhibits good performance as well.

Appendices

Appendices

1.1 Appendix A. Topology construction method proofs

Theorem 8

For every graph G with n′ ≥ 1 and n′′ ≥ 2k − 2, there is exactly one joint-minimal admissible overlay graph G′.

Proof 7

We first show that at least one joint-minimal admissible overlay graph exists, by inductively constructing joint-minimal groups (with increasing weights) one after the other and showing that these groups are stable, i.e., not destroyed during later construction steps. For i ≥ 1, let g _i be the group added in the i-th step of this construction and G _i−1 be the set of groups constructed in steps 1,…, i − 1.

For i = 1, g ₁ is the (unique) group with minimal ω(g ₁) chosen among all nodes in $\Uppi.$ Clearly, g ₁ is joint-minimal and trivially stable. For g _i , i > 1, we choose the group with minimal weight ω(g _i ) formed from groups and nodes in $G_{i-1} \cup \Uppi.$ Clearly, all members of g _i agree on this as the minimal choice. Moreover, it holds that ω(g _i ) >  ω(g _i−1): For, if g _i does not incorporate g _i−1, then it can incorporate only groups and nodes from $G_{i-2} \cup \Uppi,$ where g _i−1 is the minimum-weight choice. If g _i incorporates g _i−1, then ω(g _i ) > ω(g _i−1) according to the definition of ω(·).

It remains to be shown, however, that the choice to include some member $x\,\in\, G_{i-1} \cup \Uppi$ in g _i does not violate the joint-minimum criterion for some earlier built group g _j  ∈ G _i−1, j < i, which might already have x as a member. Since ω(g _j ) < ω(g _i ), however, this cannot happen.

Finally, since it is ensured by our weight assumptions that all regular nodes are used up before gateway nodes are considered, Theorem 1 holds also for this inductive construction. Hence, the final group can be built since at least 2k − 2 gateway nodes are available.

It thus only remains to be shown that the joint-minimal overlay graph is unique. So let us assume that there exist two admissible overlay graphs G′₁ and G′₂, which are both joint-minimal. Going up the topology tree of G′₁ and G′₂, at some depth the group structure must be different. More specifically, there must be a group member x in some group g _i in G′₁ which is member in some other (= not corresponding) group g _j in G′₂. Recall that a node or a group can only be member of at most one group. However, either g _i of G′₁ or g _j of G′₂ has lower weight, which implies that the admissible overlay graphs G′₁ and G′₂ cannot both be joint-minimal according to Definition 3. □

To show that our admissible overlay graph is k-connected, we first need an additional definition and a few preliminary lemmas.

Definition 7

A node p is nd-connected (“node-disjointly connected”) to a set $S \,\subseteq\, \Pi,$ if p is connected to every s ∈ S via node-disjoint paths. Formally, p is nd-connected to S, if there exists a set P of paths, containing a path from p to s for every s ∈ S, and p is the only node shared between any two paths in P.

Lemma 1

Let$g \,\in\, ({\mathbb{G}} \cup \Pi)$ be a group or node. In G _g , every p ∈ nodes(g) is nd-connected to T _g , the set of terminal nodes of g. Only nodes from nodes(g) are required for these paths.

Proof 8

Induction on depth d of g’s topology tree. For d = 0, g is a node. Thus, T _g  = nodes(g) = {g} and the claim follows trivially.

For some d > 0, suppose that the claim holds for every sub-group of g. Clearly, p ∈ nodes(g _a ) for some specific (direct) sub-group g _a of g and is hence nd-connected to $T_{g_a},$ resulting in a (multi-)set of k node-disjoint paths (p,…, t _a ), for every $t_a {\,\in\,}T_{g_a}.$ (If g _a is a node, all k paths are (p); if g _a is a group and $p {\,\in\,}T_{g_a},$ one of the k paths is (p).)

If g _a is a group, then, by construction, $T_{g_a}$ contains one node $t^{\prime}_a$ that is also element of T _g , and k − 1 other nodes t _a , each of which is connected to one terminal node t _b of another sub-group g _b of g. (If g _a is a node, it is both element of T _g and connected to all other sub-groups of g). Thus, we can extend k − 1 paths by following k − 1 different t _a →t _b edges (one for every “neighbor” g _b of g _a ). By applying the induction hypothesis, each of these t _b ’s is connected to all other terminal nodes of its group g _b , including the one which is also element of T _g , and, thus, each path can be continued to a terminal node of g. It is easy to see that all of these k − 1 paths are node-disjoint, since each one was extended only with nodes from “its” unique g _b .

The one remaining path does not need to be extended, since $t^{\prime}_a \in T_g.$ □

Theorem 9

B(G′) of each admissible overlay graph$G^{\prime}=G_{g_{final}}$with n′′ ≥ 2k − 2 isk-connected.

Proof 9

Let p and q be two arbitrary nodes and g ∈ groups(g _final ) be the lowest-level group such that p and q ∈ nodes(g). We show that p,q are connected via k node-disjoint paths, which implies the statement of our theorem via Menger’s Theorem [19].

Let g be a group, p ∈ nodes(g _a ) and q ∈ nodes(g _b ) with g _a ,g _b  ∈ members(g) and g _a ≠g _b . We claim that there are k node-disjoint paths between the two sets of terminals $T_{g_a}$ and $T_{g_b}.$ By the construction of a group, k − 1 pairs of nodes of $T_{g_a}\times T_{g_b}$ are connected by disjoint paths routed over at most one sub-group. According to Lemma 1, there is always a (group-internal) path between any pair of terminals of this sub-group. Thus, we have one path $(p, {\ldots, }t_a, t_b, {\ldots, }q)$ and k − 2 paths (p,…,t _a , t _c1,…,t _c2, t _b ,…, q) (with different $t_a {\,\in\,}T_{g_a}$ and $t_b \,\in\, T_{g_b}$ each). By Lemma 1, we know that all k − 1 (p,…,t _a ) and (t _b ,…,q) paths are node-disjoint (except for p and q, of course). Likewise, we have k − 2 different sub-groups g _c   ∈ members(g), where a g _c -internal path (t _c1,…,t _c2) exists for all $t_{c1}, t_{c2} \,\in\, T_{g_c}.$

Hence, we only have to show that the external connections of $T_{g_a}$ and $T_{g_b}$ are connected outside of g as well. Consider the parent group g′ of g.

• 1. If the terminal nodes $p^{\prime}\,\in\, T_{g_a}$ and $q^{\prime}\,\in\, T_{g_b}$ of group g are non-terminals in g′, they are connected in g′ via some path (p′, t _c1,…,t _c2, t _d1,…,t _d2, q′), with g _c (terminals: t _cx ) and g _d (terminals: t _dx ) being other sub-groups of g′. Lemma 1 shows that the paths (t _c1,…,t _c2) and (t _d1,…,t _d2) must exist and that they only require nodes from g _c and g _d , respectively.

• 2. If w.l.o.g. $p^{\prime} \in T_{g^{\prime}}$ but $q^{\prime}\notin T_{g^{\prime}},$ then q′ is used for some internal connection in g′, i.e., it is directly connected to some terminal t _c1 of g _c   ∈ members(g′). By Lemma 1, there must be some g _c -internal path (t _c1,…,t _c2) to some terminal t _c2 of g _c , which is also a terminal node of g′. Let q* = t _c2 and continue with (3).

• 3. If both p′ and q′ or q* are terminals in g′, go up to the parent group of g′ until either (1) or (2) applies or the final group is reached. If the final group g′ = g _final is reached, then p′ and q′ or q* are terminals in g _final and therefore gateway nodes by Definition 1. Since all gateway nodes are fully connected in B(G′), the external connection follows trivially.

Applying Menger’s Theorem, the claimed connectivity follows. □

1.2 Appendix B. Propose modules proofs

Lemma 2

The worst case message complexity of our local perfect propose module for generating a single proposal is O(n^k−1).

Proof 10

Let (G′) ^e _p be the incompletely built overlay graph at time e as viewed by node p. The maximum number of groups in (G′)^e is $< \lfloor\frac{n-1}{k-1}\rfloor = O(n)$ by Theorem 3. Since, in the worst case, p does not know about those groups (it need not be a member of any of those), O(n) messages are needed for initially collecting the required group information.

The maximum number of candidates in any initial potential member set is obviously $n+\lfloor\frac{n-1}{k-1}\rfloor.$ Assuming that search and result messages are sent to every node and leader in the worst case, we obtain a message complexity of at most $2\cdot \sum_{i=0}^{k-1} (n+\lfloor\frac{n-1}{k-1}\rfloor)^i=O(n^{k-1}).$ □

Lemma 3

The worst case time complexity of the local perfect propose module for generating a single proposal is O(n).

Proof 11

According to the proof of Lemma 2, node p has to establish information about all groups in (G′) ^e _p . Since group information is distributed over the whole system, this collection process may take up to O(n) time. pms _p may incorporate O(n) nodes and groups initially. In the search phase, each search and result message is forwarded k − 1 times in the worst case, concurrently for all nodes and groups in pms _p , resulting in a time complexity of 2·(k − 1) = O(1). Hence the claimed time complexity of O(n) follows. □

Lemma 4

The worst case message complexity of our local non-perfect propose module for generating a single proposal is O(n²).

Proof 12

Let again (G′) ^e _p be the incompletely built overlay graph at time e as viewed by node p. Due to the DFS search, at most a single search and result message is sent over any link connecting the at most $n+\lfloor\frac{n-1}{k-1}\rfloor=O(n)$ nodes/leaders in (G′) ^e _p . The worst case message complexity for generating a single proposal is hence at most O(n ²). □

Lemma 5

The worst case time complexity of the local non-perfect propose module for generating a single proposal is O(n²).

Proof 13

Since, by Lemma 4, O(n ²) messages are sent in the worst case, the worst case time complexity cannot be larger⁹ than O(n ²). □

Combining the perfect resp. the non-perfect propose modules with the basic construction algorithm leads to the worst case time complexities given by Theorem 10.

Theorem 10

The worst case time complexity for generating a joint-minimal admissible overlay network G′ with the basic topology construction algorithm using local perfect resp. local non-perfect propose modules is O(n²) resp. O(n³).

Proof 14

The time complexity of generating a single perfect proposal is O(n) by Lemma 3. Since all nodes generate proposals concurrently, after O(n) time, the first minimal proposal, which is always accepted by the construction algorithm and never destroyed, is released (among other proposals). Since there are O(n) groups to construct according to Theorem 3, the claimed time complexity of O(n ²) follows.

The result for the local non-perfect propose module is derived analogously, starting from the O(n ²) time complexity for a single proposal according to Lemma 5. □