The Energy-Aware Operational Time of Wireless Ad-Hoc Sensor Networks

Abstract

Sensor networks are deployed in numerous military and civil applications, such as remote target detection, weather monitoring, weather forecast, natural resource exploration and disaster management. Despite having many potential applications, wireless sensor networks still face a number of challenges due to their particular characteristics that other wireless networks, like cellular networks or mobile ad hoc networks do not have. The most difficult challenge of the design of wireless sensor networks is the limited energy resource of the battery of the sensors. This limited resource restricts the operational time that wireless sensor networks can function in their applications. Routing protocols play a major part in the energy efficiency of wireless sensor networks because data communication dissipates most of the energy resource of the networks. This paper studies the importance of considering neighboring nodes in the energy efficiency routing problem. After showing that the routing problem that considers the remaining energy of all sensor nodes is NP-complete, heuristics are proposed for the problem. Simulation results show that the routing algorithm that considers the remaining energy of all sensor nodes improves the system lifetime significantly compared to that of minimum transmission energy algorithms. Also, the energy dissipation of neighboring nodes accounts for a considerable amount of the total energy dissipation. Therefore, a method that reduces the energy dissipation by notifying the neighboring nodes to turn off their radio when not necessary is proposed. By reducing the unnecessary energy dissipation of the neighbors, the lifetime is increased significantly.

Appendix

1.1 Proof of the NP-completeness of problem (14) [11, 12]

Path with forbidden pairs problem (PFP)

Instance::

Consider a graph G(V, E), given a source node s and destination node d, and a collection C = {(a ₁,b ₁), ..., (a _m ,b _m )} of pairs of vertices in V.

Question::

Find a simple path from s to d that contains at most one vertex from each pair in C.

The PFP problem is known to be well-known graph theory NP-complete.

Path with remaining energy problem (RE)

A sensor network is modelled as G(V, E). All nodes can transmit at a constant i = P, ∀ i ∈ V. In other words, a node i does not transmit or transmit with power P. Every node i has the remaining energy capacity e(i). Given a source node s and a destination node d, find a simple path from s to d that e(i) ≥ c for all nodes i ∈ V.

We now give a polynomial reduction from this problem to the Path with Forbidden Pairs problem (PFP). Without loss of generality, we assume that for any node, a reception usage of one unit of energy (i.e., r _i  = 1 for any node i). We first transform an instance (G(V, E), s, d, C) of the PFP problem in an instance (G′(V′, E′), s, d, p, c) of the Remaining Energy problem by formally definition as follows, where s and d are unchanged, c is the minimum tolerable capacity at any node i and is set to an arbitrary positive value.

$$\label{eq17} V' = V \cup \{v_{xy} | (x, y) \in C\} \\ $$

(17)

$$\label{eq18} E' = E \cup \{(x, v_{xy}), (y, v_{xy}) | (x, y) \in C \} $$

(18)

e _i is the remaining energy set to:

• 1. e _i  = c if i ∈ {v _xy | (x, y) ∈ C & (x = t) || y = t}

• 2. e _i  = c + 1 if $i \in \{v_{xy} | (x, y) \in C \& (x \not= t) || y \not= t\}$

• 3. e _i  = c + |V|, otherwise.

By the definition, G′ contains all the vertices of G and m new vertices that represent a forbidden pair. Let us define F as the set of the m vertices. Each vertex of F is only connected to its two respective “forbidden” vertices and is assigned e _i  = c + 1, or e _i  = c, if the destination is part of the forbidden pair.

We now prove that a solution of this instance of the RE problem if and only if it is a solution for the original instance of the PFP problem.

It is easy to see that a solution from s to d for the RE problem in (G′(V′, E′), s, d, p, c) does not include any of the vertices in F, as any vertex i of the path (except the destination d) requires to decrement e _i by at least 2. Hence this path is also the path in G.

Conversely, given a solution path ∏ ′ of the instance (G(V, E), s, d, C), we can verify that the path is a feasible solution path for the RE problem in (G′(V′, E′), s, d, p, c). As G is a sub graph of G′, a path in G is also a path in G′. Hence we need to verify that the path in G satisfies the remaining energy constraints. As all nodes except nodes in F respect the remaining energy constraints, only nodes in F may violate the feasibility of the solution. This can be seen that none of nodes in F belong to the path as $F \not\in G$. A vertex in F can be a be neighbor of maximum a vertex of ∏ ′. As each vertex of F can only be a neighbor of its forbidden pair, the vertex cannot be a neighbor of two nodes in the path.

As the proof follows a polynomial reduction to the Path with Forbidden Pairs (PFP) problem, the RE problem is NP-complete.