On Sensor Network Reconfiguration for Downtime-Free System Migration

Abstract

Many state-of-the-art wireless sensor networks have been equipped with reprogramming modules, e.g., those for software/firmware updates. System migration tasks such as software reprogramming, however, will interrupt normal sensing and data processing operations of a sensor node. Although such tasks are occasionally invoked, the long time of such tasks may disable the network from detecting critical events, posing a severe threat to many sensitive applications. In this paper, we present the first formal study on the problem of downtime-free migration. We demonstrate that the downtime can be effectively eliminated, by partitioning the sensors into subsets, and let them perform migration tasks successively with the rest still performing normal services. We then present a series of effective algorithms, and further extend our solution to a practical distributed and localized implementation. The performance of these algorithms have been evaluated through extensive simulations, and the results demonstrate that our algorithms achieve good balance between the sensing quality and system migration time.

Appendix: Proof of Lemma 1

First, this problem is in NP: Given a partition scheme, a nondeterministic algorithm only needs to calculate the event detection capability (EDC) of each division so as to get the EDC of the network during the time interval T. And then it can verify whether this value is smaller than u or not. So now we need to prove that this problem is harder than a known NP-Complete problem.

We transform the provably NP-Complete set partition problem [7] to the decision version of the simplified sensor network reconfiguration problem. Given a set of non-negative numbers $\{q_i\}_{i=1}^n$, the set partition problem asks whether it is feasible to partition the set so that the sum of numbers in each partition is equal.

As p _ij ∈ [0, 1), we can construct an n by m matrix $\mathcal{Q}$ of which each element is defined as q _ij  = − log ₂(1 − p _ij ). We can know q _ij  > 0. Based on the property of d _ik , we get:

$$ 1 - d_{ik}p_{ij} = 2^{-d_{ik}q_{ij}}. $$

(12)

Let us construct an instance of the sensor network reconfiguration problem in which N = 2, q _ij is equal to each other given the same i and equal to the q _i in the set partition problem, and $u= 1- 2^{-(\sum_{i=1}^{n}q_i)/2}$. Now we can always have d _i1 = 1 − d _i2 because a sensor should be in either division D ₁ or division D ₂, but not in both. Also we can write q _ij as q _i without the subscript j. Therefore, we get:

$$ \begin{array}{rcl} P^\prime - u &=& \min\limits_{\forall k}\left\{\min\limits_{\forall j}\left[1 - \prod\limits_{i=1}^n(1-d_{ik}p_{ij})\right]\right\}-u\\ &=& \min\limits_{\forall k}\left[\min\limits_{\forall j}(1-2^{-\sum_{i=1}^n d_{ik}q_{ij}})\right]-u\\ &=& \min\limits_{\forall k}\left(1-2^{-\sum_{i=1}^n d_{ik}q_{i}}\right)-u\\ &=& 2^{-\frac{\sum_{i=1}^{n}q_i}{2}} -2^{-\left[\min\limits_{\forall k}(\sum_{i=1}^n d_{ik}q_{i})\right]}. \end{array} $$

(13)

If the answer to whether P ^′ ≥ u is yes, we get:

$$ \begin{array}{rcl} &&\min\limits_{\forall k}\left(\sum_{i=1}^n d_{ik}q_{i}\right) \geq \frac{\sum\limits_{i=1}^{n}q_i}{2} \\ &&\Rightarrow\left\{\begin{array} {l} \sum\limits_{i=1}^n d_{i1}q_{i} \geq \frac{\sum_{i=1}^{n}q_i}{2}\\[12pt] \sum\limits_{i=1}^n d_{i2}q_{i} = \sum\limits_{i=1}^n (1-d_{i1})q_{i} \geq \frac{\sum_{i=1}^{n}q_i}{2} \end{array}\right.\Rightarrow\\ &&\frac{\sum_{i=1}^{n}q_i}{2}\geq\sum\limits_{i=1}^n d_{i1}q_{i} \geq \frac{\sum_{i=1}^{n}q_i}{2} \Rightarrow \sum\limits_{i=1}^n d_{i1}q_{i} = \frac{\sum_{i=1}^{n}q_i}{2}. \end{array} $$

(14)

Therefore, the answer to the set partition problem is also yes.

On the other hand, if the answer to the set partition problem is yes, in the same way we can partition the sensors in the simplified sensor network reconfiguration problem so that:

$$ \begin{array}{rcl} &&\left\{\begin{array} {l} \sum\limits_{i=1}^n d_{i1}q_{i} = \frac{\sum_{i=1}^{n}q_i}{2}\\[12pt] \sum\limits_{i=1}^n d_{i2}q_{i} = \sum\limits_{i=1}^n (1-d_{i1})q_{i} = \frac{\sum_{i=1}^{n}q_i}{2} \end{array}\right.\\ &&\Rightarrow \min\limits_{\forall k}\left(\sum_{i=1}^n d_{ik}q_{i}\right)=\frac{\sum_{i=1}^{n}q_i}{2}. \end{array} $$

(15)

According to Eq. 13, P ^′ = u. Therefore, the answer to the decision version of the simplified sensor network reconfiguration problem is also yes.

The above reduction requires only O(n) steps to be completed (for calculating p _i and u with q _i ). Therefore, the decision version of the simplified sensor network reconfiguration problem is both NP-Hard and NP. Then it is NP-Complete. The lemma is proved.