[Paper Review] Minimum Power Range Assignment for Symmetric Connectivity in Sensor Networks with two Power Levels

Stefan Hoffmann, Egon Wanke|arXiv (Cornell University)|May 5, 2016

Mobile Ad Hoc Networks9 references3 citations

TL;DR

This paper proposes a parameterized approximation algorithm, Approx2LSRA_k, for the symmetric minimum power range assignment problem in wireless sensor networks with only two transmission power levels. It achieves a performance ratio arbitrarily close to π²/6 ≈ 1.645, with a specialized O(n log n) implementation for k=3 yielding a tight 7/4 approximation ratio, significantly improving on prior bounds for the two-power-level symmetric connectivity problem.

ABSTRACT

This paper examines the problem of assigning a transmission power to every node of a wireless sensor network. The goal is to minimize the total power consumption while ensuring that the resulting communication graph is connected. We focus on a restricted version of this Range Assignment (RA) problem in which there are two different power levels. We only consider symmetrical transmission links to allow easy integration with low level wireless protocols that typically require bidirectional communication between two neighboring nodes. We introduce a parameterized polynomial time approximation algorithm with a performance ratio arbitrarily close to $π^2/6$. Additionally, we give an almost linear time approximation algorithm with a tight quality bound of $7/4$.

Motivation & Objective

• To address the symmetric minimum power range assignment problem in wireless sensor networks with only two power levels, minimizing total energy consumption while ensuring bidirectional connectivity.
• To develop a polynomial-time approximation algorithm with a performance ratio arbitrarily close to π²/6 ≈ 1.645 for the two-power-level symmetric RA problem.
• To provide an efficient, almost linear-time implementation for the k=3 case with a tight 7/4 approximation guarantee.
• To establish theoretical bounds on the approximation quality of the proposed algorithm family, independent of geometric embedding or link correlation assumptions.

Proposed method

• The algorithm, Approx2LSRA_k, is based on identifying k-mergings (sets of k nodes that can be connected via max-power edges) and 2-mergings (pairs of nodes connected via max-power edges) in a greedy, component-based manner.
• It uses a union-find data structure to efficiently maintain and merge connected components of the min-power graph, enabling near-linear time complexity for k=3.
• The algorithm processes max-power edges in a way that prioritizes forming k-mergings before 2-mergings, ensuring structural optimality in the solution construction.
• The performance ratio is derived by analyzing worst-case instances where the algorithm's solution size is compared to the optimal solution size, yielding bounds dependent on k.
• The method assumes that if a node receives a signal at min-power, it will also receive it at max-power, without requiring symmetric link assumptions.
• The algorithm is implemented with explicit data structures for min- and max-power neighbor relations, enabling efficient processing in O(s_min + s_max) time, where s_min and s_max are the sizes of the neighbor relation representations.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio achievable for the symmetric two-power-level range assignment problem with a polynomial-time algorithm?
• RQ2 Can a parameterized algorithm be designed such that its performance ratio approaches π²/6 as the parameter k increases?
• RQ3 Is there a practical, near-linear time implementation for a specific k value (e.g., k=3) with a tight approximation guarantee?
• RQ4 How does the algorithm's performance compare to known lower bounds in worst-case network configurations?
• RQ5 Can the algorithm be implemented efficiently without relying on geometric embeddings or symmetric link assumptions?

Key findings

• The performance ratio of Approx2LSRA_k is bounded between (3k−2)/(2k−2) and 1/(k−1) + Σ(i=1 to k−1) 1/i², which approaches π²/6 ≈ 1.645 as k increases.
• For k=3, the algorithm achieves a tight approximation ratio of 7/4, which is the best known for this problem variant.
• The algorithm can be implemented in almost linear time using a union-find data structure, with a time complexity of O(f(s_min, s_max) · α(f(s_min, s_max), |V|)), where f ∈ O(s_min + s_max).
• The worst-case instance construction shows that the upper bound of 7/4 for k=3 is tight, confirming the optimality of the approximation factor.
• The algorithm's performance is independent of the network's geometric embedding and does not require symmetric link assumptions, relying only on min- and max-power neighbor sets.
• The theoretical analysis confirms that the algorithm is a parameterized polynomial-time approximation scheme (PPTAS) with performance ratio arbitrarily close to π²/6.