[Paper Review] Node coloring for dense wireless sensor networks

Ichrak Amdouni, Pascale Minet+1 more|arXiv (Cornell University)|Apr 11, 2011

Energy Efficient Wireless Sensor Networks21 references8 citations

TL;DR

This paper proposes a distributed 3-hop node coloring algorithm optimized for dense wireless sensor networks, introducing the Vector Method to achieve periodic, optimal colorings on grid topologies. It proves the h-hop node coloring problem is NP-complete and establishes theoretical bounds, showing the Vector Method achieves the minimal number of colors independent of grid size, with simulations confirming optimality for various transmission ranges.

ABSTRACT

Coloring is used in wireless networks to improve communication efficiency, mainly in terms of bandwidth, energy and possibly end-to-end delays. In this research report, we define the h-hop node coloring problem, with h any positive integer. We prove that the associated decision problem is NP-complete. We then present a 3-hop distributed coloring algorithm that is optimized for dense networks: a node does not need to exchange the priorities and colors of its 2-hop neighbors. Through simulation results, we highlight the impact of priority assignment on the number of colors obtained for any network. We then focus on grids and identify a color pattern that can be reproduced to color the whole grid. We show how the coloring algorithm can use regularity properties to obtain a periodic color pattern with the optimal number of colors. We then consider grids with holes and study how to extend our results.

Motivation & Objective

• Address the challenge of efficient medium access scheduling in dense wireless sensor networks (WSNs) with limited memory and energy.
• Develop a low-complexity, distributed coloring algorithm that avoids full 2-hop neighbor exchange, crucial for memory-constrained nodes.
• Establish theoretical bounds and a systematic method for achieving optimal periodic h-hop colorings on regular grid topologies.
• Improve existing protocols like SERENA by integrating theoretical results to minimize the number of colors used.
• Demonstrate that priority assignment via the Vector Method yields optimal, scalable colorings independent of grid size.

Proposed method

• Introduce the h-hop node coloring problem, proving its decision variant is NP-complete.
• Design a 3-hop distributed coloring algorithm that reduces message overhead by eliminating exchange of 2-hop neighbor colors and priorities.
• Propose the Vector Method to compute node priorities using two vectors, enabling periodic color patterns with optimal color count.
• Derive lower and upper bounds on the number of colors required for periodic h-hop colorings using geometric and determinant-based analysis.
• Use the determinant of transformation matrices to bound the number of colors, showing asymptotic optimality relative to hexagonal lattice coloring.
• Apply the Vector Method to grids with varying transmission ranges, validating results through simulations on SERENA.

Experimental results

Research questions

• RQ1 What is the computational complexity of the h-hop node coloring problem in dense wireless sensor networks?
• RQ2 How can a distributed coloring algorithm be optimized to reduce memory and communication overhead in dense networks?
• RQ3 What is the minimal number of colors required for a periodic h-hop coloring on a regular grid topology?
• RQ4 Can a systematic method be devised to assign node priorities such that the resulting coloring is optimal and periodic?
• RQ5 How does the choice of priority assignment (e.g., line, column, vector) impact the number of colors used in practical implementations?

Key findings

• The h-hop node coloring problem is proven to be NP-complete, establishing its computational hardness.
• The Vector Method produces periodic h-hop colorings with the optimal number of colors, achieving asymptotic optimality matching hexagonal lattice bounds.
• For 3-hop coloring, the Vector Method achieves 8, 16, 25, and 68 colors at radio ranges 1, 1.5, 2, and 3 grid steps, respectively, with no dependence on grid size.
• Simulations show SERENA with vector-based priority assignment achieves optimal coloring across all tested grid sizes and transmission ranges.
• The number of protocol rounds is significantly reduced (e.g., 21 rounds for 50x50 grid at R=1) when using vector priority, compared to 58 with line/column assignment.
• Lower and upper bounds on the number of colors in periodic h-hop colorings are derived, with the upper bound asymptotically matching the theoretical minimum as R→∞.