Upgraded DV-Hop algorithm based on polynomial approximation [UDV-PA] and multiverse optimization for WSN

Node localization is a fundamental challenge in Wireless Sensor Networks (WSNs), where accurate position information is crucial for data interpretation while maintaining low power consumption and minimal hardware costs. This paper presents the Upgraded DV-Hop algorithm based on Polynomial Approximation (UDV-PA), a novel three-dimensional localization method that significantly enhances accuracy while preserving the range-free characteristics essential for resource-constrained WSN deployments. Unlike existing 3D DV-Hop variants that primarily focus on hop-size refinement or optimization techniques in isolation, our approach uniquely integrates polynomial distance modeling with the Multi-Verse Optimizer (MVO) featuring dynamically bounded search spaces and adaptive hyperparameter tuning. The algorithm operates in three phases: (i) hop-count acquisition using flooding with minimum hop selection, (ii) distance estimation through polynomial fitting of hop-distance relationships followed by MVO-based refinement with constrained search bounds, and (iii) coordinate computation using optimally selected anchor nodes (ANs) based on proximity metrics. Key innovations include adaptive polynomial degree selection ( 2ⁿᵈ to 4ᵗʰ order) based on network density, dynamic MVO parameter adjustment (WEP: 0.2-1.0, TDR: 0.6-1.0) scaled to network size, and energy-efficient anchor selection limiting multilateration to the four nearest ANs. Extensive simulations in 100 × 100 × 100 m³ volumes with 100-300 nodes demonstrate average localization errors of 8.23% in dense networks and 15.67% in sparse deployments, representing 42.3% and 31.8% improvements over traditional 3D DV-Hop respectively. Computational complexity remains O(n log n) with energy consumption of 0.018J per localization round, confirming suitability for low-power WSN applications. The algorithm achieves convergence within 50-100 iterations, with processing times under 2.3ms per unknown node (UN) on resource-constrained platforms.