{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T14:14:14Z","timestamp":1762956854499,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T00:00:00Z","timestamp":1653523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>Wireless underground sensor networks (WUSNs) are being used in agricultural applications, in border patrol, and in the monitoring of remote areas. Coverage holes in WUSNs are an issue which needs to be dealt with. Coverage holes may occur due to random deployment of nodes as well as the failure of nodes with time. In this paper, a mathematical approach for hole detection using Delaunay geometry is proposed which divides the network region into Delaunay triangles and applies the laws of triangles to identify coverage holes. WUSNs comprise static nodes, and replacing underground nodes is a complex task. A simplistic algorithm for detecting coverage holes in static WSNs\/WUSNs is proposed. The algorithm was simulated in the region of interest for the initially randomly deployed network and after energy depletion of the nodes with time. The performance of the algorithm was evaluated by varying the number of nodes and the sensing radius of the nodes. Our scheme is advantageous over other schemes in the following aspects: (1) it builds a mathematical model and polynomial-time algorithm for detecting holes, and (2) the scheme does not work on centralized computation and therefore provides better scalability, (3) is energy-efficient, and (4) provides a cost-effective solution to detect holes with great accuracy and a low detection time. The algorithm takes less than 0.1 milliseconds to detect holes in a 100 m \u00d7 100 m-size network with 100 sensor nodes having a sensing radius of 8 m. The detection time shows only a linear change with an increase in the number of nodes in the network, which makes the algorithm applicable for every network size from small to large.<\/jats:p>","DOI":"10.3390\/computers11060086","type":"journal-article","created":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T23:04:31Z","timestamp":1653606271000},"page":"86","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Energy-Efficient Deterministic Approach for Coverage Hole Detection in Wireless Underground Sensor Network: Mathematical Model and Simulation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1049-3166","authenticated-orcid":false,"given":"Priyanka","family":"Sharma","sequence":"first","affiliation":[{"name":"Semi-Conductor Laboratory, Govt. of India, Mohali 160071, India"},{"name":"Department of Computer Science & Engineering, Guru Jambheshwar University of Science & Technology, Hisar 125001, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rishi Pal","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Computer Science & Engineering, Guru Jambheshwar University of Science & Technology, Hisar 125001, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Silva, A.R., and Vuran, M.C. (2010, January 23\u201327). Communication with Aboveground Devices in Wireless Underground Sensor Networks: An Empirical Study. Proceedings of the 2010 IEEE international conference on communications, Cape Town, South Africa.","DOI":"10.1109\/ICC.2010.5502315"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Vuran, M.C., and Silva, A.R. (2010). Communication through Soil in Wireless Underground Sensor Networks\u2013Theory and Practice. Sensor Networks, Springer.","DOI":"10.1155\/2010\/620307"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.adhoc.2006.04.003","article-title":"Wireless Underground Sensor Networks: Research Challenges","volume":"4","author":"Akyildiz","year":"2006","journal-title":"Ad Hoc Netw."},{"key":"ref_4","unstructured":"Yoon, S.-U., Cheng, L., Ghazanfari, E., Pamukcu, S., and Suleiman, M.T. (2011, January 5\u20139). A radio propagation model for wireless underground sensor networks. Proceedings of the 2011 IEEE Global Telecommunications Conference-GLOBECOM 2011, Houston, TX, USA."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10762-020-00753-8","article-title":"Study on on-chip antenna design based on metamaterial-inspired and substrate-integrated waveguide properties for millimetre-wave and THz integrated-circuit applications","volume":"42","author":"Alibakhshikenari","year":"2021","journal-title":"J. Infrared Millim. Terahertz Waves"},{"key":"ref_6","first-page":"e22412","article-title":"Wideband linear microstrip array antenna with high efficiency and low side lobe level","volume":"30","author":"Shirkolaei","year":"2020","journal-title":"Int. J. RF Microw. Comput. Aided Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6565","DOI":"10.1109\/JIOT.2020.3044647","article-title":"Experimental link quality analysis for LoRa-based wireless underground sensor networks","volume":"8","author":"Lin","year":"2020","journal-title":"IEEE Internet Things J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1109\/TII.2015.2471263","article-title":"Experimental link quality characterization of wireless sensor networks for underground monitoring","volume":"11","author":"Silva","year":"2015","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_9","unstructured":"Kaushik, A. (2018, January 1\u20132). Multiple Hole Detection in Wireless Underground Sensor Networks. Proceedings of the Communication and Computing Systems: Proceedings of the 2nd International Conference on Communication and Computing Systems ICCCS 2018, Gurgaon, India."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bi, K., Gu, N., Tu, K., and Dong, W. (2006, January 6\u20139). Neighborhood-Based Distributed Topological Hole Detection Algorithm in Sensor Networks. Proceedings of the IET International Conference on Wireless, Mobile and Multimedia Networks, Hangzhou, China.","DOI":"10.1049\/cp:20061186"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.pmcj.2008.02.001","article-title":"Coverage and Connectivity Issues in Wireless Sensor Networks: A Survey","volume":"4","author":"Ghosh","year":"2008","journal-title":"Pervasive Mob. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.comnet.2017.10.001","article-title":"Distributed Hole-Bypassing Protocol in WSNs with Constant Stretch and Load Balancing","volume":"129","author":"Nguyen","year":"2017","journal-title":"Comput. Netw."},{"key":"ref_13","first-page":"969501","article-title":"Hole Detection for Quantifying Connectivity in Wireless Sensor Networks: A Survey","volume":"2014","author":"Antil","year":"2014","journal-title":"J. Comput. Netw. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Watfa, M.K., and Commuri, S. (2006, January 4\u20136). Energy-Efficient Approaches to Coverage Holes Detection in Wireless Sensor Networks. Proceedings of the 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, Munich, Germany.","DOI":"10.1109\/CACSD-CCA-ISIC.2006.4776636"},{"key":"ref_15","unstructured":"De Silva, V., Ghrist, R., and Muhammad, A. Blind Swarms for Coverage in 2-D. Proceedings of the 1st Conference on Robotics: Science and Systems, Massachusetts Institute of Technology, Cambridge, MA, USA. Available online: http:\/\/math.uchicago.edu\/~shmuel\/AAT-readings\/Sensory%20networks\/blind%20swarms.pdf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1177\/0278364906072252","article-title":"Coordinate-Free Coverage in Sensor Networks with Controlled Boundaries via Homology","volume":"25","author":"Ghrist","year":"2006","journal-title":"Int. J. Robot. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"73520V","DOI":"10.1117\/12.819294","article-title":"Decentralized Detection and Patching of Coverage Holes in Wireless Sensor Networks","volume":"Volume 7352","author":"Yao","year":"2009","journal-title":"Intelligent Sensing, Situation Management, Impact Assessment, and Cyber-Sensing"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Ghosh, P., Gao, J., Gasparri, A., and Krishnamachari, B. (2014, January 28\u201330). Distributed Hole Detection Algorithms for Wireless Sensor Networks. Proceedings of the 2014 IEEE 11th International Conference on Mobile Ad Hoc and Sensor Systems, Philadelphia, PA, USA.","DOI":"10.1109\/MASS.2014.25"},{"key":"ref_19","unstructured":"Yan, F., Martins, P., and Decreusefond, L. (2011, January 5\u20139). Connectivity-Based Distributed Coverage Hole Detection in Wireless Sensor Networks. Proceedings of the 2011 IEEE Global Telecommunications Conference GLOBECOM 2011, Houston, TX, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1049\/iet-wss.2012.0076","article-title":"Hop-Based Approach for Holes and Boundary Detection in Wireless Sensor Networks","volume":"2","author":"Khan","year":"2012","journal-title":"IET Wirel. Sens. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"640","DOI":"10.1109\/TMC.2006.80","article-title":"Movement-Assisted Sensor Deployment","volume":"5","author":"Wang","year":"2006","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s11276-007-0021-1","article-title":"Localized Algorithms for Coverage Boundary Detection in Wireless Sensor Networks","volume":"15","author":"Zhang","year":"2009","journal-title":"Wirel. Netw."},{"key":"ref_23","unstructured":"Hatcher, A. (2005). Algebraic Topology, Tsinghua University Press Co., Ltd."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1109\/TMC.2011.85","article-title":"Coverage Verification without Location Information","volume":"11","author":"Bejerano","year":"2011","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Funke, S., and Klein, C. (2006, January 5\u20137). Hole Detection or: How Much Geometry Hides in Connectivity?. Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, Sedona, AZ, USA.","DOI":"10.1145\/1137856.1137911"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Li, X., and Hunter, D.K. (2008, January 19\u201323). Distributed Coordinate-Free Hole Recovery. Proceedings of the ICC Workshops-2008 IEEE International Conference on Communications Workshops, Beijing, China.","DOI":"10.1109\/ICCW.2008.41"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bi, K., Tu, K., Gu, N., Dong, W.L., and Liu, X. (November, January 29). Topological Hole Detection in Sensor Networks with Cooperative Neighbors. Proceedings of the 2006 International Conference on Systems and Networks Communications ICSNC\u201906, Tahiti, French Polynesia.","DOI":"10.1109\/ICSNC.2006.71"},{"key":"ref_28","unstructured":"Ghrist, R., and Muhammad, A. (2005, January 24\u201327). Coverage and Hole-Detection in Sensor Networks via Homology. Proceedings of the IPSN 2005 Fourth International Symposium on Information Processing in Sensor Networks, Los Angeles, CA, USA."},{"key":"ref_29","unstructured":"Corke, P., Peterson, R., and Rus, D. (2007, January 10\u201314). Finding Holes in Sensor Networks. Proceedings of the 2007 IEEE International Conference on Robotics and Automation, Roma, Italy. Available online: https:\/\/www.researchgate.net\/profile\/Daniela_Rus\/publication\/268256993_Finding_Holes_in_Sensor_Networks\/links\/56d0439f08ae059e375cfbae.pdf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1109\/TPDS.2007.1048","article-title":"Scan-Based Movement-Assisted Sensor Deployment Methods in Wireless Sensor Networks","volume":"18","author":"Yangy","year":"2007","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.1002\/sec.271","article-title":"Local Geometric Algorithm for Hole Boundary Detection in Sensor Networks","volume":"4","author":"Shirsat","year":"2011","journal-title":"Secur. Commun. Netw."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1109\/TPDS.2013.137","article-title":"Localized Movement-Assisted Sensor Deployment Algorithm for Hole Detection and Healing","volume":"25","author":"Senouci","year":"2013","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Aurenhammer, F., Klein, R., and Lee, D.-T. (2013). Voronoi Diagrams and Delaunay Triangulations, World Scientific Publishing Company.","DOI":"10.1142\/8685"},{"key":"ref_34","unstructured":"Rakovic, S., Grieder, P., and Jones, C. (2004). Computation of Voronoi Diagrams and Delaunay Triangulation via Parametric Linear Programming, ETH (Eidgen\u00f6ssische Technische Hochschule)."},{"key":"ref_35","unstructured":"(2022, April 01). Low-Energy Adaptive Clustering Hierarchy; In Wikipedia, The Free Encyclopedia; Page Version ID: 1076138763. Available online: https:\/\/en.wikipedia.org\/wiki\/Low-energy_adaptive_clustering_hierarchy."}],"container-title":["Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-431X\/11\/6\/86\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:19:26Z","timestamp":1760138366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-431X\/11\/6\/86"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,26]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["computers11060086"],"URL":"https:\/\/doi.org\/10.3390\/computers11060086","relation":{},"ISSN":["2073-431X"],"issn-type":[{"type":"electronic","value":"2073-431X"}],"subject":[],"published":{"date-parts":[[2022,5,26]]}}}