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On-chip thermal sensors are employed by dynamic thermal management (DTM) techniques to appropriately manage chip performance. However, the effectiveness of the DTM mechanisms is directly dependent on the number of placed sensors, which should be minimised, while guaranteeing accurate tracking of hot spots and full thermal characterisation. In this study, the authors propose a rigid sensor allocation and placement technique for determining the fewest number of thermal sensors and the optimal locations based on dual clustering. Initially, the authors utilise the dual clustering algorithm to devise method that can reduce the number of sensors to a great extent while satisfying an expected accuracy of hot spot temperature error. Then they identify an optimal physical location for each sensor such that the accuracy of full thermal characterisation is maximised. They also propose a flexible sensor computation technique which combines the measurements of the rigid sensors in an optimal way to precisely estimate the temperatures where no sensors are embedded, which can further improve the hot spots tracking resolution. Experimental results indicate the superiority of the authors techniques and confirm that their proposed methods are capable of accurately characterising the temperatures of microprocessors.