## 3 Preliminaries
### Evaluation Metrics  
1. Successful Reconfiguration Rate (SRR)  
In the domain of REmesh topological physical architectures with dimensions \(m\times(n+k)\), the objective is to accurately reconfigure the desired physical topology through logical topological manipulations. The success rate of topology reconfiguration is quantified as the ratio of successful reconfigurations to the total number of experimental tests. The mathematical formula for a successful reconfiguration rate (SRR) in this experiment is shown below:  
\[SRR=\frac{N_{successful}}{N_{total}}\times 100\% \tag{1}\]  
where \(N_{successful}\) represents the number of successful topology reconfigurations and \(N_{total}\) represents the total number of experimental tests conducted. SRR serves as a critical benchmark for evaluating the effectiveness of topology reconfiguration algorithms. A higher SRR indicates superior algorithm performance and enhanced computational efficiency, particularly when other performance metrics remain constant. 2. Average Recovery Time (ART)  
In the domain of REmesh topological physical architectures with dimensions \(m\times(n+k)\), the average recovery time is defined as the percentage of the total runtime of the topology reconfiguration algorithm relative to the total number of successful topology reconfigurations, compared to the overall rate of the experimental algorithms. The mathematical formula for average recovery time (ART) in this experiment is shown below:  
\[ART=\frac{T_{total}}{N_{successful}}\times 100\% \tag{2}\]  
where \(T_{total}\) represents the total runtime of the topology reconfiguration algorithm and \(N_{successful}\) represents the number of successful reconfigurations. ART quantifies the time efficiency of the algorithm, with shorter ART values indicating better performance. In the actual research field of embedded systems and networks-on-chip, the time performance is often considered the key performance indicator. Therefore, the average recovery time (ART) can be used to evaluate the time performance of the topology reconfiguration algorithm. 3. Core Reuse Rate (CRR)  
In the domain of REmesh topological physical architectures with dimensions \(m\times(n+k)\), the core reuse rate is defined as the percentage of original cores that have successfully undergone topology reconfiguration relative to the total number of cores in the initial array. The mathematical formula for core reuse rate (CRR) in this experiment is shown below:  
\[CRR=\frac{C_{reused}}{C_{total}}\times 100\% \tag{3}\]where \(C_{\textit{reused}}\) represents the number of original cores that have successfully undergone topology reconfiguration and \(C_{\textit{total}}\) represents the total number of cores in the initial array. CRR highlights the effectiveness of the reconfiguration process in maintaining similarity to the original topology. A higher CRR signifies better performance and efficiency of the topology reconfiguration algorithm, particularly in system-on-chip (SoC) research. (4) Comprehensive Evaluation Metric (CEM)  
In this experiment, we innovatively propose a comprehensive evaluation metric (CEM) that considers the weighted average of the above three performance metrics and gives different metrics with different weights. The mathematical formula for the comprehensive evaluation metric (CEM) in this experiment is shown below:  
\[\textbf{CEM}=0.4\times\textbf{SRR}+0.3\times\textbf{CRR}+0.3\times(1-\textbf{ ART}) \tag{4}\]  
The metrics reflect the relative importance of SRR, CRR, and ART in assessing overall algorithm performance. CEM provides a holistic view of the algorithm's effectiveness by balancing success reconfiguration rates, core reuse rate, and time efficiency. In this paper, we conduct experiments on different scales of network topologies based on a 2D REmesh structure, performing 1000 tests for each topology of the same scale.