## 3 Proposed Algorithm for the Generalized Local \(I_{i}\)-Mutual Inclusion
* For \(P_{i}\)'s entering the CS, \(P_{i}\) sends a \(\langle\mathsf{Request},\mathsf{MUTEX},ts_{i},P_{i}\rangle\) message to each process \(P_{j}\) in \(N_{i}\cup\{P_{i}\}\). Then, \(P_{j}\) sends a \(\langle\mathsf{Preempt},\mathsf{MUTEX},P_{j}\rangle\) message to the process \(P_{m}\) to which \(P_{j}\) sends a \(\langle\mathsf{Grant},\mathsf{MUTEX},P_{j}\rangle\) message; \(P_{m}\) sends a \(\langle\mathsf{Relianguish},\mathsf{MUTEX},P_{m}\rangle\) message back to \(P_{j}\); andsends a \(\langle\mathsf{Grant},\mathsf{MUTEX},P_{i}\rangle\) message to \(P_{i}\). After \(P_{i}\) enters the CS, \(P_{i}\) sends a \(\langle\mathsf{Release},\mathsf{MUTIN},P_{i}\rangle\) message to each process \(P_{i}\) in \(N_{i}\cup\{P_{i}\}\). Then, \(P_{j}\) sends a \(\langle\mathsf{Grant},\mathsf{MUTIN},P_{j}\rangle\) message to some process with the highest priority in \(\mathit{pendingReq}_{j}[\mathsf{MUTIN}]\). Thus, \(6(|N_{i}|+1)\) messages are exchanged. Thus, the message complexity is \(12(|N_{i}|+1)\) in the worst case. Theorem 4.1: _LKCS solves the generalized local \((l_{i},k_{i})\)-critical section problem with a message complexity of \(O(\Delta)\), where \(\Delta\) is the maximum degree of a network._

## 6 Conclusions

In this paper, we consider the generalized local \((l_{i},k_{i})\)-critical section problem, which is a new version of critical section problems. Because this problem is useful for fault-tolerance and load balancing of distributed systems, we can consider various future applications. We first proposed an algorithm for the generalized local \(l_{i}\)-mutual inclusion. Next, we showed the generalized local complementary theorem. By using this theorem, we proposed an algorithm for the generalized local \((l_{i},k_{i})\)-critical section problem. In the future, we plan to perform extensive simulations and confirm the performance of our algorithms under various application scenarios. Additionally, we plan to improve the proposed algorithm in message complexity and time complexity and to design an algorithm that guarantees exactly \(l_{i}\leq|\mathcal{CS}_{i}(C)|\leq k_{i}\) in every process. This work was supported by JSPS KAKENHI Grant Numbers 16K00018 and 26330015. [PERSON] and [PERSON] designed the algorithms. [PERSON] analyzed the algorithms and wrote the paper. The authors declare no conflict of interest. ## References

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