Consistent Channel Hopping Algorithms for the Multichannel Rendezvous Problem with Heterogeneous Available Channel Sets

We propose a theoretical framework for consistent channel hopping algorithms to address the multichannel rendezvous problem (MRP) in wireless networks with heterogeneous available channel sets. A channel selection function is called consistent if the selected channel remains unchanged when the available channel set shrinks, provided the selected channel is still available. We show that all consistent channel selection functions are equivalent to the function that always selects the smallest-index channel under appropriate channel relabeling. This leads to a natural representation of a consistent channel hopping algorithm as a sequence of permutations. For the two-user MRP, we characterize rendezvous time slots using a fictitious user and derive tight bounds on the maximum time-to-rendezvous (MTTR) and expected time-to-rendezvous (ETTR). Notably, the ETTR is shown to be the inverse of the Jaccard index when permutations are randomly selected. We also prove that consistent channel hopping algorithms maximize the rendezvous probability. To reduce implementation complexity, we propose the modulo algorithm, which uses modular arithmetic with one-cycle permutations and achieves performance comparable to locality-sensitive hashing (LSH)-based algorithms. The framework is extended to multiple users, with novel strategies such as stick-together, spread-out, and a hybrid method that accelerates rendezvous in both synchronous and asynchronous settings. Simulation results confirm the effectiveness and scalability of the proposed algorithms.