Multichannel Rendezvous Problem

• MRP is defined as the challenge of discovering a common channel among users with asynchronous clocks and heterogeneous channel sets in CRNs and IoT.
• It employs metrics such as the expected time-to-rendezvous (ETTR) and maximum time-to-rendezvous (MTTR) to evaluate the performance of random, sequence-based, and hybrid channel hopping strategies.
• Recent solutions use techniques like locality-sensitive hashing, modular arithmetic, and duty-cycle control to enhance robustness and adaptivity under adversarial conditions.

The multichannel rendezvous problem (MRP) is a foundational challenge in cognitive radio networks (CRNs) and wide-area IoT contexts, wherein two or more users independently seek to discover a shared communication channel by hopping among possibly distinct available channel sets. MRP addresses inherent disruptors such as asynchronous clocks, locally heterogeneous channel availability (due to primary user activity or interference), and the lack of globally shared identifiers. The objective is typically to minimize the time to rendezvous (TTR), using metrics such as the expected time-to-rendezvous (ETTR) and the maximum time-to-rendezvous (MTTR), under strong practical and adversarial assumptions.

1. Problem Formulation and Metrics

MRP considers $K$ users, each equipped with a set of available channels $\mathcal{C}_k \subset \mathcal{N}$ from a (possibly large) universe %%%%2%%%%. Users operate independently, often without clock synchronization, and channel hopping (CH) sequences are constructed without the presumption of external coordination. Rendezvous occurs when two users are simultaneously on a common channel.

The core performance metrics are:

• Expected Time-to-Rendezvous (ETTR): The average number of time slots until a rendezvous occurs. For random hopping,

  $$\mathrm{ETTR} = \frac{n_1 n_2}{|\mathcal{C}_1 \cap \mathcal{C}_2|},$$

  where $n_1 = |\mathcal{C}_1|$, $n_2 = |\mathcal{C}_2|$.

• Maximum Time-to-Rendezvous (MTTR): The worst-case slot within which rendezvous is guaranteed, critical in adversarial or safety-critical settings.
• Rendezvous Diversity: The number of distinct channels on which rendezvous can occur during the sequence period.

With time-varying or unobservable channel conditions, the rendezvous probability may be further modulated by stochastic channel state processes, yielding generalized formulae (see (Chang et al., 2019)).

2. Channel Hopping Strategies

Three principal paradigms have emerged:

a. Random Channel Hopping

Each user hops onto channels chosen uniformly at random from their available set per time slot. This scheme achieves minimum ETTR ($N$ when all channels are used), but provides no upper-bound guarantee: MTTR is unbounded; two users may never rendezvous.

b. Sequence-Based Channel Hopping

Each user follows a deterministic, usually ID- or clock-based, hopping sequence (e.g., cyclic permutations (Chang et al., 2019)). These sequences guarantee finite MTTR (typically scaling with $N^2$ for full rendezvous diversity), are robust to asynchrony and local set heterogeneity, and can ensure that rendezvous occurs over many (possibly all) channels. However, ETTR is typically much larger than that of the random scheme.

c. Hybrid and Consistent Algorithms

Recent frameworks advocate interleaving sequence-based and random hopping according to a wake-up schedule (binary duty-cycle control) or leveraging locality-sensitive hashing (LSH) or consistent permutation techniques, to strike a formal trade-off between low ETTR and bounded MTTR (Chen et al., 2015, Jiang et al., 2022, Liu et al., 23 Jun 2025). Consistent channel selection functions are formally those that are invariant under shrinkage of the available channel set (provided the selected channel remains available), a property shown to maximize rendezvous probability and allow for natural representation as permutation sequences (Liu et al., 23 Jun 2025).

3. Theoretical Bounds and Trade-offs

The achievable trade-off between ETTR and MTTR is a central theoretical concern. For random hopping,

$$\mathrm{ETTR}_{\text{random}} = \frac{n_1 n_2}{|\mathcal{C}_1 \cap \mathcal{C}_2|}$$

with unbounded MTTR. For sequence-based algorithms, constructions such as IDEAL-CH guarantee MTTR $\leq 2N^2$, and in the best known constructions, the period (and thus MTTR) for maximum rendezvous diversity has an asymptotic ratio of $2$ over the $N^2$ lower-bound (Chang et al., 2019).

Hybrid protocols with duty cycle $\delta$ achieve ETTR: $$\mathrm{ETTR}_{\text{hybrid}} = \delta\,\mathrm{ETTR}_{\text{seq}} + (1-\delta)\,\mathrm{ETTR}_{\text{random}},$$ guaranteeing both low average and bounded worst-case delay. Consistent channel hopping algorithms using random permutations achieve ETTR equal to the inverse of the Jaccard index $J$: $$J = \frac{|\mathcal{C}_1 \cap \mathcal{C}_2|}{|\mathcal{C}_1| + |\mathcal{C}_2| - |\mathcal{C}_1 \cap \mathcal{C}_2|}, \quad \mathrm{ETTR} = \frac{1}{J}$$ with MTTR upper bound $N - |\mathcal{C}_1 \cap \mathcal{C}_2| + 1$ when using one-cycle permutations (Liu et al., 23 Jun 2025).

4. Algorithmic Constructions and Representative Methods

a. Orthogonal and Quasi-Random Sequences

Orthogonal CH matrix structures (Chang et al., 2019), quasi-random (QR) symmetrization via strong ternary mappings (Chang et al., 2019), and modular clock with coprime periods (Chinese Remainder Theorem) are common in constructing robust and even anonymous rendezvous algorithms with tight MTTR guarantees. QR algorithms, for example, realize: $$\mathrm{MTTR} \leq 9M\lceil n_1/m_1 \rceil \lceil n_2/m_2 \rceil,$$ where $M$ relates to channel ID coding length.

b. Locality-Sensitive Hashing (LSH) and Consistent Hashing

LSH exploits available set similarity to boost the collision probability: users hash their available sets via pseudo-random permutations and then select channels accordingly. ETTR becomes inversely proportional to the Jaccard index $J$ (Jiang et al., 2022), and dimensionality reduction (mapping to multisets) supports efficiency and adaptation to asynchronous scenarios—see the LC-LSH4 and QR-LC-LSH4 variants (Cheng et al., 31 Aug 2025).

c. Modular Arithmetic Algorithms

Efficient O($n$)-time per slot algorithms arise from adopting one-cycle permutations computed via modular exponentiation (Liu et al., 23 Jun 2025). The permutation at time $t$ is

$\pi_t(i) = g^t i \bmod P,$

with $g$ a generator modulo $P=N+1$. This ensures every channel and thus every common channel is traversed within a bounded window, yielding predictable MTTR irrespective of asynchrony.

d. Multichannel Topology Discovery and Cooperative Extensions

Extensions to network topology—beyond pairwise rendezvous—require pseudo-random sweep algorithms with forward replacement and threshold-based stick-together protocols, which reduce positive correlation between consecutive rendezvous attempts, further lowering ETTD (expected time-to-discovery) (Wang et al., 23 Apr 2025). Cooperative multi-user strategies (stick-together, spread-out, hybrid) and their theoretical guarantees are formalized in (Liu et al., 23 Jun 2025).

5. Multi-User, Heterogeneous, and Asynchronous Models

All algorithmic approaches are extended or designed for asynchronous settings (arbitrary or unknown clock drift), for both symmetric and asymmetric/hierarchical user roles. Matrix-structured algorithms such as QCMS-CH (Liu et al., 2022) handle arbitrary heterogeneity by encoding rendezvous structure in channel-availability–independent sequences and ensuring strong overlap properties via number-theoretic coding. Stick-together and spread-out strategies accelerate multi-user rendezvous by propagating intersection information post-rendezvous, while hybrid methods alternate between coordinated and exploratory phases, balancing rapid convergence and exploration.

Remapping and equivalence under channel relabeling are key for robust performance in the absence of a global enumeration system or when channels are labeled by unique L-bit identifiers; consistent hashing and multiset-embedding provide practical solutions (Cheng et al., 31 Aug 2025).

6. Open Problems and Future Directions

Despite strong progress, open theoretical questions remain. For example, it is unresolved whether a periodic channel hopping sequence with period exactly $N^2$ (asymptotic ratio 1) and full rendezvous diversity is possible—current best constructions achieve ratio 2 (Chang et al., 2019). Duty cycle selection in hybrid schemes, adaptability with minimal global knowledge, and efficient embedding of LSH-based multisets into periodic CH sequences with both low ETTR and bounded MTTR are prominent research directions. Incorporating learning under unobservable or partially observable channel states has been shown effective (Exp3 bandit-based policies) and motivates further integration of reinforcement learning with the core rendezvous logic (Wang et al., 2019).

7. Comparative Analysis and Practical Implications

Algorithm Class	ETTR Scaling	MTTR Scaling	Asynchrony/Heterogeneity	Implementation Complexity
Random CH	$N$	Unbounded	High	Minimal
Sequence-based (IDEAL-CH etc.)	$>N$	$O(N^2)$	High	High (complex sequence construction)
Hybrid/Wake-up	$(B/T) \mathrm{ETTR}_{\text{seq}} + (1-B/T) N$	$O(T)$ (for $B$ awake slots)	High	Moderate (schedule management)
Consistent/randomized perms	$1/J$	$N-n_{1,2}+1$	High	Low/Moderate (O($n$) per slot)
LSH/LC-LSH-based	$1/J$	Unbounded or bounded*	High	O($\log(nK)$) per slot with K virtuals
Modular arithmetic (one-cycle)	$1/J$	$N-n_{1,2}+1$	High	O($n$) per slot

Conclusions point to a maturing theory that links combinatorial, number-theoretic, coding, and hashing approaches with performance guarantees that are explicit in ETTR, MTTR, and rendezvous diversity. Robust, low-complexity, and scalable protocols now exist for heterogeneous, asynchronous, and even globally unenumerated channel spaces, and future developments are likely to focus on closing fundamental combinatorial gaps and improving adaptive performance in dynamic wireless environments.