Rendezvous Tracking in Unknown Scenarios

• RTUS is a framework of formal methods, protocols, and algorithms that guarantees autonomous agents rendezvous under minimal environmental knowledge.
• It tackles challenges such as asynchronous start times, adversarial delays, and limited communication using memory-efficient and distance-aware techniques.
• The methodologies underpin applications in robotics, distributed computing, and space systems, with provable bounds on time, memory, and energy complexities.

Rendezvous Tracking for Unknown Scenarios (RTUS) is a class of formal methodologies, protocols, and algorithmic strategies for guaranteeing and optimizing rendezvous—i.e., the meeting of two or more autonomous agents—under minimal prior knowledge of the environment, network, or agent initialization. RTUS settings are characterized by unknown parameters such as the communication topology, graph structure, spatial embedding, or initial delays between agents, and emphasize robust, generalizable protocol design for distributed coordination without centralized infrastructure or global state. RTUS arises in mobile robotics, distributed computing, space systems, and exploration scenarios where system environment, communication, or adversarial interference is only partially observable or highly uncertain.

1. Formal Models and Problem Statement

RTUS encompasses rendezvous protocols for a range of computational models:

• Graph and Terrain Models: Agents may operate in arbitrary connected undirected graphs (finite or infinite degree, labeled/unlabeled, with or without whiteboards or communication) (Pattanayak et al., 2023), anonymous trees (Fraigniaud et al., 2011), 2D polygonal terrains with obstacles (Czyzowicz et al., 2010), or Euclidean workspaces with obstacles and sensor/communication constraints (Song et al., 14 May 2024, Silva et al., 15 Nov 2025).
• Agent Capabilities: Agents vary in sensing (distance measurement, visibility, beeping, labeling), memory (finite automata to Turing machines), and communication (none, peering, map exchange, scheduling) (Das et al., 2014, Elouasbi et al., 2017).
• Timing: Protocols consider both synchronous and asynchronous rounds, and address uncertainty in start-times including arbitrary (adversarial) delays (Miller et al., 2023).
• Objective: Agents seek to achieve rendezvous—simultaneously occupying a node, physical position, or region—sometimes with detection and/or full awareness, under worst-case guarantees on time, memory, energy, or communication complexity (Elouasbi et al., 2017, Silva et al., 2023).

A unifying RTUS goal is the design of algorithms whose correctness and performance scale with local, instance-specific parameters (e.g., degree, initial distance, number of leaves, local cost metrics), not the global size or geometry of the unknown scenario (Das et al., 2014, Fraigniaud et al., 2011, Cosson, 12 Mar 2024).

2. Memory, Time, and Communication Complexity

RTUS protocols are driven by the constraints of incomplete knowledge. Complexity analyses reveal sharp separations between what is possible in different knowledge or asynchrony regimes:

• Memory Gaps: In deterministic rendezvous in unknown trees, simultaneous start enables exponential savings in agent memory compared to arbitrary-delay (asynchronous) cases. Simultaneous start requires only $O(\log \ell + \log\log n)$ bits (with $\ell$ leaves, $n$ nodes), while arbitrary-delay requires $\Theta(\log n)$ bits—an exponential gap when $\ell = O(\operatorname{polylog} n)$ (Fraigniaud et al., 2011). This is a fundamental RTUS insight: structural asymmetry (number of leaves) dominates memory requirements when delay can be controlled.
• Time Complexity and Instance-Locality: In unknown graphs of max degree $\Delta$, rendezvous without distance-sensing can be exponential in $\Delta$; distance-aware agents reduce this to $O(\Delta(D+\log \ell))$, where $D$ is the initial distance and $\ell$ the smaller agent label (Das et al., 2014). In infinite lines, protocols achieve $O(D)$ to $O(D^2 (\log^*\ell)^3)$ rounds depending on whether agents know $D$ and their local label (Miller et al., 2023).
• Communication Constraints: Scenarios involving limited communication exploit intermittent rendezvous with scheduled map-sharing while minimizing bandwidth and adhering to local rules; complexity is analyzed both in data volume and in incremental area covered versus time (Silva et al., 15 Nov 2025, Song et al., 14 May 2024, Tellaroli et al., 18 Mar 2024).

Complexity Table: Key Instance Parameters and Bounds

Model / Scenario	Key Parameters	RTUS Complexity Upper Bound	RTUS Complexity Lower Bound
Unknown tree, sync start	$n$: nodes, $\ell$: leaves	$O(\log \ell + \log\log n)$ memory	$\Omega(\log \ell + \log\log n)$ memory (Fraigniaud et al., 2011)
Graph, dist. aware, 2 agents	$\Delta$, $D$, $\ell$	$O(\Delta (D+\log \ell))$ time	$\Omega(\Delta (D+\frac{\log \ell}{\log\Delta}))$ time (Das et al., 2014)
Line, unknown $D$	$D$, $\ell$	$O(D^2 (\log^* \ell)^3)$ time	$\Omega(D\log^*\ell)$ time (Miller et al., 2023)
Multi-robot rendezvous, MRE	$n$, map density, bandwidth	$O(\text{local instance terms})$	See (Song et al., 14 May 2024, Silva et al., 15 Nov 2025)

These results show that RTUS methodology must match protocol expressiveness and agent model to the specific unknowns and instance parameters of the environment.

3. Core Algorithms and Protocolic Mechanisms

RTUS protocols leverage variants of the following foundations:

• Memory-efficient walks: Protocols that contract degree-2 chains and exploit centrality in trees allow minimal-memory deterministic rendezvous, including exploration (Explo), resynchronization, and virtual-line reduction with prime-based symmetry breaking (Fraigniaud et al., 2011).
• Distance-aware port exploration: Subroutines such as TestPorts, BoundDegree, and CompareLabels implement port probing guided by minimal (nonmetric) distance feedback, breaking symmetries while keeping time polynomial in local parameters (Das et al., 2014).
• Scheduling and intermittent connectivity: Multi-agent systems under communication bounds deploy MILP- or JSSP-based rendezvous planning, where RTUS rules per robot use local time-to-go (slack) heuristics to trigger when exploration must be abandoned for heading toward rendezvous, ensuring compliance with a global plan without centralized control (Silva et al., 15 Nov 2025, Silva et al., 2023).
• Map, trace, and virtual frontier management: For probabilistic and multi-robot scenarios, RTUS leverages hybrid map representations (FHT-Maps), compact place descriptors, virtual frontiers via explicit decay of exploration history, or clustering/voronoi techniques for partitioned environment coverage and robust rendezvous-point decision (Song et al., 14 May 2024, Tellaroli et al., 18 Mar 2024).
• Low-overhead communication protocols: Beeping models, map exchange only at connection, and sleep-wait dynamics avoid excessive energy use, yet guarantee rendezvous with detection under adversarial wake-up or bounded energy (Elouasbi et al., 2017).

4. Applications and Evaluation in Robotics and Networks

RTUS methodologies are explicitly implemented in:

• Multi-robot exploration and intermittent communication: Protocols combining MILP/JSSP scheduling and local "rendezvous-heading" threshold rules empirically guarantee near-mission-completion rendezvous in environments with range or bandwidth constraints. Simulations in Gazebo, as well as real-world TurtleBot trials, show efficiency and low idle waiting times (Silva et al., 15 Nov 2025, Song et al., 14 May 2024).
• Collaborative mapping with decentralized map merging: Strategies such as partitioned and incomplete exploration (PIER) with light-weight FHT-Map sharing, relative-pose estimation, and sub-optimal rendezvous-point selection show reduced time to rendezvous and drastically reduced communication overhead compared to grid- or classical topological map baselines (Song et al., 14 May 2024).
• Communication-restricted exploration: RTUS integration with frontier-based strategies, extended via information decay and virtual frontiers, consistently yields higher probability of successful rendezvous and significant time savings, as demonstrated in a variety of simulated architectural environments (Tellaroli et al., 18 Mar 2024).
• Space systems and model-predictive tracking: RTUS underpins MPC-for-Tracking for rendezvous with non-cooperative, tumbling targets in space debris removal. The controller leverages coordinate transformation to maintain time-invariant linear constraints, ensuring recursive feasibility and closed-loop stability in realistic scenarios (e.g., Envisat approach) (Rebollo et al., 16 Mar 2024).

5. Lower Bounds, Impossibility, and Universality

RTUS research tightly characterizes the limits of deterministic rendezvous:

• Exponential lower bounds: In arbitrary-degree or infinite-degree graphs, rendezvous for agents without degrees or start distances can inherently require exponential time in instance weight or label-length (Pattanayak et al., 2023), and memory needs jump exponentially with increased adversarial uncertainty (Fraigniaud et al., 2011).
• Matching upper/lower bounds: In the infinite labeled line, a two-way reduction establishes equivalence between LOCAL 3-coloring and fast rendezvous, showing tightness and illuminating the fundamental combinatorial barriers (Miller et al., 2023).
• Universality constructs: UTH/URV-style algorithms explore port sequences by type, enabling universal guarantee regardless of degree growth or topology; no subexponential algorithm exists in the absence of extra structure (Pattanayak et al., 2023).

6. Key Structural Insights and Open Directions

RTUS algorithms reveal several unifying themes and ongoing areas of investigation:

• Structural asymmetry is crucial: Memory and communication savings, protocol optimality, and feasibility conditions all hinge on the presence of nonperfectly symmetric start positions or underlying graph irreducibility (Fraigniaud et al., 2011).
• Locality and decoupling: Time-, memory-, and energy-optimal protocols rely on local feedback (distance, ports, local matching), minimal shared state, and rules that drive agents to "track" rendezvous even while lacking knowledge of global topology or exact synchrony (Das et al., 2014, Silva et al., 15 Nov 2025, Tellaroli et al., 18 Mar 2024).
• Scalability by parameterization: Rather than scaling in global metrics (diameter, size), RTUS complexity bounds are parameterized by initial agent separation, node degree, number of labels, or structural descriptors (e.g., leaf count, label entropy) (Das et al., 2014, Miller et al., 2023).
• Provable integration of planning, exploration, and control: In hybrid systems (space, large robotic teams), RTUS methodologies harmonize model-predictive optimization, dynamic rescheduling, and decentralized switching rules for guaranteed convergence under real-world uncertainty and actuation limits (Rebollo et al., 16 Mar 2024, Silva et al., 15 Nov 2025).

Ongoing questions include extension to dynamic or failing topologies, multi-agent (beyond pairwise) optimality, and fine-grained tradeoffs between energy, time, and communication in adversarial or probabilistic RTUS environments.

7. Representative Protocol Taxonomy

Protocol Type	Environment	Main Techniques	Reference
Minimal-memory tree rendezvous	Anonymous tree, sync	Contraction, Explo, resynchronization	(Fraigniaud et al., 2011)
Distance-aware sweep	Unknown graph	Port probing, label comparison	(Das et al., 2014)
Decentralized MILP/JSSP RTUS	Multi-robot, unknown 2D	MILP/JSSP plan, local slack heuristics	(Silva et al., 15 Nov 2025Silva et al., 2023)
Virtual frontier exploration	Multi-robot, limited comm.	Info decay, leader-cluster consistency	(Tellaroli et al., 18 Mar 2024)
Universal traversal/rendezvous	Arbitrary graph	Lex-ordered port-sequence exploration	(Pattanayak et al., 2023)
Model-predictive rendezvous	Space, tumbling target	LTV coordinate transform, terminal constraints	(Rebollo et al., 16 Mar 2024)

These taxonomy entries illustrate the algorithmic diversity and structural rigor that characterizes contemporary RTUS research across theoretical computer science, robotics, and control domains.