[Paper] Location-Aware Dispersion on Anonymous Graphs

Source: arXiv - 2602.05948v1

Overview

The paper Location‑Aware Dispersion on Anonymous Graphs extends the classic dispersion problem for mobile robots by adding a color‑matching constraint: each robot must settle on a node whose color matches its own. The authors study this problem on anonymous, unlabeled graphs and design deterministic algorithms that achieve provable bounds on execution time and per‑robot memory, while also proving impossibility results that delineate what cannot be solved efficiently.

Key Contributions

• Formal definition of Location‑Aware Dispersion – a natural generalization of dispersion that incorporates node and robot colors.
• Deterministic algorithms for several settings (e.g., unknown number of robots k, unknown graph size n, arbitrary color sets) with explicit upper bounds on time complexity (number of synchronous rounds) and memory usage (bits per robot).
• Impossibility theorem showing that, without additional assumptions, no deterministic algorithm can guarantee dispersion when robots cannot distinguish certain symmetric configurations.
• Lower‑bound proof on the time required for any deterministic solution, establishing a gap between the new problem and classic dispersion.
• Comprehensive comparison with existing dispersion results, highlighting the extra difficulty introduced by color constraints.

Methodology

1. Model & Assumptions

   • The underlying network is an anonymous, undirected, connected graph (nodes have no unique IDs).
   • Each node carries a color from a finite set C (|C| ≤ n).
   • Robots are identical except for their own color and have no global knowledge of k (the number of robots) or n (the number of nodes).
   • Communication is limited to local observations (a robot sees the colors of its current node and neighboring ports).

2. Algorithmic Building Blocks

   • Exploration primitives based on depth‑first search (DFS) that work without node IDs, using port numbers to navigate.
   • Color‑matching handshake: a robot only settles when it reaches a node whose color equals its own and the node is currently unoccupied.
   • Conflict resolution via deterministic tie‑breaking rules (e.g., priority based on robot color ordering) to avoid two robots attempting the same node.

3. Design of Specific Algorithms

   • Baseline algorithm for the case |C| = 1 (reduces to classic dispersion) – serves as a correctness baseline.
   • Multi‑color algorithm that cycles through colors, allowing robots of each color to “claim” matching nodes in separate phases, thereby preventing cross‑color interference.
   • Memory‑optimal variant that compresses the DFS stack into O(log Δ + log t) bits (Δ = max degree, t = |C|).

4. Proof Techniques

   • Inductive invariants guaranteeing that after each phase, all settled robots occupy correctly colored nodes and no two share a node.
   • Adversarial symmetry arguments to establish impossibility when robots cannot break symmetry without extra information.
   • Counting arguments for lower bounds, showing any deterministic protocol must explore at least Ω(n) edges in the worst case.

Results & Findings

The authors demonstrate that location awareness adds a multiplicative overhead proportional to the number of colors, but the problem remains solvable deterministically with modest memory footprints.

Practical Implications

• Swarm robotics in colored environments – e.g., warehouse robots that must dock at stations of a specific type (charging, loading, maintenance). The algorithms give a blueprint for guaranteeing that each robot finds a suitable station without central coordination.
• Distributed sensor placement – deploying autonomous drones to occupy pre‑designated zones (colored by terrain type or task) while avoiding collisions.
• Network resource allocation – mapping the problem to virtual agents (e.g., micro‑services) that must claim servers with matching capabilities (CPU‑type, GPU‑type) in a data‑center modeled as an anonymous graph.
• Memory‑constrained IoT devices – the memory‑optimal variant shows that even devices with a few dozen bits of state can achieve coordinated placement, which is valuable for ultra‑low‑power edge nodes.

Overall, the work provides deterministic, provably correct protocols that can be embedded directly into robot firmware or distributed system libraries, reducing the need for heavyweight central planners.

Limitations & Future Work

• Scalability with many colors: The runtime scales linearly with t; for large color sets (e.g., fine‑grained task categories) this may become prohibitive.
• Synchronous round model: The analysis assumes globally synchronized steps; real‑world deployments often face asynchrony and message delays.
• No fault tolerance: The algorithms assume all robots remain functional; handling crashes or Byzantine behavior is left open.
• Impossibility under anonymity: The paper shows deterministic impossibility in certain symmetric graphs; incorporating randomization or weak identifiers (e.g., locally unique port numbers) could overcome these barriers.

Future research directions include randomized algorithms that break symmetry more efficiently, asynchronous extensions, and adaptive strategies that reduce the t factor by dynamically grouping colors or merging phases.

Authors

• Himani
• Supantha Pandit
• Gokarna Sharma

Paper Information

• arXiv ID: 2602.05948v1
• Categories: cs.DC, cs.DS, cs.MA, cs.RO
• Published: February 5, 2026
• PDF: Download PDF