Spanning Tree Coverage Algorithm

• Spanning Tree Coverage (STC) is a foundational algorithm that guarantees complete coverage of graphs by generating a spanning tree and systematically circumnavigating it.
• The algorithm extends to multi-robot and obstructed terrain settings using variants like ESTC and MSTC*, balancing coverage cost, congestion, and makespan through efficient partitioning and local search techniques.
• Modern STC frameworks offer provable performance guarantees with near-linear runtime and approximation bounds, making them critical for applications in robotic path planning and network design.

Spanning Tree Coverage (STC) Algorithm is a foundational algorithmic paradigm for complete coverage of graphs, with primary applications in coverage path planning, robot coordination on grid terrains, and metric structures in computational geometry. STC constructs a structured exploration path by first generating a spanning tree of a graph and then performing a systematic circumnavigation of this tree to guarantee visitation of all vertices or subregions of interest. Variants of STC handle congestion minimization in network design, optimize coverage cost under physical constraints, and extend to multi-robot and obstructed-terrain settings. Modern research investigates approximation guarantees, hereditary bisection lower bounds, and local-search-enhanced frameworks for large-scale deployment.

1. Classical STC Paradigm and Core Definitions

The original Spanning Tree Coverage (STC) paradigm operates as follows on a discretized environment such as a 2D grid or a generic undirected, connected graph $G=(V,E)$:

• A spanning tree $T$ of $G$ is computed, typically via MST or DFS, ensuring connectivity without cycles.
• The coverage tour is generated by circumnavigating $T$, often using the right-hand or left-hand rule, such that every edge and vertex associated with the spanning tree is visited according to a deterministic sequence.
• For a grid, the coverage path corresponds to a traversal that "skirts" around the constructed tree, guaranteeing that all cells are swept without omission or hole.
• In mathematical terms, for a graph $G$ and its spanning tree $T$, the congestion of $T$, $c(G,T)$, is the maximal number of original graph edges that are routed through a given edge $e\in E_T$ when all pairs $(u,v)$ are connected via their unique tree path; formally,

$$c_T(e) = |\{uv \in E : \text{the unique path between } u,v \text{ in } T \text{ passes through } e\}|$$

The minimum achievable congestion over all spanning trees of $G$ is denoted $STC(G)$ (Kolman, 2024).

This single-robot STC ensures complete coverage with a path cost that is at most twice the sum of edge weights of $T$, as each edge of $T$ is circumnavigated from both sides (Tang et al., 2021). The algorithm runs in $O(|V| + |E|)$ for typical cases.

2. Extensions: Multi-Robot, Physical Constraints, and Incomplete Terrains

The STC framework generalizes to multi-robot coverage (MCPP) and to environments with obstacles and additional cost structures:

• Multi-Robot STC (MSTC*, Balanced-MSTC*) partitions the coverage tour produced by STC into $k$ contiguous segments (one per robot), optionally applying iterative balanced-cut strategies to minimize makespan (maximum partition cost). Each robot executes its assigned tour, starting and ending at specified depots, and must cover all assigned cells. The segmentation is refined such that workloads are balanced, up to the point where no further reduction in maximum cost is possible by shifting segment boundaries (Tang et al., 2021).
• Incomplete or Obstructed Terrains: In grids where obstacles result in partially missing 2×2 blocks, the classical STC fails because circumnavigation cannot proceed as usual. Here, Extended-STC (ESTC) is applied, constructing a hypergraph where each cell (or partial cell) is collapsed into a hypervertex, and hyperedges are assigned weights reflecting the true cost difference between possible traversals around adjacent cells. ESTC finds a minimum spanning tree (MST) in this hypergraph and applies an adapted circumnavigation to guarantee that all reachable free vertices are covered (Tang et al., 2024, Tang et al., 2023).

3. Approximation, Congestion, and Theoretical Guarantees

STC is utilized in both coverage path planning and the Spanning Tree Congestion (congestion minimization) problem:

• Congestion Minimization: The goal is to find $T$ that minimizes $\max_{e \in E_T} c_T(e)$. This NP-hard problem is classically approximated via arbitrary spanning trees ($n/2$-approximation). However, an $O(\Delta \cdot \log^{3/2}n)$-approximation algorithm is achieved (where $\Delta$ is the maximum degree of $G$), a major exponential improvement for polylog-bounded-degree graphs. The approximation guarantee critically relies on hereditary bisection width $hb(G)$, defined as the maximum bisection width over all subgraphs $H\subseteq G$:

$STC(G) \geq \Omega(hb(G)/\Delta)$

The approach recursively bisects $G$ using a balanced cut routine (e.g., ARV-Cut), recurses on subcomponents, and stitches subtrees with minimal added congestion at each level (Kolman, 2024).

• Single-Robot Makespan Bound: Classical STC produces a tour with length ≤2·(minimum spanning tree cost of $H$) where $H$ is the spanning cell-graph (Tang et al., 2021).
• Completeness and Optimality: ESTC guarantees coverage of all accessible free vertices for any connected subgraph of a decomposed grid and produces minimum-cost circumnavigation among all possible spanning trees on the induced hypergraph (Tang et al., 2024, Tang et al., 2023).
• Approximation for Path Planning: ESTC provides a bounded suboptimality ratio for general coverage tours, dependent on the number of incomplete hypervertices and hyperedge weights, leading to guarantees of constant-factor approximation on fixed grids (Tang et al., 2024).

4. Algorithmic Implementations and Local Search Enhancements

The core implementations for STC and its extensions are characterized by their recursive structure and post-processing techniques:

• CongSpanTree (Congestion Minimization): Recursively finds balanced cuts, applies ARV-Cut for $O(\sqrt{\log n})$-factor cut approximation, and merges subtour components using $k-1$ edges to form a spanning tree, ensuring the depth is $O(\log n)$. Each level adds congestion at most proportional to the bisection width (Kolman, 2024).
• ESTC (Obstructed, Weighted Grids):

1. Builds a hypergraph summarizing block connectivity and assigns edge weights as per grid topology,
2. Computes an MST (possibly with negative edge weights) via Kruskal's algorithm,
3. Applies coverage order by depth-first traversal, deforming paths as needed for incomplete neighborhoods,
4. Completes in $O(m n \log(m n))$ time for $m \times n$ grid (Tang et al., 2024, Tang et al., 2023).

• Local Search Multi-Robot Path Planning (LS-MCPP):
  • Maintains a partition of the decomposed coverage graph into $k$ connected subgraphs (one per robot),
  • Iteratively applies neighborhood operators (grow, deduplicate, exchange), modeled to preserve connectivity and coverage, optimizing the makespan heuristic,
  • After solution generation, a MAPF-style (e.g., PBS+SIPP) post-process resolves path conflicts and ensures turn cost-aware, collision-free execution (Tang et al., 2023, Tang et al., 2024).
  • Empirical results demonstrate up to 35.7% reduction in makespan versus prior tree-cover MCPP baselines and comparable or better results versus optimal solutions at much lower computational cost.

5. Applications in Robotic Coverage, Metric Embeddings, and Network Design

The STC paradigm and its variants have multifaceted relevance:

Application Area	STC Variant	Primary Objective
Multi-Robot Coverage Planning	STC, ESTC, MSTC*	Complete, balanced, and cost-efficient grid exploration
Spanning Tree Congestion	CongSpanTree, STC	Minimize worst-case routing congestion
Metric Spanners/Emulators	Spanning Tree Covers	Compact structures with small stretch for all pairs

• Robotic Coverage: STC (and ESTC) in multi-robot settings is foundational for exploration, agriculture, surveillance, and cleaning robots in structured and obstacle-rich environments (Tang et al., 2021, Tang et al., 2024, Tang et al., 2023).
• Metric Structures: Spanning tree covers yield spanners and emulators with parameterizable stretch and size. Recursively constructed tree covers parameterized by treewidth or separator size enable efficient path-reporting, distance-labeling schemes, and scalable routing (Elkin et al., 9 Nov 2025).
• Network Design: STC-based approximations minimize edge congestion in communication networks and inform optimal layouts with provable guarantees (Kolman, 2024).

6. Complexity Results and Empirical Performance

Algorithmic and empirical properties include:

• Running Time: Classical STC and its single-robot variants run in linear or near-linear time for grid environments; ESTC and multi-robot STC depend linearly or quasi-linearly on the graph size and the number of local search iterations (Tang et al., 2024, Tang et al., 2021, Tang et al., 2023).
• Approximation Quality: The balanced-cut-based CongSpanTree achieves $O(\Delta \cdot \log^{3/2} n)$-approximation for STC in bounded degree graphs, with sublinear guarantees when $\Delta = o(n/\log^{3/2} n)$ (Kolman, 2024). Empirical studies observe consistent 10–40% improvements in makespan over classical approaches in MCPP (Tang et al., 2021, Tang et al., 2023).
• Practical Deployment: ESTC and LS-MCPP scale to 100+ robots on $256 \times 256$ grids within minutes; successful field deployment is reported for real robot platforms, including cost-aware and turn-aware navigation (Tang et al., 2024).

7. Related Paradigms and Ongoing Research Directions

• Generalizations and Parameterizations: The framework of spanning tree covers parameterized by graph treewidth or separator size achieves trade-offs between stretch, cover size, and average per-vertex overlap, extending the utility of STC to metric embedding, spanners, and labeling schemes (Elkin et al., 9 Nov 2025).
• Hereditary Bisection Lower Bounds: By establishing $\Omega(hb(G)/\Delta)$ as a lower bound for $STC(G)$, modern results more tightly couple coverage/path-planning hardness to underlying structural properties of $G$ (Kolman, 2024).
• Cut-Based and Minor-Closed Family Specializations: For planar and minor-closed graphs, constant-factor balanced separators yield improved approximation bounds in both congestion and coverage planning (Kolman, 2024).

STC and its modern extensions thus connect algorithmic robotics, network design, and metric graph theory, offering provable guarantees and practical scalability for a variety of coverage, congestion, and path optimization problems in graphs.