G3T*: Greedy GuILD Grafting Trees for Motion Planning

• The paper introduces a novel mechanism that repairs failed inter-tree connections through common-neighbor grafting, avoiding costly reverse search restarts.
• G3T* optimizes local densification using a greedy selection rule based on minimizing the Lebesgue measure of GuILD subsets, which refines the sampling region effectively.
• Adaptive sampling balances effort between GuILD subsets and the full informed set, ensuring fast convergence and asymptotic optimality even in complex, high-dimensional environments.

Searching arXiv for the cited G3T* and related greedy tree papers to ground the article in current records. Greedy GuILD Grafting Trees (G3T*) is a bidirectional sampling-based motion planner that addresses a specific failure mode in asymmetric bidirectional search: when a candidate edge connecting the forward and reverse trees is invalid, the planner grafts invalid edge connections at both ends to re-establish tree-based connectivity instead of restarting the reverse search. It combines this grafting mechanism with a greedy use of the minimum Lebesgue measure of guided incremental local densification (GuILD) subsets and with dynamic adjustment of the sampling distribution between the informed set and GuILD subsets based on historical and current cost improvements, with the stated goal of enabling rapid path convergence while ensuring asymptotic optimality (Zhang et al., 27 Aug 2025).

1. Problem setting and algorithmic position

G3T* is formulated for single-query, asymptotically optimal motion planning in environments where bidirectional planning is advantageous but conventional asymmetric bidirectional methods can lose efficiency when the connection between the two trees fails. The motivating comparison class explicitly includes BIT*, AIT*, EIT*, and FIT*, which are described as asymmetric bidirectional methods that may restart the reverse lazy search at higher effort or resolution after a collision on a candidate connecting edge, causing expansion delays and redundant work, especially in narrow passages or under uneven sampling (Zhang et al., 27 Aug 2025).

The planner is therefore organized around three coupled ideas. First, invalid inter-tree edges are repaired through grafting rather than by immediate reverse-tree restart. Second, path optimization is focused through GuILD subsets, with beacon selection chosen greedily by minimizing combined Lebesgue measure. Third, the algorithm does not remain permanently confined to those local subsets; instead, it splits sampling effort between GuILD and the full informed set according to current and historical cost-improvement statistics, thereby preserving asymptotic optimality (Zhang et al., 27 Aug 2025).

A common misconception is to reduce G3T* to a generic bidirectional RRT-style planner with a local repair heuristic. The description in the source is narrower and more structured: the grafting rule is defined through common neighbors within an RGG-style connection radius, the local densification region is derived from GuILD hyperspheroids, and the exploration–exploitation balance is explicitly history-aware rather than fixed.

2. Grafting invalid inter-tree edges

The defining mechanism of G3T* is activated when an edge $(v_s,v_t)$ that should connect the forward and reverse trees is found to be invalid because of collision. Rather than restarting the reverse search, G3T* searches for common neighbors within a defined radius of both endpoints. The paper defines the common-neighbor set as

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$

where $N(v)$ contains neighbors within radius $r(m)$ (Zhang et al., 27 Aug 2025).

Each common neighbor $v'$ induces a detour edge-pair $(v_s,v')$ and $(v',v_t)$. Candidate detours are ranked greedily by estimated cost or estimated effort, namely by

$\widehat{c}(v_s, v') + \widehat{c}(v', v_t)$

or

$\overline{e}(v_s, v') + \overline{e}(v', v_t),$

and the best detour is evaluated first. The evaluation is two-stage: a lazy check is applied before full collision checking. If a valid grafting edge-pair is found, the forward search continues through the new connection; otherwise, the next-ranked candidate is examined (Zhang et al., 27 Aug 2025).

The connection radius follows random geometric graph theory:

$$r(m):= \gamma \left( \frac{\log m}{m} \right)^{1/n},$$

with $$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$0 determined by free space and problem dimension. This embeds the grafting rule within the usual connectivity regime used in asymptotically optimal sampling-based planning (Zhang et al., 27 Aug 2025).

In algorithmic terms, the planner alternates among batch sampling, lazy reverse search, forward search, and grafting. The high-level pseudocode in the source places the grafting subroutine directly inside the loop that handles forward-search edges: if the forward-search edge collides, grafting is attempted; if grafting fails, only then is the lazy reverse search updated (Zhang et al., 27 Aug 2025). This ordering is central to the claimed reduction in redundant reverse expansion.

3. Greedy GuILD subsets and beacon selection

GuILD, or Guided Incremental Local Densification, is used in G3T* as a path-optimization mechanism that concentrates sampling in two hyperellipsoids, denoted front and back, partitioned by a beacon along the current solution path. G3T* modifies this scheme through a greedy beacon-selection rule. Rather than selecting the beacon heuristically, it chooses the path state $$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$1 that minimizes the combined Lebesgue measure of the front and back hyperspheroids:

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$2

Here, $$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$3 and $$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$4 are the ellipsoids between start–beacon and beacon–goal, respectively (Zhang et al., 27 Aug 2025).

The source characterizes this as a minimum-Lebesgue-measure criterion. It further states that G3T* narrows the sampling region by identifying the “most difficult” nodes along each subpath, specifically those maximizing the sum of distances from start to beacon and from beacon to goal, and then constructing the greedy GuILD set as the union of the corresponding minimized-volume ellipsoidal subsets (Zhang et al., 27 Aug 2025).

This construction has two immediate consequences. First, it makes the local densification region smaller than the standard informed set associated with Informed RRT*, which the paper presents as a mechanism for focused local refinement. Second, it gives a concrete meaning to the “greedy” component of the algorithm’s name: greediness is not only in candidate graft selection, but also in the minimization of the local densification volume around the current solution structure.

A plausible implication is that G3T* uses greediness at two distinct scales: local topological repair through edge-pair choice and local measure minimization through subset selection. The source material supports this interpretation, although it presents the two components separately rather than as a single formal principle.

4. Adaptive sampling, completeness, and optimality

A planner that overcommits to narrow local subsets risks a local trap, a concern stated explicitly in the source. G3T* addresses this by dynamically allocating samples between GuILD subsets and the full informed set on the basis of current and historical cost improvement (Zhang et al., 27 Aug 2025).

The paper defines

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$5

for current cost improvement, and

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$6

for historical cost improvement. If $$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$7, interpreted in the source as local optimization stalling, the planner increases sampling outside GuILD but within the full informed set; otherwise, it continues to focus sampling locally (Zhang et al., 27 Aug 2025).

The theoretical claims attached to this strategy are explicit. The planner is stated to be probabilistically complete:

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$8

and asymptotically optimal:

$$Q(v_s, v_t) = \{v' \in \mathcal{X}_{sample} \mid v' \in N(v_s) \cap N(v_t)\},$$9

The source attributes the asymptotic-optimality statement to a formulation leveraging results from Karaman and Frazzoli and ties the guarantee to the fact that uniform exploration is periodically injected rather than permanently excluded (Zhang et al., 27 Aug 2025).

The paper also states a grafting-specific guarantee: as the neighbor radius shrinks and the sample density increases, the probability that a valid grafting neighbor exists approaches $N(v)$0. In the context of difficult obstacle configurations, this is the formal counterpart of the algorithm’s claim that it can bypass failed direct inter-tree connections without systematic restart (Zhang et al., 27 Aug 2025).

5. Empirical behavior and reported benchmarks

The reported empirical study covers Dividing wall-gaps (DW) and Random Rectangles (RR) environments in $N(v)$1, $N(v)$2, and $N(v)$3, with comparisons against RRT-Connect, Informed RRT*, BIT*, AIT*, EIT*, and FIT* (Zhang et al., 27 Aug 2025). The source summarizes the outcome as follows: success rate is higher, the initial solution is found faster, and quality is better in G3T*; the accompanying boxplots are described as showing faster convergence and improved robustness as dimension grows or as environments become more complex (Zhang et al., 27 Aug 2025).

Scenario	FIT*	G3T*
DW-4D	$N(v)$4 s, $N(v)$5	$N(v)$6 s, $N(v)$7
DW-8D	$N(v)$8 s, $N(v)$9	$r(m)$0 s, $r(m)$1

For these scenarios, the source highlights “+32.14% time” in DW-4D and “+33.15% time” in DW-8D for G3T* relative to FIT* (Zhang et al., 27 Aug 2025). The article does not normalize these figures further; it presents them as benchmark highlights.

A separate real-world evaluation uses a kitchen-manipulation task with an 8-DoF robot arm (DARKO) in a cluttered narrow space, over 30 runs with a 1 s timeout. The reported results are FIT*: 70% success with median cost approximately 22.7; EIT*: 33% success with median cost approximately 20.8; and G3T*: 80% success with median cost approximately 17.9 (Zhang et al., 27 Aug 2025). The paper’s conclusion from this experiment is that G3T* finds initial feasible paths more quickly and also returns shorter or smoother solutions more consistently.

6. Terminological scope and relation to other tree literature

The name G3T* can invite confusion because both “greedy trees” and “grafting” already have established meanings in other technical literatures. In metric geometry and computational geometry, a greedy tree is an added structure on a greedy permutation that tracks the approximate nearest predecessor. Such greedy trees have applications in proximity search and topological data analysis, and recent work gives deterministic constructions in $r(m)$2 time together with a linear-time merge of two approximate greedy trees in doubling metrics (Chubet et al., 2024). That literature concerns greedy permutations, Gonzalez orderings, farthest point traversals, finite Voronoi diagrams, bucket heaps with backburner, and merge-based construction; it is not the motion-planning construction used in G3T*, although the practical summary of that work explicitly notes implications for proximity-search and learning infrastructures “based on greedy-like ball trees” (Chubet et al., 2024).

Likewise, in Gaussian graphical testing, a grafting operation is a local topological transformation in which a subtree is cut from one location and reattached to another, and it is analyzed through Chernoff information and generalized eigenvalues (Li et al., 2017). This use of “grafting” is mathematically precise but belongs to a different problem class. The shared terminology suggests only a broad structural analogy: a local reconnection operation on a tree-like object. It does not imply shared algorithms, guarantees, or objective functions.

Within motion planning itself, the relevant contrast is instead with planners that handle invalid inter-tree edges by re-expanding or restarting lazy reverse search. G3T* is distinguished in the source by replacing that behavior with common-neighbor grafting, by selecting GuILD subsets through minimum Lebesgue measure, and by adapting sampling according to $r(m)$3 and $r(m)$4 so that local densification does not eliminate global exploration (Zhang et al., 27 Aug 2025).