Monotonic Relative Neighborhood Graph (MRNG)

• MRNG is a formally defined proximity graph that guarantees strictly decreasing, monotonic paths in greedy best-first search for ANNS.
• It is constructed by ordering neighbors by increasing Euclidean distance and selecting edges based on an empty-lune condition to maintain constant out-degree.
• MRNG underpins scalable, efficient graph-based ANN indices and inspires extensions like block-aware MRNG for disk-resident data environments.

A Monotonic Relative Neighborhood Graph (MRNG) is a formally defined proximity graph designed to guarantee strictly decreasing distance paths in greedy best-first search, primarily for approximate nearest neighbor search (ANNS) in high-dimensional spaces. MRNG ensures connectedness, compactness (constant out-degree), minimal search complexity, and forms the theoretical core for practical graph-based ANN indices, including recent disk-aware extensions.

1. Formal Definition and Construction

MRNG is constructed over a finite set $S \subset \mathbb{R}^d$ with the metric $\delta(p,q) = \|p-q\|_2$. For each pair $(p, q)$, define the lune

$$\mathrm{lune}_{p,q} = B(p, \delta(p,q)) \cap B(q, \delta(p,q)),$$

where $B(x, r) = \{y : \delta(x, y) < r\}$. The MRNG is the directed graph $(S, E)$ with

$$(p \to q) \in E \iff \mathrm{lune}_{p,q} \cap S = \varnothing \;\;\text{or}\;\; \forall r \in \mathrm{lune}_{p,q}:\ (p \to r) \notin E.$$

Equivalently, for construction: order $S \setminus \{p\}$ by increasing $\delta(p, \cdot)$, initialize the out-neighbor list $L_p$ with the nearest neighbor, and for each successive candidate $q$, include $q$ if no $r \in L_p$ satisfies $\delta(r, q) < \delta(p, q)$.

Pseudocode (all-pairs naive construction, $O(n^2\log n + n^2c)$ time):

for each p in S: R ← S \ {p}, sorted by δ(p, ·) ascending L ← { R[1] } for q in R[2:]: conflict ← false for r in L: if δ(r, q) < δ(p, q): conflict ← true break if not conflict: L.add(q) for q in L: add edge (p→q) to MRNG

MRNG is unique and edge-minimal with respect to monotonicity: removing any edge breaks the monotonic path guarantee (Zhu et al., 2021).

2. Monotonicity and Connectivity Properties

A path $v_1 \to v_2 \to \dots \to v_k$ from $p$ to $q$ is monotonic if

$$\delta(v_i, q) > \delta(v_{i+1}, q) \quad \forall\ i = 1, \dots, k-1.$$

MRNG is a Monotonic Search Network (MSNET), meaning for every pair $(p, q)$ there exists a monotonic path from $p$ to $q$. Greedy best-first search in MRNG—always moving to the neighbor closest to the query that is strictly closer—never backtracks, with each hop reducing the distance to the target. This ensures strong connectivity: for any $p, q \in S$, there is a directed path from $p$ to $q$ (Fu et al., 2017, Zhu et al., 2021).

MRNG is the unique minimal graph structure guaranteeing this property. Any deletion of edges would violate monotonic search coverage for some node pairs (Zhu et al., 2021).

3. Degree Bound and Space Complexity

The out-degree in MRNG is bounded only by the ambient dimension $d$, not by the cardinality $n$; for $d$-dimensional Euclidean space, by sphere-packing arguments, the degree is

$\Delta \leq O((1 + 6/\pi)^d).$

Empirical statistics show average node degrees plateau with increasing $n$; for $d=10$, mean degree $\approx 10$, max $\approx 30$, and for $d=100$ mean degree $\approx 40$, max $\approx 200$ (Zhu et al., 2021). The total number of edges is $O(n)$, yielding a compact index size (Fu et al., 2017).

4. Search-Time Complexity and Greedy Guarantees

For query $q$ and source $p$, let $L(p,q)$ denote the length of the monotonic path in MRNG, and $c$ the average node out-degree, which is $O(1)$. The greedy search procedure inspects at most $c$ candidates per hop, with total time

$$T_{\mathrm{search}} = O(c\,\mathbb{E}_{p,q}[L(p,q)]).$$

Under uniform random distribution in $\mathbb{R}^d$,

$$\mathbb{E}[L] = O\Big(n^{1/d} \log n^{1/d} / \Delta r\Big)$$

where $\Delta r$ is the minimal difference in triangle side-lengths in $S$. In high dimensions, this growth is close to logarithmic in $n$ (Fu et al., 2017). Greedy monotonic search in MRNG always terminates at the true nearest neighbor in at most $n$ steps; empirical scaling closely matches or exceeds alternatives like k-NN graphs (Zhu et al., 2021).

5. Generalizations, Approximation Techniques, and Hidden Structures

To accommodate large $n$, generalizations of MRNG restrict neighbor candidate sets and cap the out-degree:

• Degree cap $m$ ($|N_x| \leq m$)
• Candidate pool $U_x \subset S \setminus \{x\}$

Generic MRNG (GenMRNG) applies identical neighbor construction, limited to candidates from $U_x$ and up to $m$ neighbors. With $|U_x| = \ell$, build time is $O(n\ell\log\ell)$ and storage $O(nm)$ (Zhu et al., 2021).

MRNG possesses hidden structure termed conflicting nodes: for edge $v \to u$, the conflicting node set $C(v \to u)$ contains all $w$ such that $u \in \mathrm{lune}_{v,w}$—these are nodes "blocked" during construction. Recording conflict lists per edge facilitates escape from local minima in search, enabling improved recall and faster query time. The conflict-search subroutine, upon encountering a local minimum, scans $C(v \to u)$ for nodes closer to $q$; this strictly accelerates search convergence (Zhu et al., 2021).

Empirical findings indicate that capping degree at roughly half the mean degree of MRNG retains nearly full recall, with a phase transition where adding more edges yields negligible improvement (Zhu et al., 2021).

6. Block-Aware Monotonic Extensions for Disk-Based ANNS

The Block-aware Monotonic Relative Neighborhood Graph (BMRNG) extends MRNG principles to disk-resident graphs where disk I/O cost dominates. Vertex set $V$ is partitioned into blocks $\mathcal{B}$; block-aware pruning jointly considers both geometric and storage layout to ensure monotonic search over block transitions.

• Intra-block edges: Retain the full block-wise MRNG.
• Cross-block edges: Pruned aggressively; only kept if no intra-block monotonic shortcut exists.

A monotonic I/O path is defined as a sequence of block-transitions and node steps, with strictly decreasing distances to $q$ at every step. BMRNG guarantees such monotonic I/O paths for every pair $(u,q)$. The expected number of block transitions is

$$O\left(\frac{n-c}{n-1} \cdot \frac{n^{1/d}\log n^{1/d}}{\Delta r}\right)$$

(Li et al., 3 Sep 2025).

The Block-Aware Monotonic Graph (BAMG) is a practical, linear-time index that approximates BMRNG by integrating block layout into edge selection and leverages in-memory multi-layer navigation graphs for efficient search entry. BAMG achieves up to 2.1× higher throughput and 13–52% fewer disk I/Os compared to prior approaches, with only a small index size overhead (Li et al., 3 Sep 2025).

7. Practical Implications and Empirical Performance

MRNG serves as a theoretical foundation for scalable ANNS indices such as Navigating Spreading-out Graphs (NSG), which are practical MRNG approximations with nearly $O(n^{1.3})$ build cost and billion-node scalability (Fu et al., 2017).

Guidance for implementation includes using genMRNG with degree caps at approximately half the empirical mean, candidate pools drawn from k-NN graphs or random samples, and conflict lists for heavy edges, optimizing both index build and search times (Zhu et al., 2021).

Compared to classical Relative Neighborhood Graphs (RNG), MRNG relaxes the empty-lune condition, enforcing forward-only monotonic greedy paths and precluding backtracking. BMRNG/BAMG further extend this property to disk-based environments, making graph construction and block assignment a joint optimization problem, resulting in significant query throughput and I/O gains while preserving monotonicity and recall (Li et al., 3 Sep 2025).