---
title: 'BMRNG: Block-aware Monotonic Relative Graph'
url: https://www.emergentmind.com/topics/block-aware-monotonic-relative-neighborhood-graph-bmrng
type: topic
---

# BMRNG: Block-aware Monotonic Relative Graph

Searching arXiv for the cited BMRNG/MRNG papers to ground the article in the latest indexed records.
Block-aware Monotonic Relative Neighborhood Graph (BMRNG) is a proximity-graph model for disk-based approximate nearest neighbor search (ANNS) in which graph construction is defined jointly with a fixed storage layout over disk blocks. In contrast to classical monotonic proximity graphs, which reason only at the node level, BMRNG introduces the notion of a monotonic I/O path and requires that, for any pair of nodes, navigation can proceed through a sequence of block accesses and intra-block graph steps that strictly decrease distance to the target at every edge traversal and at every block transition [2509.03226]. The concept extends the Monotonic Relative Neighborhood Graph (MRNG), whose theoretical role in graph-based ANNS is to provide a unique edge-minimal monotonic graph under a lens-based directed edge rule in Euclidean space [2107.13052]. BMRNG retains the monotonic-neighborhood perspective of MRNG, but its occlusion rules are modified so that “one disk I/O plus an intra-block monotone walk” acts as the relevant primitive for pruning cross-block edges [2509.03226].

## 1. MRNG as the theoretical precursor

MRNG is defined on a finite set \(S \subseteq \mathbb{R}^d\) with Euclidean distance
\[
\delta(x,y)=\|x-y\|_2,
\]
and uses the lune
\[
lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)
\]
to determine directed edges [2107.13052]. For a directed graph \(G\) with \(V(G)=S\), monotonicity means that for every pair \(p,q\in V(G)\) there exists a path \(v_1,\dots,v_l\) from \(p\) to \(q\) such that
\[
\delta(v_i,q)>\delta(v_{i+1},q),\quad \forall i=1,\dots,l-1.
\]
This property underlies greedy search: the standard `closer-and-go` procedure repeatedly moves to a strictly closer out-neighbor and, on a monotonic graph, is guaranteed to reach the target node [2107.13052].

The MRNG edge rule is lens-based and asymmetric. A directed edge \(\overrightarrow{xy}\) is present iff for every \(z\in lune(x,y)\cap S\), the edge \(\overrightarrow{xz}\) is absent. The 2021 analysis established that, for any finite \(S\subseteq \mathbb{R}^d\), there exists a unique MRNG on \(S\), and that this graph is edge-minimal monotonic: removing any edge destroys monotonicity [2107.13052]. This gives MRNG a canonical theoretical status among monotonic proximity graphs: it guarantees greedy reachability while using the fewest edges among monotonic graphs on the same point set.

The same work also showed that MRNG has bounded maximum out-degree for fixed dimension,
\[
\Delta(G)\le O\bigl((1+6/\pi)^d\bigr),
\]
reflecting angular separation induced by the lens rule [2107.13052]. The bound is exponential in dimension in the worst case, but the experiments reported in the paper indicate more moderate degree behavior on synthetic data, together with skewed high-dimensional degree distributions that produce hubs. This suggests why degree-bounded and approximate variants are practically relevant even when the exact theoretical object is well understood.

## 2. Block assignment and monotonic I/O paths

BMRNG begins from a proximity graph \(G=(V,E)\) together with a block assignment
\[
\mathcal{B}=(V,\mathcal{L}),\qquad \mathcal{L}:V\to\{1,\dots,m\},
\]
which partitions the vertex set into disk blocks
\[
B_i=\{v\in V:\mathcal{L}(v)=i\}
\]
[2509.03226]. In the intended disk-resident model, nodes in the same block are fetched by a single I/O, so the storage partition is not an auxiliary annotation but part of the index semantics.

The central object is a monotonic I/O path from \(u\) to \(q\). Such a path is a sequence of blocks \(P=[B_0,B_1,\dots,B_L]\) equipped with intra-block node sequences \([v_{i,1},\dots,v_{i,l_i}]\) satisfying graph-consistency and strict distance decrease both within blocks and across blocks. For each block \(B_i\), the node sequence forms an intra-block path, adjacent blocks are connected by a cross-block edge, and the endpoints satisfy \(v_{0,1}=u\) and \(v_{L,l_L}=q\). Monotonicity requires
\[
\delta(v_{i,j},q)>\delta(v_{i,j+1},q)
\]
within each block and
\[
\delta(v_{i,l_i},q)>\delta(v_{i+1,l_{i+1}},q)
\]
across consecutive block transitions [2509.03226].

This definition is stronger than the node-level monotonicity studied for MRNG. Classical MRNG guarantees a strictly decreasing sequence of node distances to the target; BMRNG requires that the sequence of I/O operations itself be monotone in the sense that each new block access moves to a block whose best terminal node on the path is closer to the target [2509.03226]. A plausible implication is that BMRNG is designed not merely to support navigability in geometric space, but to align navigability with the actual cost model of disk-based retrieval.

## 3. Edge occlusion rules that define BMRNG

Given \(G=(V,E)\) and a block assignment \(\mathcal{B}=(V,\mathcal{L})\), \(G\) is a BMRNG if for any pair \(u,q\in V\), there exists a monotonic I/O path from \(u\) to \(q\) [2509.03226]. The definition is operationalized through two distinct occlusion rules, one for intra-block edges and one for cross-block edges.

For intra-block edges, if \(\mathcal{L}(u)=\mathcal{L}(q)\) and \((u,q)\) is an MRNG edge in the MRNG induced by block \(B_{\mathcal{L}(u)}\), then \((u,q)\) is kept in BMRNG. Thus each block’s induced subgraph is itself an MRNG [2509.03226]. This imports into each block the classical monotonic graph structure analyzed in the MRNG literature [2107.13052].

For cross-block edges, a candidate \((u,q)\) with \(\mathcal{L}(u)\neq\mathcal{L}(q)\) is occluded if there exists a node \(v\) such that \((u,v)\in BMRNG\) and either of two cases holds [2509.03226]. In Case 1, \(v\) lies in the same block as \(u\) and is in the lune of \(u\) and \(q\):
\[
\mathcal{L}(u)=\mathcal{L}(v),\qquad v\in lune_{u q}.
\]
This is the classical relative-neighborhood occlusion condition, but restricted to intra-block neighbors.

In Case 2, \(v\) lies in a different block, yet within block \(B_{\mathcal{L}(v)}\) there exists a monotonic path
\[
[v_1=v,v_2,\dots,v_l]
\]
ending at some \(v_l\in lune_{u q}\), with
\[
\delta(v_{i+1},q)<\delta(v_i,q),\quad \forall i.
\]
Here the occluding witness is not merely a single neighbor of \(u\), but a reachable node obtained via one cross-block step followed by an intra-block monotone walk [2509.03226]. This is the defining block-aware modification: relative-neighborhood occlusion is generalized from single-hop local geometry to a storage-aware multi-hop criterion.

This construction differs from standard MRNG in a precise way. In MRNG, only immediate outgoing neighbors can block a candidate edge from a node under the lens rule [2107.13052]. In BMRNG, the blocking witness may be a node accessible after one block access plus a monotone walk inside that block [2509.03226]. This suggests that BMRNG treats intra-block traversal as comparatively cheap once the corresponding page has been loaded, and therefore permits more aggressive pruning of cross-block edges than an ordinary monotonic graph would.

## 4. Theoretical guarantees and asymptotic implications

The foundational theoretical statement for BMRNG is Theorem 1: if a graph satisfies the intra-block and cross-block rules above, then for any two nodes \(u,q\), there exists a monotonic I/O path from \(u\) to \(q\) [2509.03226]. The proof argument mirrors the classical monotonic descent logic of MRNG but adapts it to block transitions. If \(u\) and \(q\) are in the same block, the induced intra-block MRNG ensures a strictly monotone path. If they are in different blocks and the direct edge \((u,q)\) is absent, the occlusion rules imply the existence of a valid witness that yields either an intra-block monotone step or a cross-block step followed by a monotone intra-block walk to a node in the lune, thereby strictly reducing distance to \(q\) [2509.03226].

The same work derives an expected I/O path length under a random block assignment model. Assume nodes are partitioned uniformly at random into \(m=\lceil n/c\rceil\) blocks of exactly \(c\) nodes, and let \(P=\{B_1,\dots,B_k\}\) be a monotonic I/O path. If \(p=[v_1,\dots,v_{k'}]\) is a strictly distance-decreasing node subsequence with \(k'\ge k\), and
\[
X=\sum_{i=1}^{k'-1}\mathbf{1}\{v_i\text{ and }v_{i+1}\text{ are in the same block}\},
\]
then
\[
\mathbb{E}[X\mid k']=(k'-1)\frac{c-1}{n-1},
\]
and hence
\[
\mathbb{E}[k-1]=\frac{n-c}{n-1}\mathbb{E}[k'-1].
\]
Using the adapted NSG path-length analysis, the paper states
\[
\mathbb{E}[k'-1]=O\left(\frac{n^{1/d}\log n^{1/d}{\Delta r}\right),
\]
which yields
\[
\mathbb{E}[|P|]=O\left(\frac{n-c}{n-1}\cdot\frac{n^{1/d}\log n^{1/d}{\Delta r}}\right)
\]
[2509.03226]. The formal conclusion is that increasing the number of nodes per block decreases expected I/O path length.

The exact construction of BMRNG is, however, expensive. The paper describes a conceptual pipeline consisting of block assignment, intra-block MRNG construction, and cross-block edge evaluation under the block-aware occlusion criterion. Because cross-block edge selection requires candidate consideration outside each node’s block, with sorting and occlusion checks dominating cost, the total time is stated to be at least
\[
O(n^2\log n),
\]
which is prohibitive at large scale [2509.03226]. The same source also notes that block assignment itself is NP-hard as a balanced graph partitioning problem. Accordingly, the exact BMRNG serves primarily as a theoretical object, not as the directly deployed index.

## 5. Practical approximation: BAMG

The practical realization associated with BMRNG is BAMG, the Block-Aware Monotonic Graph, which is introduced as an efficient approximation that can be constructed in linear time from a monotonic graph while considering storage layout [2509.03226]. BAMG starts from an NSG graph, derives a block assignment using Starling’s BNF block shuffling, and then performs block-aware pruning to produce a graph with fewer cross-block edges and preserved intra-block navigability [2509.03226].

The construction is deliberately asymmetric between intra-block and cross-block structure. BAMG keeps all intra-block NSG edges rather than rebuilding an MRNG within each block, because the paper argues that the requirement of “MRNG inside each block” may force unnatural intra-block edges when block assignment is geometrically imperfect [2509.03226]. Cross-block edges are then pruned using a bounded-depth intra-block search inspired by BMRNG Rule 2 Case 2. For each node \(u\), neighbors are partitioned into intra-block neighbors and cross-block candidates. Each cross-block candidate \(q\) is tested against already accepted cross-block neighbors \(v\in R_{\text{out}}\) by invoking `search_within_block(B_{L(v)}, v, q, C, \alpha)`, which explores up to \(\alpha\) hops along a monotone path within the candidate neighbor’s block [2509.03226].

The pruning condition is
\[
Prune(u,q)\iff \exists l\le \alpha,\ \exists [v_1,\dots,v_l]\text{ monotone towards }q,
\]
with
\[
v_0=u,\qquad \delta(v_{i+1},q)<\delta(v_i,q)\ \forall i,
\]
and
\[
\delta(v_l,q)\cdot \beta<\delta(u,q),
\]
where \(\alpha\) is the maximum intra-block search depth and \(\beta\ge 1\) is a closeness threshold [2509.03226]. Larger \(\alpha\) allows more intra-block paths to act as occluders, while larger \(\beta\) makes pruning harder to satisfy, resulting in a denser graph [2509.03226]. The paper also states that when \(\mathcal{L}(v)=\mathcal{L}(q)\) after such search, BAMG adds intra-block edges \((v,q)\) and \((q,v)\), and additionally may add edges between multiple neighbors of \(u\) residing in the same block to reduce duplicate block accesses [2509.03226].

Under the assumption that NSG degree is constant in expectation and \(\alpha\) is a small constant, BAMG construction is stated to run in overall
\[
O(n)
\]
time, in contrast to the \(O(n^2\log n)\) exact BMRNG construction [2509.03226]. The practical distinction is therefore clear: BMRNG is the exact storage-aware monotonic graph model with formal I/O-monotonic guarantees, whereas BAMG is a scalable approximation that preserves the intended bias toward intra-block progress and reduced cross-block degree without reproducing the full exact criterion.

## 6. Search, storage organization, and empirical behavior

BAMG is accompanied by a decoupled storage design and a block-first search strategy, both motivated by the BMRNG cost model [2509.03226]. The graph index stores compact node records containing OID, VID, and neighbor OIDs, while raw vectors are stored separately in contiguous raw-vector blocks that respect the same ordering as the graph blocks [2509.03226]. This decoupling reduces graph-node size, allowing more nodes per graph block and thus increasing the effective block occupancy parameter \(c\), which the theoretical BMRNG analysis links to shorter expected I/O paths [2509.03226].

The query algorithm performs block-first exploration. After obtaining entry nodes from a multi-layer in-memory navigation graph, the search maintains a candidate pool ranked by PQ-coded approximate distances \(\hat{\delta}(\cdot,q)\). When the closest unchecked node \(v\) is selected, the algorithm loads the corresponding graph block and calls `search_within_block(B, v, q, C, \alpha)`, which explores promising same-block neighbors before relying on cross-block edges [2509.03226]. The search thus attempts to exploit each disk I/O fully by saturating useful intra-block traversal while the page is already resident. A plausible implication is that the BMRNG notion of monotonic I/O path is not only a structural guarantee but also a direct guide for scheduling graph expansion under disk latency constraints.

The paper also introduces a multi-layer navigation graph built recursively by selecting representative nodes from each block based on in-degree and reachability, continuing until the top layer has size at most \(\gamma\) [2509.03226]. The resulting hierarchy resembles a block-aware entry-selection mechanism rather than an independent graph family: each layer is again built as a BAMG on the selected representatives, so the storage-aware monotonic design persists across scales.

Empirically, on six real datasets—DEEP1M, SIFT1M, GIST, MSONG, CRAWL, and GLOVE—BAMG is reported to achieve up to \(2.1\times\) higher throughput and up to 52% fewer I/O reads than the compared state-of-the-art methods while maintaining comparable recall [2509.03226]. The paper further reports that BAMG has lower cross-block out-degree than Starling on all datasets, while total degree is similar or slightly lower, supporting the intended effect of BMRNG-inspired pruning: most cross-block edges can be removed without harming connectivity or search quality [2509.03226]. The ablation study states that removing BMRNG-inspired pruning degrades QPS and increases NIO, and removing the navigation graph also hurts performance, particularly at lower recall regimes [2509.03226].

## 7. Relation to neighboring graph-index paradigms and limitations

BMRNG and BAMG occupy a specific position within the broader graph-based ANN literature. MRNG and NSG supply node-level monotonicity, HNSW uses hierarchical layers without explicitly enforcing monotone paths, DiskANN focuses on disk-resident ANN with compressed codes and raw-vector separation but does not co-design graph edges with block boundaries, and Starling optimizes storage layout for an existing graph through block shuffling [2509.03226]. BMRNG differs by making block assignment part of the edge definition itself; BAMG differs by pruning cross-block edges explicitly in response to that block structure [2509.03226].

The relationship to MRNG is especially direct. MRNG offers the clean theoretical model of a unique edge-minimal monotonic graph on points in Euclidean space [2107.13052]. BMRNG preserves the monotonic-graph ethos but changes the granularity of monotonicity from nodes to disk I/Os, while relaxing the occlusion logic so that a reachable intra-block monotone path can substitute for a direct cross-block edge [2509.03226]. This does not supersede MRNG; rather, it reinterprets monotonicity under a different objective function, namely disk access count instead of only graph-hop descent.

Several limitations are explicit. The formal guarantees apply to exact BMRNG, whose construction is expensive and depends on block assignment quality; BAMG only approximates those guarantees [2509.03226]. The theoretical expected-I/O analysis assumes uniformly random block assignment, whereas the implemented system uses BNF block shuffling, so the asymptotic result is primarily interpretive rather than a direct predictor of realized path lengths [2509.03226]. The quality of block assignment remains critical, PQ-based distance estimates may misrank candidates, and SSD-based evaluation does not necessarily characterize behavior on other storage media [2509.03226].

These constraints clarify a common misconception. BMRNG is not merely an MRNG stored in blocks, nor is BAMG simply a storage-layout optimization of an existing graph. The defining claim is stronger: BMRNG formalizes a graph whose edges are chosen with block access as the primitive cost, and BAMG is a practical approximation that attempts to preserve that block-aware monotonic structure while remaining scalable [2509.03226].

Source: https://www.emergentmind.com/topics/block-aware-monotonic-relative-neighborhood-graph-bmrng