BMRNG: Block-aware Monotonic Relative Graph

• Block-aware Monotonic Relative Neighborhood Graph (BMRNG) is a proximity graph model that enhances disk-resident approximate nearest neighbor search by jointly considering graph connectivity and fixed disk block layouts.
• It extends classical MRNG by introducing occlusion rules where intra-block monotone walks can substitute for cross-block edges, thereby reducing disk I/O costs.
• A practical approximation called BAMG leverages BMRNG-inspired block-aware pruning to achieve higher throughput and fewer I/O reads, as confirmed on several real-world datasets.

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 (Li et al., 3 Sep 2025). 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 (Zhu et al., 2021). 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 (Li et al., 3 Sep 2025).

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 (Zhu et al., 2021). 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 (Zhu et al., 2021).

The MRNG edge rule is lens-based and asymmetric. A directed edge $\delta(x,y)=\|x-y\|_2,$0 is present iff for every $\delta(x,y)=\|x-y\|_2,$1, the edge $\delta(x,y)=\|x-y\|_2,$2 is absent. The 2021 analysis established that, for any finite $\delta(x,y)=\|x-y\|_2,$3, there exists a unique MRNG on $\delta(x,y)=\|x-y\|_2,$4, and that this graph is edge-minimal monotonic: removing any edge destroys monotonicity (Zhu et al., 2021). 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(x,y)=\|x-y\|_2,$5

reflecting angular separation induced by the lens rule (Zhu et al., 2021). 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 $\delta(x,y)=\|x-y\|_2,$6 together with a block assignment

$\delta(x,y)=\|x-y\|_2,$7

which partitions the vertex set into disk blocks

$\delta(x,y)=\|x-y\|_2,$8

(Li et al., 3 Sep 2025). 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 $\delta(x,y)=\|x-y\|_2,$9 to $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$0. Such a path is a sequence of blocks $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$1 equipped with intra-block node sequences $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$2 satisfying graph-consistency and strict distance decrease both within blocks and across blocks. For each block $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$3, the node sequence forms an intra-block path, adjacent blocks are connected by a cross-block edge, and the endpoints satisfy $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$4 and $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$5. Monotonicity requires

$$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$6

within each block and

$$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$7

across consecutive block transitions (Li et al., 3 Sep 2025).

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 (Li et al., 3 Sep 2025). 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 $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$8 and a block assignment $$lune(x,y)=B_{\delta(x,y)}(x)\cap B_{\delta(x,y)}(y)$$9, $G$0 is a BMRNG if for any pair $G$1, there exists a monotonic I/O path from $G$2 to $G$3 (Li et al., 3 Sep 2025). 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 $G$4 and $G$5 is an MRNG edge in the MRNG induced by block $G$6, then $G$7 is kept in BMRNG. Thus each block’s induced subgraph is itself an MRNG (Li et al., 3 Sep 2025). This imports into each block the classical monotonic graph structure analyzed in the MRNG literature (Zhu et al., 2021).

For cross-block edges, a candidate $G$8 with $G$9 is occluded if there exists a node $V(G)=S$0 such that $V(G)=S$1 and either of two cases holds (Li et al., 3 Sep 2025). In Case 1, $V(G)=S$2 lies in the same block as $V(G)=S$3 and is in the lune of $V(G)=S$4 and $V(G)=S$5: $V(G)=S$6 This is the classical relative-neighborhood occlusion condition, but restricted to intra-block neighbors.

In Case 2, $V(G)=S$7 lies in a different block, yet within block $V(G)=S$8 there exists a monotonic path

$V(G)=S$9

ending at some $p,q\in V(G)$0, with

$p,q\in V(G)$1

Here the occluding witness is not merely a single neighbor of $p,q\in V(G)$2, but a reachable node obtained via one cross-block step followed by an intra-block monotone walk (Li et al., 3 Sep 2025). 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 (Zhu et al., 2021). In BMRNG, the blocking witness may be a node accessible after one block access plus a monotone walk inside that block (Li et al., 3 Sep 2025). 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 $p,q\in V(G)$3, there exists a monotonic I/O path from $p,q\in V(G)$4 to $p,q\in V(G)$5 (Li et al., 3 Sep 2025). The proof argument mirrors the classical monotonic descent logic of MRNG but adapts it to block transitions. If $p,q\in V(G)$6 and $p,q\in V(G)$7 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 $p,q\in V(G)$8 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 $p,q\in V(G)$9 (Li et al., 3 Sep 2025).

The same work derives an expected I/O path length under a random block assignment model. Assume nodes are partitioned uniformly at random into $v_1,\dots,v_l$0 blocks of exactly $v_1,\dots,v_l$1 nodes, and let $v_1,\dots,v_l$2 be a monotonic I/O path. If $v_1,\dots,v_l$3 is a strictly distance-decreasing node subsequence with $v_1,\dots,v_l$4, and

$v_1,\dots,v_l$5

then

$v_1,\dots,v_l$6

and hence

$v_1,\dots,v_l$7

Using the adapted NSG path-length analysis, the paper states

$v_1,\dots,v_l$8

which yields

$v_1,\dots,v_l$9

(Li et al., 3 Sep 2025). 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

$p$0

which is prohibitive at large scale (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025).

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 (Li et al., 3 Sep 2025). Cross-block edges are then pruned using a bounded-depth intra-block search inspired by BMRNG Rule 2 Case 2. For each node $p$1, neighbors are partitioned into intra-block neighbors and cross-block candidates. Each cross-block candidate $p$2 is tested against already accepted cross-block neighbors $p$3 by invoking search_within_block(B_{L(v)}, v, q, C, \alpha), which explores up to $p$4 hops along a monotone path within the candidate neighbor’s block (Li et al., 3 Sep 2025).

The pruning condition is

$p$5

with

$p$6

and

$p$7

where $p$8 is the maximum intra-block search depth and $p$9 is a closeness threshold (Li et al., 3 Sep 2025). Larger $q$0 allows more intra-block paths to act as occluders, while larger $q$1 makes pruning harder to satisfy, resulting in a denser graph (Li et al., 3 Sep 2025). The paper also states that when $q$2 after such search, BAMG adds intra-block edges $q$3 and $q$4, and additionally may add edges between multiple neighbors of $q$5 residing in the same block to reduce duplicate block accesses (Li et al., 3 Sep 2025).

Under the assumption that NSG degree is constant in expectation and $q$6 is a small constant, BAMG construction is stated to run in overall

$q$7

time, in contrast to the $q$8 exact BMRNG construction (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). This decoupling reduces graph-node size, allowing more nodes per graph block and thus increasing the effective block occupancy parameter $q$9, which the theoretical BMRNG analysis links to shorter expected I/O paths (Li et al., 3 Sep 2025).

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 $$\delta(v_i,q)>\delta(v_{i+1},q),\quad \forall i=1,\dots,l-1.$$0. When the closest unchecked node $$\delta(v_i,q)>\delta(v_{i+1},q),\quad \forall i=1,\dots,l-1.$$1 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 (Li et al., 3 Sep 2025). 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 $$\delta(v_i,q)>\delta(v_{i+1},q),\quad \forall i=1,\dots,l-1.$$2 (Li et al., 3 Sep 2025). 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 $$\delta(v_i,q)>\delta(v_{i+1},q),\quad \forall i=1,\dots,l-1.$$3 higher throughput and up to 52% fewer I/O reads than the compared state-of-the-art methods while maintaining comparable recall (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025).

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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025).

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 (Zhu et al., 2021). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025). 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 (Li et al., 3 Sep 2025).

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 (Li et al., 3 Sep 2025).