Asymmetric Grid Quorum Systems

• Asymmetric grid quorum systems are specialized constructions on grid-based substrates where quorum shapes, sizes, or families vary while ensuring non-empty intersections.
• They are applied in adaptive cognitive-radio rendezvous, flexible Paxos protocols, and topology-aware consensus to optimize load balance and energy efficiency.
• Evaluations reveal that strategic asymmetry enhances rendezvous probability, reduces time-to-rendezvous, and accommodates heterogeneous trust with robust intersection guarantees.

Searching arXiv for recent and foundational papers on asymmetric grid quorum systems and related quorum geometry. Searching for "asymmetric grid quorum systems" and related quorum constructions on arXiv. Asymmetric grid quorum systems are quorum constructions on a shared geometric substrate—most commonly a grid, hypergrid, or wall—in which quorum shape, size, or family is not globally uniform. In the literature, this asymmetry appears in several distinct but related forms: load-dependent row–column selections for cognitive-radio rendezvous, phase-specific row/column families for Paxos-style consensus, initiator-dependent tier paths in topology-aware quorum geometry, and per-process subjective quorum choices derived from heterogeneous trust assumptions. Across these settings, the unifying idea is that intersection is preserved while quorum geometry varies by user, process, phase, or tier (Jarrah et al., 2017, Howard et al., 2016, Mason, 20 Mar 2026, Senn et al., 16 Sep 2025).

1. Classical grid quorums and the structural basis of asymmetry

A quorum system is a family of subsets with a non-empty intersection property. In the grid-based formulation used by Maekawa and subsequent work, the universe is arranged as an $n \times n$ matrix, and a quorum is the union of a full row and a full column. In its basic form, the defining condition is

$$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$

The cognitive-radio literature imports this structure from classical distributed systems and power-saving protocols, but applies it to time slots or channel-time pairs rather than to sites. The thesis version makes the classical vocabulary explicit: besides non-empty intersection, it recalls the equal work property $|C_i| = k$ for all $i$, and the equal responsibility property that each site appears in exactly $k$ quorums. It then notes that, in the rendezvous setting, the same structure is applied over time slots or channel-time pairs rather than over sites (Jarrah, 2016).

Grid construction in the rendezvous setting is straightforward. With $N$ licensed channels, one chooses

$n = \lceil \sqrt{N} \rceil$

and organizes a cycle of $n^2$ time slots as an $n \times n$ grid $G=\{g_{i,j}\}$, mapping each cell to a channel. A user quorum is the union of selected rows and columns. In the basic $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$0 case, one row and one column yield quorum size $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$1, and any two such quorums intersect in at least two positions in the ideal symmetric design. This classical row–column geometry is the point of departure for all later asymmetric variants (Jarrah et al., 2017).

2. Forms of asymmetry in grid quorum literature

The literature uses “asymmetry” in several technically distinct senses.

Quorum size asymmetry: in adaptive rendezvous schemes, some users select $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$2 row and $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$3 column, others $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$4 rows and $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$5 column, and others $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$6 rows and $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$7 columns. The resulting quorum sizes differ, so the set of active slots is not uniform across users (Jarrah et al., 2017).

Pattern asymmetry: even when users share the same global grid and the same channel mapping, they may choose different row/column patterns. In the adaptive $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$8-Quorum RDV Scheme, the choice among $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$9, $|C_i| = k$0, and $|C_i| = k$1 depends on local traffic estimates, so asymmetry is algorithmic and per-user rather than geometric (Jarrah, 2016).

Cross-phase asymmetry: Flexible Paxos separates quorum families into $|C_i| = k$2 and $|C_i| = k$3, and requires only cross-phase intersection,

$|C_i| = k$4

The paper explicitly identifies a grid construction in which Phase-1 quorums are rows and Phase-2 quorums are columns; rows need not intersect other rows, and columns need not intersect other columns, but every row intersects every column (Howard et al., 2016).

Initiator-dependent asymmetry: the crumbling-wall construction in topology-aware consensus assigns different Phase-1 quorum families to different tiers. A proposer at tier $|C_i| = k$5 must obtain at least one respondent from its own tier and every tier below, so the quorum family depends on where the proposer is located in the wall (Mason, 20 Mar 2026).

Subjective asymmetry: asymmetric Byzantine quorum systems allow each process to choose its own fail-prone system and, correspondingly, its own quorum family. Several papers note that classical grid constructions can be instantiated in this per-process manner, yielding asymmetric grid quorums in which different processes trust different row/column combinations while still satisfying pairwise consistency and availability conditions (Alpos et al., 2019, Amores-Sesar et al., 11 Sep 2025).

These usages are compatible rather than contradictory. They identify different axes along which grid geometry can cease to be uniform while preserving the intersection structure required by the target protocol.

3. Adaptive row–column quorums for cognitive-radio rendezvous

The most explicit operational use of asymmetric grid quorums appears in channel-hopping coordination for cognitive radio networks. The network has $|C_i| = k$6 licensed channels $|C_i| = k$7, a cycle of $|C_i| = k$8 synchronized time slots with $|C_i| = k$9, and a shared channel-to-grid mapping

$i$0

For user $i$1, with selected row indices $i$2 and column indices $i$3, the quorum is

$i$4

Three concrete variants are used:

$i$5

$i$6

$i$7

In the thesis formulation, the corresponding quorum sizes are

$i$8

For $i$9-selection, any pair of users intersects in at least two slots in the ideal case; the paper illustrates this with a $k$0 grid and notes that users can exchange control information in at least two channels per cycle. The more rows and columns are selected, the larger the intersection multiplicity becomes, increasing the number of possible rendezvous points per cycle (Jarrah et al., 2017, Jarrah, 2016).

The adaptive scheme binds quorum geometry to traffic load. Low traffic uses $k$1-selection, moderate traffic uses $k$2-selection, and high traffic uses $k$3-selection. The rationale given is operational rather than combinatorial: low traffic permits more active slots to increase the average number of successful rendezvous, whereas high traffic makes energy expenditure more important, so a smaller quorum is preferred while preserving rendezvous guarantees. Because different users may classify traffic differently at the same time, the deployed quorum family is asymmetric in practice even though the underlying grid and mapping are shared (Jarrah et al., 2017).

The protocol assumes a single-hop CRN, the same licensed-channel set for all secondary users, synchronized slots and cycles, and two half-duplex transceivers per secondary user—one for quorum-based control exchange and one for data transmission. Under these assumptions, rendezvous occurs at some $k$4 within each cycle if the corresponding channel is idle (Jarrah, 2016).

4. Rendezvous probability, time-to-rendezvous, and energy trade-offs

The adaptive CRN work evaluates asymmetric grid quorums with MATLAB simulations. One configuration uses $k$5 channels, hence $k$6, with 26 or 50 CR users. Primary-user activity follows a two-state ON–OFF Markov model with average idle period $k$7 and busy period $k$8, yielding

$k$9

The principal metrics are average number of successful rendezvous per time slot, average time to rendezvous (TTR), normalized energy per successful rendezvous, and forced blocking probability (Jarrah et al., 2017).

For fixed designs, the ordering is consistent. In average successful rendezvous per time slot,

$N$0

For average TTR,

$N$1

The thesis further reports that, for grid size $N$2 and increasing $N$3, $N$4-selection improves average rendezvous per slot by up to $N$5 over $N$6, and that the same qualitative ordering persists for $N$7 and $N$8 grids (Jarrah, 2016).

Energy behavior is explicitly non-monotone with respect to rendezvous performance. Normalized energy per successful rendezvous is defined as total number of active slots multiplied by one unit of energy, divided by total number of successful rendezvous. Larger quorums raise rendezvous density but also enlarge the active fraction of the cycle. Consequently, $N$9-selection has the highest energy per successful rendezvous, $n = \lceil \sqrt{N} \rceil$0-selection the lowest, and $n = \lceil \sqrt{N} \rceil$1-selection lies between them. The adaptive scheme is reported to achieve rendezvous performance close to $n = \lceil \sqrt{N} \rceil$2 with lower energy: in the journal paper, average RDV per slot and average TTR are approximately those of $n = \lceil \sqrt{N} \rceil$3, while normalized energy per RDV is lower than for $n = \lceil \sqrt{N} \rceil$4 (Jarrah et al., 2017).

Forced blocking probability decreases as $n = \lceil \sqrt{N} \rceil$5 increases and shows similar qualitative behavior across $n = \lceil \sqrt{N} \rceil$6, $n = \lceil \sqrt{N} \rceil$7, $n = \lceil \sqrt{N} \rceil$8, and adaptive schemes. This matters because the combinatorial intersection guarantee is not, by itself, a guarantee of successful rendezvous on a licensed channel: the slot-channel pair must also be primary-user idle. The papers therefore separate guaranteed intersection from realized rendezvous success (Jarrah, 2016).

5. Consensus reinterpretations: flexible phases, relaxed intersection, and topology-aware walls

Consensus papers use grid asymmetry differently. Flexible Paxos argues that majority quorums in both phases are conservative: safety requires only that every Phase-1 quorum intersect every Phase-2 quorum. In the simple uniform case, the size condition is

$n = \lceil \sqrt{N} \rceil$9

The paper explicitly gives a grid construction with $n^2$0 columns and $n^2$1 rows, $n^2$2, taking Phase-1 quorums as rows of size $n^2$3 and Phase-2 quorums as columns of size $n^2$4. Each row–column pair intersects in exactly one cell, while distinct rows and distinct columns may be disjoint. This is an asymmetric grid quorum system in the strict sense that two quorum families, not one globally intersecting family, determine safety (Howard et al., 2016).

Relaxed Paxos weakens the usual intersection reading further. It models acceptors as holding write-once registers by round, tracks which quorums may still decide which values, and shows that if a proposer learns that a non-$n^2$5 value was already proposed in some previous round, its Phase-1 quorum no longer needs to intersect the Phase-2 quorums from that round or from any previous rounds. The result is a more state-dependent notion of safe asymmetry: different rounds may use different quorum systems, and quorums across rounds may even be disjoint, provided the protocol’s decision-table rules preserve agreement (Howard et al., 2022).

The topology-aware literature moves from grids to walls. In the crumbling-wall construction, tiers are ordered as Earth, LEO, Moon, and Mars, with

$n^2$6

For tier $n^2$7, the Phase-1 family is

$n^2$8

while strict Phase-2 is anchored to Earth,

$n^2$9

Every Phase-1 quorum intersects every Phase-2 quorum because every wall path reaches Earth. The construction is explicitly described as an asymmetric, topology-aware grid quorum system. Its asymmetry is initiator-dependent: Earth quorums are shortest and most numerous, Mars quorums longest and least flexible. Exhaustive enumeration gives $n \times n$0 valid Phase-1 quorums for Earth, $n \times n$1 for LEO, $n \times n$2 for Moon, and $n \times n$3 for Mars. The same paper argues that symmetric grid quorums cannot express this leadership cost gradient (Mason, 20 Mar 2026).

These consensus results show that “asymmetric grid quorum system” need not mean different row counts on a common matrix. It may instead mean cross-phase row/column families, round-dependent quorum sets, or tier-dependent wall paths, as long as the required intersection relation for safety is preserved.

6. Subjective trust, heterogeneous processes, and general asymmetric-grid theory

A second major line of work studies asymmetry as subjective trust. In asymmetric Byzantine quorum systems, each process $n \times n$4 has its own fail-prone system $n \times n$5 and quorum family $n \times n$6. The consistency condition becomes pairwise and subjective:

$n \times n$7

and availability is required per process. Existence is governed by the asymmetric $n \times n$8-condition. This framework does not itself define a specific grid construction, but it explicitly states how classical row–column systems can be instantiated asymmetrically, with different processes selecting different rows, columns, or grid regions while preserving pairwise consistency and availability (Alpos et al., 2019).

Later work argues that classical consistency and availability are too weak in the asymmetric model for reliable broadcast and consensus. One paper proves that there is no quorum-based algorithm solving depth-1 reliable broadcast, introduces the recursive notion of process depth, and presents algorithms for $n \times n$9 and $G=\{g_{i,j}\}$0. In that account, grid instantiations are possible, but what matters is not merely row–column intersection; it is whether the recursive depth conditions are satisfied by the chosen local quorums (Amores-Sesar et al., 11 Sep 2025).

The DAG-consensus literature reaches a related conclusion from a different direction. Simply replacing threshold quorums with asymmetric quorums in the classical constant-round gather protocol fails to preserve the common-core property. The paper then introduces a new asymmetric gather using quorums and kernels, and proves an expected-wave bound for DAG-based consensus of

$G=\{g_{i,j}\}$1

where $G=\{g_{i,j}\}$2 is the size of the smallest quorum in the system. This should be distinguished from grid geometry itself: the crucial object is the asymmetric quorum system, and a grid is one possible structured instantiation of it (Amores-Sesar et al., 23 May 2025).

The most direct recent construction of asymmetric grid quorum systems for heterogeneous processes makes the grid itself semantic. Processes are indexed by qualitative attributes, with

$G=\{g_{i,j}\}$3

where $G=\{g_{i,j}\}$4 is the value set of attribute $G=\{g_{i,j}\}$5. Each attribute induces a fail-prone system $G=\{g_{i,j}\}$6 that tolerates fewer than $G=\{g_{i,j}\}$7 fully failed attribute values and fewer than $G=\{g_{i,j}\}$8 partial failures per remaining value, and the canonical quorums $G=\{g_{i,j}\}$9 are then assembled into an asymmetric grid quorum system

$$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$00

The paper proves that every per-attribute system is $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$01, and that the resulting asymmetric collection is $$\forall Q_i, Q_j \in \mathcal{Q},\quad Q_i \cap Q_j \neq \emptyset.$$02, so processes may choose an attribute-based quorum system independently and “without coordination,” while compatibility is guaranteed by construction (Senn et al., 16 Sep 2025).

Across these papers, two clarifications recur. First, asymmetry should be distinguished from heterogeneous channel view: in adaptive rendezvous, the channel set is common and the asymmetry lies in quorum shape, not channel visibility. Second, “all quorums intersect” is not the universal safety condition once operations, phases, or subjective views are separated; several consensus papers explicitly weaken it. A plausible implication is that asymmetric grid quorum systems are best understood not as a single construction, but as a design space of structured quorum geometries in which intersection is preserved under non-uniform roles, costs, or trust assumptions.