Tree Search Algorithms Applied to the BD-RIS Configuration in MU-MISO Communication Systems

Abstract: The reconfigurable intelligent surface (RIS) has attracted considerable attention of both academia and industry in recent years, given its capacity to dynamically manipulate the reflection of incident electromagnetic waves. Although the research developed for the RIS may have reached its maturity, there are still contentious aspects and limitations regarding its potential benefits for the next generation of wireless communications. In order to improve upon the the RIS technology, the beyond diagonal reconfigurable intelligent surface (BD-RIS) was recently proposed as an promising alternative. The BD-RIS boasts a more sophisticated circuit topology that is capable of providing more combinations of different adjustments or configurations for signal reflection. However, to aptly reap the benefits of the BD-RIS, the added degrees-of-freedom of its configuration must be leveraged accordingly. Therefore, in this work we propose a depth-first tree search algorithm for configuring the BD-RIS in multi-user multiple-input single-output (MU-MISO) communication systems. Taking advantage of the tree search exploration, the proposed algorithm achieves a remarkable trade-off between channel strength maximization performance and computational complexity scalability.

• The paper introduces a depth-first tree-search algorithm with phase quantization, symmetric-unitary projection, and multi-stage pruning for configuring fully connected BD-RIS matrices in MU-MISO systems.
• The method reaches the SISO analytical channel-gain upper bound with 16 discrete phases and significantly outperforms a first-order baseline when the direct link is obstructed.
• The algorithm has a worst-case complexity of O(|Q|N⁵), but branch pruning reduces measured runtime to approximately O(N²), creating a tunable performance–complexity trade-off.

Motivation and problem setting

Reconfigurable intelligent surfaces (RIS) have moved from exploratory research toward standardization consideration, but their diagonal-only phase-shift matrices impose a structural ceiling on the achievable passive beamforming gain. Beyond-diagonal RIS (BD-RIS) addresses this by interconnecting reflecting elements through tunable impedances, so that the configuration is a general unitary matrix $\boldsymbol{\Theta} \in \mathbb{C}^{N \times N}$ rather than a diagonal one. The added degrees of freedom, however, make configuration combinatorially hard: an exhaustive search over discrete phase-shifts for a fully-connected BD-RIS costs $\mathcal{O}(|\mathcal{Q}|^{N^2})$, which is NP-hard. The paper under review proposes a depth-first tree search with multi-stage pruning to configure the BD-RIS matrix for channel strength maximization in MU-MISO systems, and claims—plausibly, based on the cited literature—to be the first work integrating tree search algorithms with BD-RIS configuration matrices.

The system model considers an $L$-antenna base station serving $K$ single-antenna UEs via a cascade channel $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$, where the unitary constraint on $\boldsymbol{\Theta}$ ensures no reflected power amplification. The optimization objective is the squared Frobenius norm of this overall channel. The authors note that manifold-based methods can solve this problem optimally without relaxing the unitary constraint (Fidanovski et al., 10 Nov 2025, Santamaria et al., 20 Jan 2026), but at high computational cost; the tree search is positioned as a scalable heuristic alternative.

SISO primer and algorithmic core

The algorithm is first developed for the obstructed-channel SISO case, where the objective reduces to maximizing $$\|\boldsymbol{h}\boldsymbol{\Theta}\boldsymbol{\upsilon}\|_2^2$$ subject to unitarity and symmetry ($\boldsymbol{\Theta} = \boldsymbol{\Theta}^\top$, justified by reciprocal impedance components). By Cauchy–Schwarz, the upper bound equals $$\|\boldsymbol{h}\|_2^2\|\boldsymbol{\upsilon}\|_2^2$$, attained when the normalized channel vectors are aligned. This yields a scalar alignment metric $$0 \leq \Re\{\bar{\boldsymbol{h}}\boldsymbol{\Theta}\bar{\boldsymbol{\upsilon}}\} \leq 1$$ that drives the search.

The depth-first search maps each tree level to an entry of the discretized configuration matrix (diagonal entries first), explores candidate phase-shifts from a uniform grid $\mathcal{O}(|\mathcal{Q}|^{N^2})$0, and prunes branches whose alignment score falls below a threshold $\mathcal{O}(|\mathcal{Q}|^{N^2})$1. Two design details are notable:

• Symmetric unitary projection: since assigning a value to entry $\mathcal{O}(|\mathcal{Q}|^{N^2})$2 must be mirrored at $\mathcal{O}(|\mathcal{Q}|^{N^2})$3, the algorithm evaluates all reciprocal-entry candidates and projects the resulting matrix onto the set of symmetric unitary matrices via averaging followed by SVD (solving the orthogonal Procrustes problem).
• Pruning semantics: if a candidate exceeds $\mathcal{O}(|\mathcal{Q}|^{N^2})$4, deeper exploration of alternatives at that level is skipped; only when no candidate qualifies does the branch degenerate into exhaustive evaluation.

The validation result here is strong: with $\mathcal{O}(|\mathcal{Q}|^{N^2})$5 (16 discrete phases), the average channel gain practically attains the analytical upper bound of $\mathcal{O}(|\mathcal{Q}|^{N^2})$6 regardless of $\mathcal{O}(|\mathcal{Q}|^{N^2})$7, whereas $\mathcal{O}(|\mathcal{Q}|^{N^2})$8 incurs a visible loss. This establishes both correctness of the approach and a coarse-to-fine complexity/performance knob.

Extension to MU-MISO

For the multi-user case, the objective decomposes into three terms involving $\mathcal{O}(|\mathcal{Q}|^{N^2})$9, $L$0, and $L$1. The suitability metric $L$2 is normalized using two bounds: Cauchy–Schwarz for the quadratic term when the direct link is obstructed, and von Neumann's trace inequality ($L$3) when it is not. A second pruning mechanism, branch pruning, terminates execution prematurely when improvement between adjacent levels falls below $L$4, allowing fewer than $L$5 levels to be explored; an optional counter parameter $L$6 delays termination.

Numerical findings

Simulations use Rayleigh fading with distance-dependent path loss ($L$7 dB at 1 m), UEs uniformly placed within 20 m of a point 150 m from the BS, averaged over 500 realizations, against the low-complexity baseline of Fang and Mao [10.1109/LCOMM.(2023.33334)11].

Scenario	Result
Unobstructed direct link	Performance close to baseline; near-negligible gain from $L$8 exhaustive search; visible penalty from branch pruning for $L$9
Obstructed direct link	Proposed algorithm significantly outperforms the baseline, whose first-order approximation degrades sharply
Complexity	Asymptotic upper bound $K$0 confirmed numerically; with branch pruning, empirical cost drops to approximately $K$1

Two claims deserve emphasis. First, the obstructed-link superiority is the most consequential result, since weak or blocked direct channels are precisely the deployment scenario motivating RIS technology; the baseline's failure there stems from its first-order approximation. Second, the empirical $K$2 runtime under pruning indicates that the pruning parameters dominate practical scalability far more than the asymptotic worst case suggests. Runtime was also found largely invariant across the tested $K$3 configurations ($K$4, $K$5, $K$6). Heatmap analysis of entry usage confirms that $K$7 and $K$8 finely control how many matrix entries (tree levels) are actually explored.

Limitations and open problems

Several caveats temper these results. All experiments assume ideal CSI; the authors argue that channel hardening of $K$9 and $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$0 for large $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$1 and $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$2 may confer robustness under obstruction, but this remains unverified. Only the fully-connected, symmetric BD-RIS architecture is treated—group-, tree-, and forest-connected topologies and non-symmetric matrices are left open, as are alternative level-to-entry mappings beyond the successive diagonal one. Active beamforming is not jointly optimized; integration with sum-rate maximization while preserving scalability is identified as a needed extension. Finally, the performance penalty of branch pruning for large $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$3 in unobstructed scenarios shows the trade-off is not free, and the heuristic thresholds $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$4 and $$h = \boldsymbol{G}^\dagger + \boldsymbol{H}^\dagger \boldsymbol{\Theta}\boldsymbol{\Upsilon}$$5 are set empirically without adaptive selection rules.

Conclusion

This work provides the first application of depth-first tree search to BD-RIS configuration, achieving exact attainment of the SISO upper bound with 16-phase quantization, polynomial—and empirically near-quadratic—complexity under pruning, and markedly superior performance over a first-order low-complexity baseline when the direct channel is obstructed. Its principal contribution is a flexible, parametrized performance–complexity trade-off mechanism rather than global optimality guarantees, which remain the province of manifold optimization methods.