All-for-One Algorithm: Unified Multi-Task Framework

• All-for-One Algorithm is a unified framework that leverages shared representations and adaptive methods to address diverse computational tasks efficiently.
• It simplifies complex processes by reusing samples and integrating operations, thereby reducing the need for multiple specialized algorithms.
• Its applications span geometric processing, graph querying, probabilistic estimation, algorithm selection, federated learning, and cognitive modeling, offering practical and theoretical benefits.

The All-for-One Algorithm refers to a class of unified frameworks and algorithms in which a single methodology is employed to simultaneously address multiple tasks, perform multiple operations, or approximate multiple objectives—often by leveraging shared representations, adaptive mechanisms, or sample reuse. In contemporary computational research, the All-for-One paradigm appears in geometric processing, path querying, probabilistic value estimation, algorithm selection, distributed learning, and general cognition, characterized by an emphasis on integration, efficiency, and universality.

1. Unified Geometric Algorithms for Deformable Models

A prototypical All-for-One approach in geometry processing is the “All-In-One Geometric Algorithm for Cutting, Tearing, and Drilling Deformable Models” (Kamarianakis et al., 2021). This framework utilizes Conformal Geometric Algebra (CGA) to represent points, planes, spheres, and all elementary deformations (rotations, translations, dilations) as uniform multivectors. Within CGA, geometric objects and deformations operate in the same algebraic space, eliminating the traditional need to convert between disparate representations (such as matrices, quaternions, and dual-quaternions).

Three algorithms—cutting, tearing, and drilling—are formulated entirely within CGA:

• Cutting: The mesh is sliced along a user-defined plane by evaluating intersection points using CGA operations. Intersections are integrated into the animated mesh through updated bone weights and a deformation equation of the form

$$C_k[m] = \sum_{n\in I_m} w_{m,n} (M_{n,k} B_n) c[m] (M_{n,k} B_n)^{\ast}$$

where transformations, offsets, and inverses are handled as multivectors.

• Tearing: Partial incisions (e.g., surgical cuts) are captured by tracking a moving tool, introducing and separating duplicated vertices to facilitate independent deformations post-tear.
• Drilling: Meshes are modified within a defined cylindrical drill region by computing intersections with cylinder boundaries via quadratic equations in the blending parameter, followed by robust re-triangulation.

This unified, algebraic formulation greatly simplifies both code base and maintenance, streamlines optimization (especially for GPU parallelization), and enables real-time or near–real-time physical simulation needed in fields such as medical VR.

2. Universal Linear Algebraic Path Query Evaluation

Another instance of the All-for-One paradigm is the linear-algebra-based algorithm for evaluating both regular and context-free path queries (RPQs and CFPQs) over graphs (Shemetova et al., 2021). The approach models both the input graph and the query automaton/grammar as Boolean matrices. The central steps are:

• Representation: Each edge label in the graph and transition in the query (whether finite state or recursive state machine form) is encoded as a Boolean adjacency matrix.
• Intersection via Kronecker Product: The “intersection” product

$$(A \otimes B)[u m_2 + v, p n_2 + q] = A[u,p] \wedge B[v,q]$$

combines the structure of the query automaton and the graph into a composite reachability matrix.

• Incremental Transitive Closure: New reachability relations are discovered via dynamic updates and the Four Russians' method accelerates closure computation, leading to overall $O(n^3/\log n)$ time complexity.

The resulting index not only records reachability facts for RPQs and CFPQs but also supports extraction of all paths. This unification is significant for large-scale graph databases and static program analysis, as it enables efficient, general-purpose evaluation compatible with GPU acceleration and distributed computing.

3. Unified Probabilistic Value Estimation through Sample Reuse

The “One Sample Fits All” framework (Li et al., 31 Oct 2024) addresses the exponential complexity of computing various probabilistic values (Shapley, Beta Shapley, weighted Banzhaf, etc.) for feature attribution and data valuation. The key innovation is to design a sampling scheme whereby every sampled subset is simultaneously informative for all value approximations, in accordance with two principles:

• Maximum Sample Reuse: Each sample is reused across all objectives, exploiting a shared intermediate decomposition:

$$\phi_i = \sum_{s=1}^n m_s \left[\varphi^+_{i,s} - \varphi^-_{i,s-1}\right]$$

Intermediate quantities are sampled using an optimized sampling vector $q$ such that, for any probabilistic value parameterized by a vector $p$, the conversion does not require extra rescaling.

• Sampling Vector Optimization: Choosing $q$ via

$$q_{s-1}^{\mathrm{OFA-A}} \propto 1 / \sqrt{s(n - s)}$$

ensures convergence rate $O(n \log n)$ for a broad class of probabilistic values; further parameter refinement enables optimal performance for individual cases.

This all-in-one estimator reduces computational cost by orders of magnitude and, by establishing a formal equivalence with least squares datamodels, allows simultaneous solution of multiple interpretability or valuation problems.

4. Universal Algorithm Selection via the Comb Operator

AlgoSelect (Yao, 17 Jun 2025) represents a general-purpose All-for-One framework for automated algorithm selection. The central mechanism is the Comb Operator, which interpolates between candidate algorithms:

• For two algorithms, e.g., systematic $A_{sys}$ and randomized $A_{ran}$, the hybrid algorithm is defined as

$A_t = (1-t) \cdot A_{sys} \oplus t \cdot A_{ran}$

where $t = \sigma(\theta^T \phi(\omega))$ is a sigmoid-gated selection parameter adapted from instance features $\phi(\omega)$. For $N$ algorithms, the selection lies on the simplex, allowing continuous blending among all strategies.

• Universal Approximation and Learnability: The framework is proven to universally approximate any selector function, and its selection thresholds converge almost surely to the population optimum via Borel–Cantelli arguments, yielding information-theoretic optimality.
• Empirical Results: In a 20×20 problem–algorithm paper, AlgoSelect achieved over 99.9% selection accuracy with few samples; for most structured domains, the mapping from problem features to best algorithm is deterministic (conditional entropy $H(\text{Algorithm}|\text{Problem}) \approx 0$).

This universal selection mechanism adapts to context, reduces manual tuning, and is directly applicable to AutoML, resource allocation, and hybrid AI systems.

5. Adaptive Federated and Decentralized Learning

The All-for-One algorithm in federated learning (Even et al., 2022, Philippenko et al., 9 Jul 2025) refers to a strategy in which each participant dynamically filters and aggregates stochastic gradients from others to minimize its local objective:

• Gradient Filtering: Each client at iteration $t$ computes a weight for each peer $k$ based on the similarity of gradients:

$$r_{ik}^t = \left[1 - \frac{\|\nabla R_k(\theta_i^{t-1}) - \nabla R_i(\theta_i^{t-1})\|}{\|\nabla R_i(\theta_i^{t-1})\|}\right]_+, \quad \alpha_{ik}^t = \phi(r_{ik}^t) (\sigma_{i,\psi}^t / \sigma_k)^2$$

The function $\phi$ controls strictness: binary thresholding or continuous weighting adaptively forms peer groups by shared gradient directionality.

• Variance Reduction and Convergence: The weighted aggregation serves as a variance reduction mechanism, offering superior convergence rates (linear phase up to a precision limit dictated by effective variance) and better generalization than FedAvg, especially in heterogeneous regimes.
• Theoretical Guarantees: With dynamic adaptation, optimal sample complexity and robust behavior are retained even without prior knowledge of the target accuracy.

This approach outperforms standard federated averaging and several state-of-the-art personalized decentralized learning methods on both synthetic and real benchmarks.

6. All-for-One Approaches in Graph Representation Learning

In graph learning, the OFA (“One for All”) framework (Liu et al., 2023) illustrates the All-for-One philosophy by designing a single graph model—across domains and tasks:

• Text-Attributed Graphs (TAGs): All nodes and edges are described with human-readable text, and a LLM encodes these into a common embedding space, facilitating integration of highly heterogeneous graph data (from citation networks to molecular graphs).
• Graph Prompting Paradigm: Tasks—node, link, or graph classification—are cast as in-context learning jobs by appending prompt substructures to the input graph. Nodes-of-Interest and class prompts allow the use of a unified GNN readout, regardless of task type.
• Performance and Generality: Supervised, few-shot, and zero-shot evaluations confirm strong performance across varied datasets. The model is the first to enable general-purpose, cross-domain graph classification without fine-tuning.

This framework is foundational for universal graph intelligence, enabling transfer and multitask learning across domains.

7. Cognitive Integration: The Ouroboros Model

A distinctive all-in-one cognitive algorithmic proposal is the Ouroboros Model (Thomsen, 7 Mar 2024), which advocates a general cognitive loop integrating:

• Compartmentalized Memory (Schemata): Non-strict hierarchies of concepts, from low-level perception to abstract thought.
• Consumption Analysis: A cybernetic feedback process comparing current activation with schemata, driving both attention (short-term focus) and emotional bias (long-term adaptation). Discrepancies between expectation and observation serve as the learning signal, akin to biologically-inspired error correction and Bayesian updating.

By framing cognition as an ever-adaptive control process—simultaneously logical and intuitive—the Ouroboros model suggests routes to overcoming symbol grounding and hierarchical representation challenges in AI.

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The All-for-One Algorithm, as reflected in contemporary research, denotes a unifying paradigm characterized by algorithmic integration, adaptive sample or operation reuse, universal selection or approximation mechanisms, and flexible representations. Its applications span computational geometry, graph querying, data valuation, algorithm selection, distributed learning, representation learning, and models of cognition, with the chief benefits being flexibility, efficiency, and theoretical guarantees across problem classes.