Parallel Sliding Block Architecture

• Parallel Sliding Block Architecture is a dual-mode framework combining statistical sliding-window computations and modular reconfiguration techniques to ensure high throughput and maintained connectivity.
• Statistical window operations leverage recurrence-based sliding sums that reduce redundant computation, achieving significant speedups on GPU-accelerated platforms.
• Modular sliding-square reconfiguration uses phased, collision-free moves that guarantee optimal O(P) makespan while preserving the connected structure of robotic assemblies.

The parallel sliding block architecture encompasses algorithmic frameworks and hardware implementations that enable efficient, concurrent computation or coordinated reconfiguration using sliding blocks as either algorithmic motifs or physical modules. Two paradigmatic lines of research are prominent: (1) high-throughput, windowed statistical operations on n-dimensional arrays using parallelized sliding windows, and (2) parallel reconfiguration of discrete, grid-aligned assemblages of sliding squares (modules) while maintaining structural connectivity. These approaches are distilled in, respectively, sum-based GPU algorithms for correlation (Poyda et al., 2018) and parallel modular robotic reconfiguration algorithms with optimal makespan (Akitaya et al., 2024).

1. Problem Definition and Model Classes

Parallel sliding block (or sliding window) architectures admit two primary formalizations: statistical sliding-window operators on multi-dimensional tensors, and physical (or abstracted) sliding block reconfiguration on an integer lattice.

• Statistical Sliding Window: Given tensors $X, Y$ of matching shape, compute local windowed operators (e.g., Pearson correlation) at every position $p$, each over a neighborhood $W(p)$ of fixed shape/size. The naïve computation, which recomputes all statistics per window, suffers from redundant work and high complexity, especially for small step sizes and high-dimensional data (Poyda et al., 2018).
• Physical Sliding Block Reconfiguration: Given start and target configurations $C_1, C_2$ of $n$ square modules on $\mathbb{Z}^2$, compute a schedule of parallel "slide" and "convex" moves transforming $C_1$ to $C_2$, maintaining connectivity (i.e., the weak dual graph remains connected), and optimize the makespan (number of parallel steps) (Akitaya et al., 2024).

Both models permit parallelism: GPU thread-level for sliding sum algorithms; synchronous modular moves under non-collision and connectivity constraints for block reconfiguration.

2. Algorithmic Principles and Sliding-Sum Recurrences

The core computational principle for statistical window operations is to exploit spatial overlap between adjacent blocks. At each dimension, a moving sum (or sliding window sum) is computed via a recurrence relation that updates each window sum by adding in the entering value and subtracting out the exiting value as the window advances by one position.

For 1D arrays, the sliding sum for position $j$ is: $R[j] = R[j-1] + A[j + (k-1)/2] - A[j - (k+1)/2]$ For multi-dimensional data, this is generalized by applying such recurrences hierarchically in each axis: first rows, then columns, and so forth, enabling $O(n \cdot \mathrm{TotalPoints})$ complexity independent of window size. This reduces redundant memory accesses and arithmetic over naïve $O(|W|\cdot \mathrm{TotalPoints})$ methods (Poyda et al., 2018).

In modular reconfiguration, the principle is to decompose the global parallel move schedule into phases—gathering modules onto a backbone (skeleton), constructing an exoskeleton, sweeping a separator line, and morphing into target histograms—each making maximal use of simultaneously movable modules while avoiding collisions and disconnectivity (Akitaya et al., 2024).

3. Parallelization Schemes and Hardware Mapping

Statistical Sliding-Block Architecture

The parallel architecture for the n-dimensional correlation algorithm follows a distinct multistage GPU pipeline (Poyda et al., 2018):

• Product calculation: A thread per voxel computes $x_i y_i$, $x_i^2$, $y_i^2$ and stores these to intermediate arrays.
• Horizontal sliding sum: Arrays are partitioned into row-wise tiles; each tile is mapped to a CUDA block, which loads its segment and a halo into shared memory, then updates running sums in parallel.
• Vertical sliding sum: Repeat for columns on outputs of the previous step.
• Pearson correlation computation: Each thread evaluates the correlation formula using the five necessary block sums.
• Synchronization: Each block uses $\_\_syncthreads()$ for intra-block sync, with independent blocks operating on disjoint regions.

This design exploits fast shared memory and minimizes global memory bandwidth by reusing loaded elements multiple times within block-local computations.

Modular Sliding-Square Reconfiguration

The parallel algorithm for sliding square modules uses synchronous phases in which numerous atomic moves execute per step, under strict collision and connectivity constraints (Akitaya et al., 2024):

• Moves are decomposed into transformation steps (slides or convex pivots), prioritized so that no two interfere.
• The schedule leverages per-step maximal concurrency (makespan-optimality), guaranteeing $O(P)$ makespan where $P$ is the bounding box perimeter.
• The algorithmic phases (exoskeleton construction, scaffolding, sweep-line extraction, histogram morphing) admit localized, parallel moves. Meta-modules (e.g., $3 \times 3$ blocks) are introduced to permit constant-time morphs at coarser granularities.

4. Complexity Analysis and Performance Metrics

A principal benefit of sliding block architectures is the decoupling of per-window operation count from window size or module count, achieved via recurrence-based updates and parallel decomposition.

Statistical Window Processing

• Naïve approach: $O(n^2 \cdot A \cdot B)$ for a window of size $n \times n$ on an $A \times B$ image.
• Optimized sliding-sum approach: $O(n \cdot \mathrm{TotalPoints})$, with $n$ the number of dimensions.
• On a 12 MPixel image, a GPU implementation achieved $\sim 60 \times$ acceleration over serial computation and $4\times$ over optimized CPU code. GPU compute phase (excluding I/O and initialization) dominated by parallel sliding-sum kernels (Poyda et al., 2018).
• Algorithm scales linearly with data volume, not with window size, and achieves near-maximal hardware occupancy.

Modular Block Reconfiguration

• Sequential model: $O(n^2)$ or $O(nP)$ moves, all sequential.
• Parallel model: $O(P)$ makespan is both achievable and provably optimal (via minimum assignment bottleneck) (Akitaya et al., 2024).
• Deciding makespan-1 feasibility (unlabeled case) is NP-complete; makespan-2 feasibility (labeled) is NP-complete, while makespan-1 feasibility in the labeled case is polynomial-time decidable.

5. Architectural Structure and Scheduling Strategies

In both algorithmic classes, careful architectural structuring is essential for high parallel throughput.

Paradigm	Key Structuring Principle	Computational Stages
Sliding-sum correlation (Poyda et al., 2018)	Recurrence-based runs, hierarchical axiswise passes	Product computation, sliding sum passes, final reduction
Sliding squares reconfig. (Akitaya et al., 2024)	Skeleton/exoskeleton phase splitting, phased sweep	Skeleton extraction, exoskeleton, sweep-line & histogram morphing

Contextually, in statistical block computation, the critical bottleneck is memory locality versus arithmetic redundancy. The decomposition into shared-memory tile computations overcomes this, while in modular block reconfiguration, the challenge is to orchestrate maximal parallel progression subject to global invariants (connectivity, collision), achieved through novel phase-based move scheduling.

6. Extensions, Applicability, and Generalizations

Extensions of the parallel sliding block architecture include:

• Windowed computation of any operator expressible as sums over local blocks (variance, covariance, local histograms), for which n-pass sliding sums are applicable.
• Median, mode, and nonlinear operators use similar tiling and halo strategies for blockwise GPU processing, though with different recurrence logic (Poyda et al., 2018).
• Higher-dimensional (3D/4D) data: sliding-sum recurrences extend naturally by applying 1D passes per axis, either sequentially or via fused GPU kernels.
• Distributed/heterogeneous architectures: global data domain partitioned into subdomains with inter-domain halo exchanges.
• Modular robotics: the skeleton/exoskeleton and sweep strategies guide distributed coordination for robot swarms, given the mechanical constraint of maintaining a connected subassembly (Akitaya et al., 2024).

A plausible implication is that the skeleton/exoskeleton decomposition paradigm could extend to higher-genus surfaces or non-rectangular lattices, subject to analogous connectivity constraints.

7. Limitations, Complexity, and Open Questions

NP-hardness results for minimal-makespan feasibility (even for constant makespan on general start/end pairs) exclude the possibility of drastic further accelerations in worst-case parallel reconfiguration beyond the achieved $O(P)$ makespan for modular squares (Akitaya et al., 2024). Similarly, in blockwise statistical algorithms, optimality is hardware- and memory-bandwidth limited; further improvements would necessitate architectural breakthroughs in hardware or fundamentally different, e.g., randomized, algorithms.

Open avenues for research include designing in-place, memory-minimal sliding-sum recurrences for heterogeneous clusters, and analyzing reconfiguration strategies on non-planar or constrained grids.

• Poyda & Zhizhin, "Optimization of the n-dimensional sliding window inter-channel correlation algorithm for multi-core architecture," (Poyda et al., 2018).
• "Sliding Squares in Parallel," (Akitaya et al., 2024).