Hardware-Aware Parallel Scan Algorithm

• The hardware-aware parallel scan algorithm is a parallel prefix-sum method that adapts to specific hardware features for optimized local and inter-unit communication.
• It employs strategies like pipelined interleaving, warp-shuffle, and matrix-based scans to minimize latency and maximize throughput on GPUs, AI accelerators, and FPGAs.
• Empirical results show improvements such as up to 25.7 billion operations/sec on GPUs and significant latency reductions over generic scan implementations.

A hardware-aware parallel scan algorithm is a parallel prefix-sum primitive that is designed to minimize latency, maximize throughput, and exploit the specific architectural features of its target hardware substrate—typically multicore processors, GPUs, AI accelerators, or FPGA-based network devices. Hardware awareness in this context refers to tuning data partitioning, memory access, and inter-thread/block/core communication to the salient features of the memory hierarchy, instruction set, and communication protocols native to the platform. Recent research demonstrates that such co-design yields substantial improvements over generic PRAM-like or vendor-supplied scan implementations, both in theoretical complexity and realized performance (Liu et al., 2016, Zouzias et al., 2024, Wróblewski et al., 21 May 2025, Arap et al., 2014).

1. Fundamental Approaches and Platform-Specific Strategies

Prefix-sum (scan) operations admit a variety of parallelization schemes. In the context of hardware-aware design, the principal approaches include:

• Hybrid scan with pipelined interleaving (GPU/Manycore): The LightScan algorithm targets CUDA-enabled GPUs using a hybrid model where independent thread blocks process interleaved data blocks in a cyclic fashion, performing local intra-block prefix scans followed by lightweight inter-block communication via globally coherent L2 cache (Liu et al., 2016).
• Matrix-engine (Tensor Core) scan: MatMulScan generalizes the scan circuit to the TCU model, wherein batches of $s$ elements are scanned in one $s \times s$ matrix-multiply, minimizing instruction count, register pressure, and synchronization (Zouzias et al., 2024). This paradigm appears also in Ascend AI Accelerators, where cube units perform local tile scans as matrix multiplications and vector units handle the reduction and downsweep steps (Wróblewski et al., 21 May 2025).
• Network-offloaded scan (FPGA/NIC): MPI_Scan offloading on NetFPGA employs in-network scan with small pipelined FSMs, leveraging a bespoke packet format and hardware state to perform scan collectives using recursive doubling or binomial trees in the network logic rather than host software (Arap et al., 2014).

Each approach aims to maximize on-chip resource utilization, minimize inter-unit communication latency, and avoid bottlenecks due to synchronization or global memory access.

2. Intra-Block/Unit Local Scan Mechanisms

Platform-specific optimizations in the local (“tile” or “block”) scan phase include:

• GPU warp-shuffle scan (CUDA): Within blocks, scan is performed by decomposing computation into warps (32 threads), loading contiguous $K$-element segments per thread, and implementing the Hillis–Steele scan entirely in registers via PTX __shfl_up instructions, with carry-propagation across rows using __shfl. This produces a coalesced prefix sum in $O(\log_{2}32)$ steps, avoiding shared memory and barriers (Liu et al., 2016).
• Matrix multiplication-based tile scan (TCU, Ascend): Both the TCU model and Ascend's cube units use small $s \times s$ or $s \times s^2$ tiles, applying matrix multiplication against fixed lower- or upper-triangular matrices respectively to effect a local prefix sum across multiple elements in a single high-throughput operation. This reduces register traffic and leverages matrix accumulate datapaths (Zouzias et al., 2024, Wróblewski et al., 21 May 2025).

3. Inter-Block/Unit Communication, Accumulation, and Downsweep

Efficient propagation of partial results across blocks or cores is critical:

• L2-coherent cache lines (CUDA): LightScan orchestrates inter-block communication by placing partial sums in a globally visible array using ld.cg and st.cg instructions (bypassing L1), enabling SMs to communicate directly via the coherent L2 cache rather than atomics or global barriers (Liu et al., 2016).
• Reduction trees and broadcast matrices (TCU/Ascend): The MatMulScan and Ascend MCScan algorithms use “upsweep” and “downsweep” phases, where local block sums are aggregated via reduction trees (using vector ops or block-level matrix multiplies), and then block sum prefixes are distributed back (“broadcasted”) into each sub-block using specialized matrix-broadcast or vector-add operations (Zouzias et al., 2024, Wróblewski et al., 21 May 2025).
• FPGA pipelining (NIC offload): NetFPGA implements the inter-node scan using pipeline-parallel FSMs that exchange partial sums using optimized packet headers, recursive doubling, or binomial tree collectives, ensuring minimal buffering and deterministic progress even in the presence of variable link delays (Arap et al., 2014).

4. Theoretical Complexity and Hardware-Tuned Performance

Hardware-awareness yields not just raw speedups, but also improved work and depth bounds contingent on platform features:

• Work and depth on GPUs/TCU: LightScan performs $O(N)$ work and achieves depth $O(L/32)$ due to the constant time local scan and $O(1)$ inter-block communication; matrix-based approaches (MatMulScan, Ascend) perform $O(n/s^2)$ matrix multiplies with depth $2\lfloor\log_s n\rfloor$, leveraging wide fan-in at each reduction/broadcast (Liu et al., 2016, Zouzias et al., 2024, Wróblewski et al., 21 May 2025).
• Empirical throughput: LightScan reports peak throughput up to 25.7 billion float elements/sec and speedups of $2.0\times$–$2.4\times$ over Thrust, CUDPP, and 8.4–8.9$\times$ over Intel TBB on CPUs, confirming hardware-coupling benefits (Liu et al., 2016). Ascend MCScan achieves $37.5\%$ peak memory bandwidth and $9.6\times$ speedup over vector-only scan for large $N$ (Wróblewski et al., 21 May 2025).
• Network offload performance: NetFPGA scan offloading reduces MPI_Scan latency by $2\times$–$3\times$ for small messages, with optimized pipeline depth per rank and bandwidth scaling to line rate for larger messages (Arap et al., 2014).

5. Application Domains and Algorithmic Extensions

The hardware-aware scan design pattern undergirds:

• AI/ML operators: Sorting (radix sort via MCScan), masking/compress/select, top-$k$/top-$p$ sampling, weighted sampling/multinomial are all reduced to one or more scan calls, with performance gains directly linked to the scan primitive's hardware mapping (Wróblewski et al., 21 May 2025).
• Collective communication: MPI_Scan, Reduce_scatter, and similar collectives in HPC see marked latency and scalability improvements when scan is offloaded or implemented in-network with topology-aware pipelining and flow control (Arap et al., 2014).
• Atomics and synchronization minimization: Register-level and matrix-multiplication techniques can eliminate the need for costly atomics or global synchronizations typical of naive implementations (Liu et al., 2016, Zouzias et al., 2024).

6. Trade-offs, Design Best Practices, and Architectural Implications

Key best practices in hardware-aware scan design are:

• Tile/block size and occupancy: Use the largest tile fitting in on-chip memory/L0 (e.g., $L=32$K on Kepler GPUs, $s=128$ on Ascend 910B4) to maximize local utilization and minimize cross-block communication (Liu et al., 2016, Wróblewski et al., 21 May 2025).
• Instruction selection and pipeline: Prefer warp shuffle, matrix-multiply/accumulate, and SRAM adder pipelines over general memory accesses; design per-block/SM/FPGA logic with minimal FSM state (Liu et al., 2016, Zouzias et al., 2024, Arap et al., 2014).
• Synchronization minimization: Global barriers are restricted to phase boundaries (after local scan & reduction), not per-step; avoid excessive pipelining depth unless required by timing closure on FPGAs (Arap et al., 2014).
• Bandwidth and core utilization: Schedule vector and matrix/cube units in concert to maintain load/store and compute occupancy balance; aggressive double-buffering and kernel fusion can improve effective bandwidth (Wróblewski et al., 21 May 2025).
• Precision choices: Where supported, use lower precision (e.g., int8 accumulation in INT32) to enhance throughput for boolean/mask scans (Wróblewski et al., 21 May 2025).

7. Comparative Summary: Major Implementations

Platform	Local Scan Method	Inter-Block/Node Comm	Peak Speedup (vs. baseline)
CUDA GPU (LightScan)	Warp shuffle & registers	L2-coherent cache (ld.cg)	2.0–2.4× (GPU libs), 8.9× (CPU) (Liu et al., 2016)
TCU/AI Accelerator	$s \times s$ matmul (cube)	Tree/broadcast in vector/cube	9.6× (vector), 15× (single-core) (Wróblewski et al., 21 May 2025)
FPGA/NIC (NetFPGA)	FSM per rank, pipeline adder	On-chip pipeline, UDP/multicast	2.5–3.2× (MPI scan) (Arap et al., 2014)

Optimal hardware-aware parallel scan algorithms are characterized by adapting the division of labor between local computation, inter-block communication, and global coordination so as to respect and saturate the hardware’s structural and temporal affordances. The state of the art demonstrates that matching the scan primitive to the fine structure of GPU SIMT execution, matrix core utilization, or NIC/FPGAs’ data plane yields orders-of-magnitude improvements over architecture-agnostic approaches.