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PALMA: A Lightweight Tropical Algebra Library for ARM-Based Embedded Systems

This paper introduces PALMA, a lightweight, SIMD-accelerated C library that enables efficient tropical algebra computations for optimization problems like shortest paths and scheduling directly on resource-constrained ARM-based embedded systems.

Original authors: Gnankan Landry Regis N'guessan

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gnankan Landry Regis N'guessan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive puzzle, but instead of using a giant table in a library (like a powerful desktop computer), you are trying to solve it on a tiny, battery-powered wristwatch (an embedded device like a Raspberry Pi).

For decades, a special kind of math called Tropical Algebra has been the "secret sauce" for solving complex puzzles like finding the fastest route, scheduling tasks, or managing traffic flow. It turns messy, non-linear problems into neat, straight-line equations. However, until now, this "secret sauce" was too heavy and complicated to fit on a wristwatch. It only lived on big, powerful servers.

Enter "Palma."

The paper introduces Palma, a lightweight, new software library that finally brings this powerful math to small, embedded devices. Think of Palma as a "travel-sized" version of a heavy-duty math toolkit, specifically designed to fit in the pocket of a drone, a smart sensor, or a factory robot.

Here is how it works, using simple analogies:

1. The Magic of "Tropical" Math

In normal math, you add numbers to get a bigger number. In Tropical Algebra, the rules change:

Addition becomes taking the maximum (or minimum).
Multiplication becomes normal addition.

The Analogy: Imagine you are a tour guide planning a trip.

Normal Math: You add up the cost of every hotel to get the total price.
Tropical Math: You only care about the most expensive hotel on the path (the bottleneck) or the cheapest path. By changing the rules, a complex "find the best path" problem becomes as simple as multiplying two matrices (a grid of numbers). It turns a tangled knot into a straight line.

2. The Problem: The "Heavy Suit"

Existing tools for this math are like wearing a full winter coat in the desert. They are designed for big computers with lots of memory and power. If you try to run them on a small device (like a drone or a smart thermostat), the device overheats, runs out of battery, or simply can't handle the weight.

3. The Solution: The "Swiss Army Knife" (Palma)

Palma is built from the ground up to be lightweight and efficient.

No Heavy Lifting: It doesn't need to download huge external programs. It's written in a simple, clean language (C99) that runs anywhere.
The Super-Speed Boost (NEON): The paper explains that Palma uses a special feature built into ARM chips (the brains of most phones and small computers) called NEON.
- Analogy: Imagine a regular worker who can carry one box at a time. NEON is like giving that worker a forklift that can carry four boxes at once. Palma uses this "forklift" to do four math calculations simultaneously, making it significantly faster.
Smart Packing (Sparse Matrices): Sometimes, your puzzle has a lot of empty spaces (like a map with very few roads). Palma knows how to ignore the empty spaces and only carry the important parts. This saves a huge amount of memory, like packing a suitcase by only bringing clothes you actually need, rather than the whole closet.

4. What Can It Do? (The Real-World Tests)

The authors tested Palma on a Raspberry Pi 4 (a small, cheap computer) and showed it can solve real problems instantly:

Drone Control: They used it to schedule the tasks of a drone (reading sensors, calculating balance, moving motors). Palma figured out the schedule in 6 microseconds (millionths of a second). This is fast enough to keep a drone stable in mid-air without it crashing.
IoT Routing: They used it to find the best path for data to travel through a network of 50 sensors. It did this in under a millisecond, saving over 50% of the memory compared to older methods.
Factory Lines: They modeled a factory production line to find the "bottleneck" (the slowest machine). Palma calculated the maximum speed the factory could run and showed that if you speed up one machine, the whole factory speeds up.

5. The Results

Speed: Palma is up to 11.9 times faster than the old standard method (Bellman-Ford) for finding the shortest path.
Efficiency: It can solve complex scheduling problems in less than 10 microseconds, which is fast enough for real-time control systems.
Accessibility: It is free, open-source software, meaning anyone can use it to build smarter, faster embedded devices.

Summary

Palma is the bridge that finally brings advanced, "super-math" optimization to the small devices we use every day. It takes complex problems that usually require a supercomputer and shrinks them down so they can run efficiently on a tiny chip, using clever tricks to be fast and memory-friendly. It proves that you don't need a giant server to solve the world's hardest scheduling and routing puzzles; sometimes, a small, smart tool is all you need.

Technical Summary: PALMA – A Lightweight Tropical Algebra Library for ARM-Based Embedded Systems

1. Problem Statement

Tropical algebra (including max-plus, min-plus, and related idempotent semirings) offers a unifying framework where nonlinear optimization problems—such as shortest paths, scheduling, and throughput analysis—become linear algebraic operations. Despite theoretical maturity, existing implementations (e.g., ScicosLab MaxPlus, polymake) are designed for desktop or server environments with heavy dependencies and high resource requirements.

This creates a critical gap for embedded systems (robotics, IoT, autonomous vehicles, industrial controllers), which are resource-constrained (limited RAM, lower clock frequencies, strict real-time deadlines) yet require exactly these types of optimization. Current tools are inaccessible on these platforms, forcing developers to use isolated, non-unified algorithms that lack the efficiency and predictability required for embedded deployment.

2. Methodology

The paper presents Palma (Parallel Algebra Library for Max-plus Applications), a lightweight, dependency-free C library designed specifically for ARM-based embedded systems.

2.1 Mathematical Foundation

Palma implements a generic semiring abstraction supporting five tropical semirings:

Max-Plus ( $\mathbb{R}_{\max}$ ): For scheduling and longest paths.
Min-Plus ( $\mathbb{R}_{\min}$ ): For shortest paths (Dijkstra/Bellman-Ford).
Max-Min: For bottleneck/bandwidth paths.
Min-Max: For reliability paths.
Boolean: For reachability and connectivity.

The library treats these operations as matrix products and fixed-point iterations. Key algorithms include:

Tropical Closure (Kleene Star): Computes all-pairs shortest paths (equivalent to Floyd-Warshall in min-plus).
Single-Source Shortest Paths: A tropical variant of Bellman-Ford using matrix-vector multiplication.
Spectral Analysis: Computation of the tropical eigenvalue (maximum cycle mean) using Karp's algorithm to determine system throughput and cycle times.

2.2 Implementation Strategy

Palma is written in approximately 2,000 lines of portable C99 code with no external dependencies. Key design choices include:

Integer Arithmetic: Uses 32-bit signed integers (int32_t) instead of floating-point to ensure determinism, simplify comparisons near infinity, and leverage efficient integer SIMD instructions.
Memory Efficiency: Supports both Dense (row-major) and Sparse (CSR) matrix representations to accommodate memory-constrained environments.
SIMD Acceleration: Leverages ARM NEON (128-bit vector registers) to process four 32-bit integers in parallel. The library automatically detects NEON support at compile time.
API Design: Provides a three-level API: low-level semiring operations, core matrix operations, and high-level application solvers (scheduling, graph algorithms).

3. Key Contributions

Mathematical Foundations: A self-contained treatment of tropical algebra, including rigorous definitions, proofs of key theorems (e.g., convergence of Kleene star, Karp's algorithm), and the connection to classical graph algorithms.
Library Design: A complete, portable C library supporting five semirings, dense/sparse matrices, and high-level APIs for scheduling and spectral analysis.
Optimized Implementation: Detailed implementation of ARM NEON-optimized kernels for matrix-vector multiplication and matrix multiplication, alongside sparse matrix techniques.
Experimental Evaluation: Comprehensive benchmarks on a Raspberry Pi 4 (ARM Cortex-A72), analyzing performance across matrix sizes, semirings, and sparsity levels.
Case Studies: Validation through three real-world embedded scenarios:
- UAV Control: Scheduling a 12-task drone control loop.
- IoT Routing: Optimizing latency and bandwidth in a 50-node sensor network.
- Manufacturing: Analyzing throughput and bottlenecks in a production line.

4. Experimental Results

Evaluated on a Raspberry Pi 4 (1.5 GHz, 8 GB RAM), Palma demonstrates the following:

Performance: Achieved a peak performance of 2,274 MOPS (million operations per second) using the max-min semiring on $64 \times 64$ matrices.
NEON Speedup: Vectorized operations provided up to 1.80 $\times$ speedup over scalar code, with peak efficiency observed when matrices fit within the L2 cache ( $64 \times 64$ to $128 \times 128$ ).
Algorithmic Efficiency:
- Single-Source Shortest Path: Palma's tropical iteration was up to 11.9 $\times$ faster than classical Bellman-Ford due to better cache locality in matrix-vector products.
- Sparse vs. Dense: Sparse (CSR) representation becomes advantageous at ~50% sparsity, offering up to 67.7 $\times$ speedup and 94% memory savings for highly sparse graphs (95% sparsity).
Real-Time Feasibility:
- Solving scheduling problems for 12 tasks took 6.28 $\mu$ s.
- Routing computations for 50-node networks completed in under 1 ms (678.4 $\mu$ s for shortest paths), enabling real-time route adaptation.
- Matrix sizes up to $64 \times 64$ are feasible for 1 kHz control loops (1 ms deadline).

5. Significance and Claims

The paper claims that Palma fills a significant gap at the intersection of tropical mathematics and embedded systems. Its primary significance lies in:

Unification: It demonstrates that diverse embedded optimization problems (routing, scheduling, throughput) can be solved using a single, unified algebraic framework, reducing code complexity and size—a critical factor for embedded platforms.
Accessibility: By providing a lightweight, dependency-free C library, it makes tropical algebra accessible on resource-constrained hardware where it was previously unavailable.
Predictability: The use of integer arithmetic and deterministic algorithms ensures predictable execution times, essential for real-time control systems.
Practicality: The case studies prove that tropical algebra is not just a theoretical construct but a practical tool for on-device optimization in cyber-physical systems, enabling high-frequency control loops and efficient network routing directly on edge hardware.

The authors position Palma as a foundational tool that decouples algebraic semantics from hardware-specific optimizations, allowing for future extension to other architectures (e.g., RISC-V, DSPs) while maintaining a consistent mathematical core.