Fast Road Path Algorithms With Tropical Geometry

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Fast Road Path Algorithms With Tropical Geometry

ISEF Category: Mathematics

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Subcategory: Other  ·  Difficulty: Advanced  ·  Setup: Home Setup  ·  Time: 1 to 2 Months

The Hook

GPS finds a route in seconds, but the math behind that speed can get heavy fast. Your project asks a real question: can a newer kind of geometry make road routing quicker without losing too much accuracy? That means you can test speed, error, and tradeoffs on real map data. This is a strong fit if you like algorithms and clean comparisons.

What Is It?

Shortest-path algorithms try to find the best route through a network. Think of roads as dots and lines on a map. Each line has a cost, like distance or travel time. Dijkstra’s algorithm checks routes in a careful, step-by-step way. A* does something similar, but it uses a guess about which direction leads toward the goal, so it often moves faster.

Tropical geometry sounds fancy, but you can think of it as a math system where some operations behave like taking a minimum or maximum instead of ordinary addition and multiplication. In this project, the idea is to use that structure to build a faster approximate routing method. Approximate means the route may not be the exact best one, but it should stay close. Your job is to test how close, how fast, and under what map conditions.

Why This Is a Good Topic

This is a strong science fair topic because you can measure real things, like runtime, path length error, and performance on different road networks. You do not need a wet lab, but you do need careful thinking, clean code, and honest benchmarks. The project connects to navigation, delivery routing, and emergency response, where speed matters but bad routes cost time. You can learn algorithm design, graph theory, and how to judge tradeoffs with data.

Research Questions

  • How does the tropical-geometry algorithm compare with Dijkstra’s algorithm in runtime on the same road network?
  • How does the tropical-geometry algorithm compare with A* in path-length error across different map regions?
  • What is the effect of graph size on the speedup from the tropical-geometry algorithm?
  • To what extent does road density change the accuracy of the approximate route?
  • Which types of origin-destination pairs create the largest error in the approximate route?
  • How does changing the heuristic or approximation rule affect the tradeoff between speed and path quality?
  • To what extent does the algorithm’s performance change when tested on urban, suburban, and mixed road networks?

Basic Materials

  • Laptop or desktop computer with enough memory to run graph algorithms.
  • Python 3 with NetworkX for graph handling.
  • OpenStreetMap data for one or more test regions.
  • A text editor or code editor such as VS Code.
  • CSV or JSON export tool for saving route results.
  • Spreadsheet software for organizing runtime and error data.

Advanced Materials

  • Workstation-class computer or university lab desktop for larger graph runs.
  • Python with NetworkX, NumPy, and SciPy for analysis.
  • OSMnx for downloading and cleaning OpenStreetMap road graphs.
  • GeoPandas for map-based analysis and region comparison.
  • A profiling tool such as cProfile or line_profiler for runtime breakdowns.
  • A plotting package such as Matplotlib or Seaborn for performance graphs.

Software & Tools

  • Python: Runs the algorithm, computes route metrics, and automates trials.
  • NetworkX: Builds and analyzes road graphs from map data.
  • OSMnx: Downloads and preprocesses OpenStreetMap street networks.
  • Matplotlib: Plots runtime, error, and speedup comparisons.
  • GeoPandas: Helps compare performance across different geographic regions.

Experiment Steps

  1. Define the exact routing problem you will study, including whether you optimize distance, travel time, or another cost.
  2. Choose one approximation rule for the tropical-geometry method and write down what counts as a fair comparison.
  3. Build a benchmark set of origin-destination pairs across several road networks so your tests are repeatable.
  4. Plan your control algorithms, usually Dijkstra and A*, and decide which metrics you will record for each run.
  5. Design an error measure that compares approximate routes with exact routes in a way that matches your research question.
  6. Organize a testing plan that checks whether the method improves speed without failing on certain map shapes or graph sizes.

Common Pitfalls

  • Comparing runtimes on different machines or with different graph sizes, which makes the speed results unfair.
  • Using only one city or one neighborhood, which hides how the algorithm behaves on other road layouts.
  • Measuring only runtime and ignoring path error, which makes a fast but poor route look better than it is.
  • Letting map cleaning choices change the graph structure without documenting them, which can alter the benchmark.
  • Testing too few origin-destination pairs, which creates noisy results and weak conclusions.

What Makes This Competitive

A competitive version of this project would go beyond a simple speed test. You would compare multiple map types, report both runtime and approximation error, and explain why the method behaves differently across graphs. Strong entries often include careful controls, confidence intervals, and a clear picture of where the new method helps and where it fails. A deeper theoretical argument, paired with real benchmark data, would make the project much stronger.

Project Variations

  • Test the algorithm on walking networks instead of driving networks to see whether graph structure changes the speedup.
  • Compare the method on small-town maps, large-city maps, and mixed suburban regions to study how road density affects accuracy.
  • Replace distance cost with travel-time cost to see whether the approximation behaves differently under weighted edges.

Learn More

  • MIT OpenCourseWare, Introduction to Algorithms: Search for graph algorithms lectures and problem sets that cover Dijkstra, A*, and shortest paths.
  • NetworkX documentation: Read the graph shortest-path sections to learn how to run exact routing tests in Python.
  • OSMnx documentation: Find guidance on downloading and cleaning OpenStreetMap street networks for research projects.
  • OpenStreetMap Wiki: Learn how road data are structured and what kinds of network filters matter.
  • SIAM Journal on Discrete Mathematics: Search for papers on shortest paths, approximation algorithms, and tropical methods in graph theory.
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