feat: add kuhn_munkres_complexity example to experimentally measure algorithm complexity

Co-authored-by: samueltardieu <[email protected]>

Author: Copilot

examples/kuhn_munkres_complexity.rs

@@ -0,0 +1,117 @@
+//! This example demonstrates the experimental complexity analysis of the Kuhn-Munkres algorithm.
+//!
+//! The Kuhn-Munkres algorithm (also known as the Hungarian algorithm) is documented to have
+//! O(n³) time complexity, where n is the size of the input matrix. This example runs the
+//! algorithm with various input sizes and measures the execution time to experimentally
+//! verify this complexity.
+
+use pathfinding::prelude::{Matrix, kuhn_munkres};
+use rand::{Rng, SeedableRng};
+use rand_xorshift::XorShiftRng;
+use std::time::Instant;
+
+/// Test sizes to use for complexity analysis
+const TEST_SIZES: &[usize] = &[5, 10, 20, 30, 40, 50, 60, 80, 100, 120, 150, 200];
+
+/// Number of iterations to average for each size
+const ITERATIONS: usize = 20;
+
+/// Structure to hold timing results for a specific size
+struct TimingResult {
+    size: usize,
+    avg_time_micros: f64,
+}
+
+fn main() {
+    println!("Kuhn-Munkres Algorithm - Experimental Complexity Analysis");
+    println!("=========================================================\n");
+    println!("Testing with matrix sizes: {TEST_SIZES:?}");
+    println!("Iterations per size: {ITERATIONS}\n");
+
+    let mut results = Vec::new();
+
+    // Run tests for each size
+    for &size in TEST_SIZES {
+        println!("Testing size {size}x{size}...");
+
+        let mut total_time_micros = 0u128;
+
+        // Use a fixed seed for reproducibility
+        let mut rng = XorShiftRng::from_seed([
+            3, 42, 93, 129, 1, 85, 72, 42, 84, 23, 95, 212, 253, 10, 4, 2,
+        ]);
+
+        for iteration in 0..ITERATIONS {
+            // Generate a random square matrix with weights between 1 and 100
+            let weights = Matrix::square_from_vec(
+                (0..(size * size))
+                    .map(|_| rng.random_range(1..=100))
+                    .collect::<Vec<_>>(),
+            )
+            .unwrap();
+
+            // Measure execution time
+            let start = Instant::now();
+            let _result = kuhn_munkres(&weights);
+            let elapsed = start.elapsed();
+
+            total_time_micros += elapsed.as_micros();
+
+            if iteration == 0 {
+                println!("  First iteration: {elapsed:.2?}");
+            }
+        }
+
+        #[expect(clippy::cast_precision_loss)]
+        let avg_time_micros = total_time_micros as f64 / ITERATIONS as f64;
+        println!("  Average time: {avg_time_micros:.2} μs\n");
+
+        results.push(TimingResult {
+            size,
+            avg_time_micros,
+        });
+    }
+
+    // Analyze complexity
+    println!("\nComplexity Analysis");
+    println!("===================\n");
+    println!("Expected complexity: O(n³)");
+    println!(
+        "{:>6} {:>15} {:>15} {:>15}",
+        "Size", "Time (μs)", "n³", "Time/n³"
+    );
+    println!("{}", "-".repeat(54));
+
+    for result in &results {
+        #[expect(clippy::cast_precision_loss)]
+        let n_cubed = (result.size.pow(3)) as f64;
+        let ratio = result.avg_time_micros / n_cubed;
+        println!(
+            "{:>6} {:>15.2} {:>15.0} {:>15.6}",
+            result.size, result.avg_time_micros, n_cubed, ratio
+        );
+    }
+
+    // Calculate growth rates between consecutive sizes
+    println!("\nGrowth Rate Analysis");
+    println!("====================\n");
+    println!("If the algorithm is O(n³), when size doubles, time should increase by ~8x (2³ = 8)");
+    println!(
+        "{:>10} {:>15} {:>15}",
+        "Size Ratio", "Time Ratio", "Expected"
+    );
+    println!("{}", "-".repeat(42));
+
+    for i in 1..results.len() {
+        #[expect(clippy::cast_precision_loss)]
+        let size_ratio = results[i].size as f64 / results[i - 1].size as f64;
+        let time_ratio = results[i].avg_time_micros / results[i - 1].avg_time_micros;
+        let expected_ratio = size_ratio.powi(3);
+        println!("{size_ratio:>10.2}x {time_ratio:>14.2}x {expected_ratio:>14.2}x");
+    }
+
+    println!("\nConclusion");
+    println!("==========");
+    println!("The Time/n³ ratio should remain relatively constant if the algorithm is O(n³).");
+    println!("The actual time ratio should be close to the expected ratio for size doublings.");
+}