Matrix Chain Multiplication in C#

using System;
// Matrix Chain Multiplication in C#
//
// PURPOSE
// Matrix-chain DP chooses the parenthesization requiring the fewest scalar multiplications
// without changing matrix order.
//
// METHOD
// For each interval length, try every split and combine both subchain costs with the
// multiplication cost of their results.
//
// COMPLEXITY
// O(n cubed) time and O(n squared) space for n matrices.
//
// CONTRACT
// dimensions has n+1 positive entries; matrix i has dimensions[i] by dimensions[i+1]. n>=1.
// All candidate costs must be < 1000000000.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 4500.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int matrix_chain(ReadOnlySpan<int> dimensions) {
        checked {
            int n = (dimensions.Length - 1);
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((dimensions.Length == (n + 1)));
            alg_require((n >= 1));
            for (int alg_i = 0; alg_i < dimensions.Length; alg_i++) {
                int alg_value = dimensions[alg_i];
                alg_require(((alg_value > 0) && (alg_value <= 1000)));
            }
            int[] dp = new int[alg_size((n * n))];
            int length = 2;
            int left = 0;
            int right = 0;
            int split = 0;
            int candidate = 0;
            while ((length <= n)) {
                left = 0;
                while (((left + length) <= n)) {
                    right = ((left + length) - 1);
                    dp[((left * n) + right)] = 1000000000;
                    split = left;
                    while ((split < right)) {
                        candidate = ((dp[((left * n) + split)] + dp[(((split + 1) * n) + right)]) + ((dimensions[left] * dimensions[(split + 1)]) * dimensions[(right + 1)]));
                        dp[((left * n) + right)] = Math.Min(dp[((left * n) + right)], candidate);
                        split = (split + 1);
                    }
                    left = (left + 1);
                }
                length = (length + 1);
            }
            return dp[(n - 1)];
        }
    }

    static void Main() {
        int[] dimensions = new int[] {10, 30, 5, 60};
        int result = matrix_chain(dimensions);
        Console.WriteLine(result);
    }
}