Jump Search in C#

using System;
// Jump Search in C#
//
// PURPOSE
// Jump search skips fixed-size blocks in a sorted sequence, then scans the candidate block.
//
// METHOD
// Choose a block width near square root of n. Jump until the block endpoint is at least the
// target, then scan the block.
//
// COMPLEXITY
// O(sqrt(n)) comparisons and O(1) auxiliary space. Binary search is usually faster on
// random-access arrays.
//
// CONTRACT
// Ascending input; returns first matching index or -1.
// 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: 1.
// 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 jump_search(ReadOnlySpan<int> a, int target) {
        checked {
            int n = a.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((a.Length == n));
            int step = 1;
            while ((step <= (n / step))) {
                step = (step + 1);
            }
            int start = 0;
            int end = Math.Min(step, n);
            while (((start < n) && (a[(end - 1)] < target))) {
                start = end;
                end = Math.Min((end + step), n);
            }
            while ((start < end)) {
                if ((a[start] == target)) {
                    return start;
                }
                start = (start + 1);
            }
            return (-1);
        }
    }

    static void Main() {
        int[] a = new int[] {1, 3, 3, 7};
        int target = 3;
        int result = jump_search(a, target);
        Console.WriteLine(result);
    }
}