Grid Path Counting in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Grid Path Counting in C++
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts must fit the integer type.
// 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++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 6.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int grid_paths(int rows, int cols) {
    alg_require(((rows >= 1) && (cols >= 1) && (rows <= 1000000) && (cols <= 1000000)));
    std::vector<int> dp(alg_size(cols), 0);
    int c = 0;
    int r = 1;
    while ((c < cols)) {
        dp[c] = 1;
        c = alg_add(c, 1);
    }
    while ((r < rows)) {
        c = 1;
        while ((c < cols)) {
            dp[c] = alg_add(dp[c], dp[alg_sub(c, 1)]);
            c = alg_add(c, 1);
        }
        r = alg_add(r, 1);
    }
    return dp[alg_sub(cols, 1)];
}

int main(void) {
    int rows = 3;
    int cols = 3;
    int result = grid_paths(rows, cols);
    printf("%d\n", result);
    return 0;
}