Algorithms: Dynamic Programming

function minimum_path_sum(array $grid, int $rows, int $cols): int {
    alg_require(array_is_list($grid));
    alg_require((count($grid) == alg_mul($rows, $cols)));
    alg_require((($rows >= 1) && ($cols >= 1) && ($rows <= 1000000) && ($cols <= 1000000)));
    $dp = array_fill(0, alg_size($cols), 0);
    $r = 0;
    $c = 0;
    while (($r < $rows)) {
        $c = 0;
        while (($c < $cols)) {
            if ((($r == 0) && ($c == 0))) {
                $dp[$c] = $grid[0];

    static int palindromic_subsequence(int[] a) {
        java.util.Objects.requireNonNull(a, "a");
        int n = a.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((a.length == n));
        int[] dp = new int[alg_size(Math.addExact(n, 1))];
        int left = Math.subtractExact(n, 1);
        int right = 0;
        int diagonal = 0;
        int old = 0;
        while ((left >= 0)) {
            dp[left] = 1;

function palindromic_subsequence(a) {
    alg_require(a instanceof Int32Array);
    let n = a.length;
    alg_require(((n >= 0) && (n <= 1000000)));
    alg_require((a.length == n));
    let dp = new Int32Array(alg_size(alg_add(n, 1)));
    let left = alg_sub(n, 1);
    let right = 0;
    let diagonal = 0;
    let old = 0;
    while ((left >= 0)) {
        dp[left] = 1;

def palindromic_subsequence(a: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    dp: list[int] = [0] * alg_size(n + 1)
    left: int = n - 1
    right: int = 0
    diagonal: int = 0
    old: int = 0
    while left >= 0:
        dp[left] = 1

int palindromic_subsequence(const int *a, size_t a_length) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (a_length > 1000000 || (a_length && !a)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int n = (int)a_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((n >= 0) && (n <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)a_length == n))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }