Problem 83

Path sum: four ways

NOTE: This problem is a significantly more challenging version of Problem 81.

In the $5$ by $5$ matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to $2297$.

$$
\begin{pmatrix}
\color{red}{131} & 673 & \color{red}{234} & \color{red}{103} & \color{red}{18}\\
\color{red}{201} & \color{red}{96} & \color{red}{342} & 965 & \color{red}{150}\\
630 & 803 & 746 & \color{red}{422} & \color{red}{111}\\
537 & 699 & 497 & \color{red}{121} & 956\\
805 & 732 & 524 & \color{red}{37} & \color{red}{331}
\end{pmatrix}
$$

Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down in matrix.txt (right click and “Save Link/Target As…”), a 31K text file containing a $80$ by $80$ matrix.

路径和：四个方向

注意：这是第81题的究极升级版。

在如下的$5\times 5$矩阵中，从左上角到右下角任意地向上、向下、向左或向右移动的最小路径和为$2297$，由标注红色的路径给出。

在文本文件matrix.txt中包含了一个$80\times80$的矩阵，求从该矩阵的左上角到右下角任意地向上、向下、向左或向右移动的最小路径和。