Problem $2$: Shortest Cost Route to Navigate a Grid

Consider a grid where each cell has a different cost to travel across the regions. Assume we can only travel and stop in straight lines between the corners of these cells. Note that the cost to travel along a border between two cells is the cheapest of the two. We want to find the cheapest route from the lower$-$left corner to the upper$-$right corner of the grid under these constraints. For example in the following $3$xx$3$ grid, one of the cheapest routes of cost $2+3+4+2=11$ is highlighted. We will read in a sequence of problem instances. The first line will contain two positive integers n and m$,$ both at most $400,$ denoting the dimensions of the grid; here the number of rows is n and the number of columns is m$.$ We then are given n lines of m non$-$negative integers representing the costs for the cells. All integers will be separated by spaces. The last problem instance will have values of n$=$m$=0,$ which is not processed.

The input should be taken from keyboard$/$stdin$/$System$.$in$.$

Sample Input:

$33$

$062$

$184$

$237$

$35$

$13991$

$510184$

$27826$

$00$

The output for each instance should be a single integer $($one per line$)$ denoting the minimum cost to travel. Print these to the console$/$stdout$/$System$.$out.

Sample Output:

$11$

$17$

The above samples are a snippet of a larger whole sample that the code has to go through.

Full sample input:

$33$

$062$

$184$

$237$

$35$

$13991$

$510184$

$27826$

$810$

$38121851710141014$

$121511510159590$

$1879451121713$

$715012161711559$

$134122010190$

$1312515144616172$

$100111118107932$

$121514521591375$

$810$

$277866889955828916363381261284$

$226664429489287881367505552582$

$94199353651191871852592965$

$817930189908903258163320794366$

$497826706679882737829890576666$

$507644536937649238386320127283$

$597715487287312122874611842696$

$94176049950230840057227851253$

$810$

$4861517389170772010821897391217591898$

$172837143636239427173287615154327493339$

$23263315447135111137341336138836983749$

$3209163629253327216619032239202615803479$

$26221105928291618361623256015651699811$

$2875720397428731799932363260919033721$

$3511172947359592457615244428822671$

$22452425368618831427136350430922833736$

$610$

$403519372321301524$

$2329232201834221127$

$13132237301618322625$

$7103512832335334$

$25232626272016362740$

$1010736281319294022$

$68$

$35311322053033$

$142152930372422$

$36392112312911$

$44273032311412$

$3563174202527$

$579214181924$

$67$

$12456476$

$13101161228$

$4101225510$

$3961112412$

$127010891$