2016-11-28 13:24

Let the light guide us

Description
Plain of despair was once an ancient battlefield
where those brave spirits had rested in peace for thousands of years.
Actually no one dare step into this sacred land until the rumor that
“there is a huge gold mine underneath the plain” started to spread.

Recently an accident destroyed the eternal tranquility. Some greedy
fools tried using powerful bombs to find the hidden treasure. Of
course they failed and such behavior enraged those spirits–the
consequence is that all the human villages nearby are haunted by
ghosts.

In order to stop those ghosts as soon as possible, Panda the Archmage
and Facer the great architect figure out a nice plan. Since the plain
can be represented as grids of N rows and M columns, the plan is that
we choose ONLY ONE cell in EACH ROW to build a magic tower so that
each tower can use holy light to protect the entire ROW, and finally
the whole plain can be covered and all spirits can rest in peace
again. It will cost different time to build up a magic tower in
different cells. The target is to minimize the total time of building
all N towers, one in each row.

“Ah, we might have some difficulties.” said Panda, “In order to
control the towers correctly, we must guarantee that every two towers
in two consecutive rows share a common magic area.”

“What?”

“Specifically, if we build a tower in cell (i,j) and another tower in
cell (i+1,k), then we shall have |j-k|≤f(i,j)+f(i+1,k). Here, f(i,j)
means the scale of magic flow in cell (i,j).”

“How?”

“Ur, I forgot that you cannot sense the magic power. Here is a map
which shows the scale of magic flows in each cell. And remember that
the constraint holds for every two consecutive rows.”

“Understood.”

“Excellent! Let’s get started!”

Would you mind helping them?

Input There are multiple test cases.

Each test case starts with a line containing 2 integers N and M
(2<=N<=100,1<=M<=5000), representing that the plain consists N rows
and M columns.

The following N lines contain M integers each, forming a matrix T of
N×M. The j-th element in row i (Tij) represents the time cost of
building a magic tower in cell (i, j). (0<=Tij<=100000）

The following N lines contain M integers each, forming a matrix F of
N×M. The j-th element in row i (Fij) represents the scale of magic
flows in cell (i, j). (0<=Fij<=100000）

For each test case, there is always a solution satisfying the
constraints.

The input ends with a test case of N=0 and M=0.

Output For each test case, output a line with a single integer, which
is the minimum time cost to finish all magic towers.

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