It's similar to how human solve Sudoku.

- create a hash table (dictionary)
`val`

to store possible values in every location. - Each time, start from the location with fewest possible values, choose one value from it and then update the board and possible values at other locations. If this update is valid, keep solving (DFS). If this update is invalid (leaving zero possible values at some locations) or this value doesn't lead to the solution, undo the updates and then choose the next value.

Since we calculated `val`

at the beginning and start filling the board from the location with fewest possible values, the amount of calculation and thus the runtime can be significantly reduced:

The run time is 48-68 ms on LeetCode OJ, which seems to be among the fastest python solutions here.

The `PossibleVals`

function may be further simplified/optimized, but it works just fine for now. (it would look less lengthy if we are allowed to use numpy array for the board lol).

```
def solveSudoku(self, board):
self.board = board
self.val = self.PossibleVals()
self.Solver()
def PossibleVals(self):
a = "123456789"
d, val = {}, {}
for i in xrange(9):
for j in xrange(9):
ele = self.board[i][j]
if ele != ".":
d[("r", i)] = d.get(("r", i), []) + [ele]
d[("c", j)] = d.get(("c", j), []) + [ele]
d[(i//3, j//3)] = d.get((i//3, j//3), []) + [ele]
else:
val[(i,j)] = []
for (i,j) in val.keys():
inval = d.get(("r",i),[])+d.get(("c",j),[])+d.get((i/3,j/3),[])
val[(i,j)] = [n for n in a if n not in inval ]
return val
def Solver(self):
if len(self.val)==0:
return True
kee = min(self.val.keys(), key=lambda x: len(self.val[x]))
nums = self.val[kee]
for n in nums:
update = {kee:self.val[kee]}
if self.ValidOne(n, kee, update): # valid choice
if self.Solver(): # keep solving
return True
self.undo(kee, update) # invalid choice or didn't solve it => undo
return False
def ValidOne(self, n, kee, update):
self.board[kee[0]][kee[1]] = n
del self.val[kee]
i, j = kee
for ind in self.val.keys():
if n in self.val[ind]:
if ind[0]==i or ind[1]==j or (ind[0]/3,ind[1]/3)==(i/3,j/3):
update[ind] = n
self.val[ind].remove(n)
if len(self.val[ind])==0:
return False
return True
def undo(self, kee, update):
self.board[kee[0]][kee[1]]="."
for k in update:
if k not in self.val:
self.val[k]= update[k]
else:
self.val[k].append(update[k])
return None
```