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path-with-minimum-effort.py
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# Time: O(m * n * log(m * n))
# Space: O(m * n)
import heapq
# Dijkstra algorithm solution
class Solution(object):
def minimumEffortPath(self, heights):
"""
:type heights: List[List[int]]
:rtype: int
"""
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
dst = (len(heights)-1, len(heights[0])-1)
dist = [[float("inf")]*len(heights[0]) for _ in xrange(len(heights))]
min_heap = [(0, 0, 0)]
lookup = [[False]*len(heights[0]) for _ in xrange(len(heights))]
while min_heap:
d, r, c = heapq.heappop(min_heap)
if lookup[r][c]:
continue
lookup[r][c] = True
if (r, c) == dst:
return d
for dr, dc in directions:
nr, nc = r+dr, c+dc
if not (0 <= nr < len(heights) and 0 <= nc < len(heights[0]) and not lookup[nr][nc]):
continue
nd = max(d, abs(heights[nr][nc]-heights[r][c]))
if nd < dist[nr][nc]:
dist[nr][nc] = nd
heapq.heappush(min_heap, (nd, nr, nc))
return -1
# Time: O(m * n * log(m * n) + m * n * α(m * n)) = O(m * n * log(m * n))
# Space: O(m * n)
import collections
class UnionFind(object): # Time: (n * α(n)), Space: O(n)
def __init__(self, n):
self.set = range(n)
self.rank = range(n)
def find_set(self, x):
stk = []
while self.set[x] != x: # path compression
stk.append(x)
x = self.set[x]
while stk:
self.set[stk.pop()] = x
return x
def union_set(self, x, y):
x_root, y_root = map(self.find_set, (x, y))
if x_root == y_root:
return False
if self.rank[x_root] < self.rank[y_root]: # union by rank
self.set[x_root] = y_root
elif self.rank[x_root] > self.rank[y_root]:
self.set[y_root] = x_root
else:
self.set[y_root] = x_root
self.rank[x_root] += 1
return True
# union find solution
class Solution2(object):
def minimumEffortPath(self, heights):
"""
:type heights: List[List[int]]
:rtype: int
"""
def index(n, i, j):
return i*n + j
diffs = []
for i in xrange(len(heights)):
for j in xrange(len(heights[0])):
if i > 0:
diffs.append((abs(heights[i][j]-heights[i-1][j]), index(len(heights[0]), i-1, j), index(len(heights[0]), i, j)))
if j > 0:
diffs.append((abs(heights[i][j]-heights[i][j-1]), index(len(heights[0]), i, j-1), index(len(heights[0]), i, j)))
diffs.sort()
union_find = UnionFind(len(heights)*len(heights[0]))
for d, i, j in diffs:
if union_find.union_set(i, j):
if union_find.find_set(index(len(heights[0]), 0, 0)) == \
union_find.find_set(index(len(heights[0]), len(heights)-1, len(heights[0])-1)):
return d
return 0
# Time: O(m * n * logh)
# Space: O(m * n)
# bi-bfs solution
class Solution3(object):
def minimumEffortPath(self, heights):
"""
:type heights: List[List[int]]
:rtype: int
"""
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
def check(heights, x): # bi-bfs
lookup = [[False]*len(heights[0]) for _ in xrange(len(heights))]
left, right = {(0, 0)}, {(len(heights)-1, len(heights[0])-1)}
while left:
for r, c in left:
lookup[r][c] = True
new_left = set()
for r, c in left:
if (r, c) in right:
return True
for dr, dc in directions:
nr, nc = r+dr, c+dc
if not (0 <= nr < len(heights) and
0 <= nc < len(heights[0]) and
abs(heights[nr][nc]-heights[r][c]) <= x and
not lookup[nr][nc]):
continue
new_left.add((nr, nc))
left = new_left
if len(left) > len(right):
left, right = right, left
return False
left, right = 0, 10**6
while left <= right:
mid = left + (right-left)//2
if check(heights, mid):
right = mid-1
else:
left = mid+1
return left
# Time: O(m * n * logh)
# Space: O(m * n)
import collections
# bfs solution
class Solution4(object):
def minimumEffortPath(self, heights):
"""
:type heights: List[List[int]]
:rtype: int
"""
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
def check(heights, x):
lookup = [[False]*len(heights[0]) for _ in xrange(len(heights))]
q = collections.deque([(0, 0)])
while q:
r, c = q.popleft()
if (r, c) == (len(heights)-1, len(heights[0])-1):
return True
for dr, dc in directions:
nr, nc = r+dr, c+dc
if not (0 <= nr < len(heights) and
0 <= nc < len(heights[0]) and
abs(heights[nr][nc]-heights[r][c]) <= x and
not lookup[nr][nc]):
continue
lookup[nr][nc] = True
q.append((nr, nc))
return False
left, right = 0, 10**6
while left <= right:
mid = left + (right-left)//2
if check(heights, mid):
right = mid-1
else:
left = mid+1
return left
# Time: O(m * n * logh)
# Space: O(m * n)
# dfs solution
class Solution5(object):
def minimumEffortPath(self, heights):
"""
:type heights: List[List[int]]
:rtype: int
"""
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
def check(heights, x):
lookup = [[False]*len(heights[0]) for _ in xrange(len(heights))]
stk = [(0, 0)]
while stk:
r, c = stk.pop()
if (r, c) == (len(heights)-1, len(heights[0])-1):
return True
for dr, dc in directions:
nr, nc = r+dr, c+dc
if not (0 <= nr < len(heights) and
0 <= nc < len(heights[0]) and
abs(heights[nr][nc]-heights[r][c]) <= x and
not lookup[nr][nc]):
continue
lookup[nr][nc] = True
stk.append((nr, nc))
return False
left, right = 0, 10**6
while left <= right:
mid = left + (right-left)//2
if check(heights, mid):
right = mid-1
else:
left = mid+1
return left