Best Meeting Point
要点:
- 题本身不难理解,manhattan distance。follow up就变成weighted了(因为一个地方可以有多个住户)
- 注意input是grid的形式,一种方法是2d iterate,然后用两个数组分别存x,y,只需要对column sort,row是按顺序的iterate的。最后直接取中
- 这题之所以是Hard,因为有另一种方法:不用sort,不用找median,类似于nested weighted sum II,只不过变成双向而非单向
- 假设到某一点i和j分别对应左右的某点(注意,和另一边无关),left表示所有i左边的人,right表示所有j右边的人。如果i右移,j左移,每移动一步所有当前left,right的距离d都会增加1。显然,在每一步选最小的增加距离最合算(到目前为止,不同层总的增加数是不同的)。而如果当前i/j上有人,人口也会增加(即left/right增加)。
- 为什么是i<j?当i==j的时候,这个位置的新增人(either left or right)距离都是0,所以不需要计入距离。推广到一般的过程,所有新增人口,都不会对当前轮的d有影响。所以要d+=left/right是上一轮的
- 另外,也利用了each row/col的sum,i.e.,把行列浓缩成一个值来降维。
- 显然,这个方法也可以处理weighted的情况。所以time complexity: O(mn)
- 最后结果不是/2
# A group of two or more people wants to meet and minimize the total travel distance. You are given a 2D grid of values 0 or 1, where each 1 marks the home of someone in the group. The distance is calculated using Manhattan Distance, where distance(p1, p2) = |p2.x - p1.x| + |p2.y - p1.y|.
# For example, given three people living at (0,0), (0,4), and (2,2):
# 1 - 0 - 0 - 0 - 1
# | | | | |
# 0 - 0 - 0 - 0 - 0
# | | | | |
# 0 - 0 - 1 - 0 - 0
# The point (0,2) is an ideal meeting point, as the total travel distance of 2+2+2=6 is minimal. So return 6.
# Hint:
# Try to solve it in one dimension first. How can this solution apply to the two dimension case?
# Hide Company Tags Twitter
# Hide Tags Math Sort
# Hide Similar Problems (H) Shortest Distance from All Buildings
class Solution(object):
def minTotalDistance(self, grid):
"""
:type grid: List[List[int]]
:rtype: int
"""
row_num = [sum(row) for row in grid]
col_num = [sum(col) for col in zip(*grid)]
def minDist(sum_list):
l, r = -1, len(sum_list)
left,right=0,0
d=0
while l<r:
if left<right:
d+=left
l+=1
left+=sum_list[l]
else:
d+=right
r-=1
right+=sum_list[r]
return d
return minDist(row_num)+minDist(col_num)
sol = Solution()
assert sol.minTotalDistance([[1,0,0,0,1],[0,0,0,0,0],[0,0,1,0,0]])==6