120. Triangle
Description
Given a triangle
array, return the minimum path sum from top to bottom.
For each step, you may move to an adjacent number of the row below. More formally, if you are on index i
on the current row, you may move to either index i
or index i + 1
on the next row.
Example 1:
Input: triangle = [[2],[3,4],[6,5,7],[4,1,8,3]] Output: 11 Explanation: The triangle looks like: 2 3 4 6 5 7 4 1 8 3 The minimum path sum from top to bottom is 2 + 3 + 5 + 1 = 11 (underlined above).
Example 2:
Input: triangle = [[-10]] Output: -10
Constraints:
1 <= triangle.length <= 200
triangle[0].length == 1
triangle[i].length == triangle[i - 1].length + 1
-104 <= triangle[i][j] <= 104
Follow up: Could you do this using only O(n)
extra space, where n
is the total number of rows in the triangle?
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ as the minimum path sum from the bottom of the triangle to the position $(i, j)$. Here, the position $(i, j)$ refers to the position in the $i$th row and $j$th column of the triangle (both starting from $0$). Then we have the following state transition equation:
$$ f[i][j] = \min(f[i + 1][j], f[i + 1][j + 1]) + triangle[i][j] $$
The answer is $f[0][0]$.
We notice that the state $f[i][j]$ is only related to the states $f[i + 1][j]$ and $f[i + 1][j + 1]$, so we can use a one-dimensional array instead of a two-dimensional array, reducing the space complexity from $O(n^2)$ to $O(n)$.
The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the number of rows in the triangle.
Furthermore, we can directly reuse the triangle
as the f
array, so there is no need to create an additional f
array, reducing the space complexity to $O(1)$.
1 2 3 4 5 6 7 8 |
|
1 2 3 4 5 6 7 8 9 10 11 12 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 14 |
|
1 2 3 4 5 6 7 8 9 10 |
|
1 2 3 4 5 6 7 8 9 10 |
|
1 2 3 4 5 6 7 8 9 10 11 12 |
|
Solution 2
1 2 3 4 5 6 7 8 |
|
1 2 3 4 5 6 7 8 9 10 11 12 |
|
1 2 3 4 5 6 7 8 9 10 11 |
|
1 2 3 4 5 6 7 8 |
|
1 2 3 4 5 6 7 8 |
|
1 2 3 4 5 6 7 8 9 10 11 |
|
Solution 3
1 2 3 4 5 6 7 8 9 |
|