https://leetcode.com/problems/largest-plus-sign/solution/
public int orderOfLargestPlusSign(int N, int[][] mines) {
Set<Integer> banned = new HashSet();
int[][] dp = new int[N][N];
for (int[] mine: mines)
banned.add(mine[0] * N + mine[1]);
int ans = 0, count;
for (int r = 0; r < N; ++r) {
count = 0;
for (int c = 0; c < N; ++c) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = count;
}
count = 0;
for (int c = N-1; c >= 0; --c) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
}
}
for (int c = 0; c < N; ++c) {
count = 0;
for (int r = 0; r < N; ++r) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
}
count = 0;
for (int r = N-1; r >= 0; --r) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
ans = Math.max(ans, dp[r][c]);
}
}
return ans;
}
https://leetcode.com/problems/largest-plus-sign/discuss/113314/JavaC++Python-O(N2)-solution-using-only-one-grid-matrix
X. Brute Force
In a 2D
grid
from (0, 0) to (N-1, N-1), every cell contains a 1
, except those cells in the given list mines
which are 0
. What is the largest axis-aligned plus sign of 1
s contained in the grid? Return the order of the plus sign. If there is none, return 0.
An "axis-aligned plus sign of
1
s of order k" has some center grid[x][y] = 1
along with 4 arms of length k-1
going up, down, left, and right, and made of 1
s. This is demonstrated in the diagrams below. Note that there could be 0
s or 1
s beyond the arms of the plus sign, only the relevant area of the plus sign is checked for 1s.
Examples of Axis-Aligned Plus Signs of Order k:
Order 1: 000 010 000 Order 2: 00000 00100 01110 00100 00000 Order 3: 0000000 0001000 0001000 0111110 0001000 0001000 0000000
Example 1:
Input: N = 5, mines = [[4, 2]] Output: 2 Explanation: 11111 11111 11111 11111 11011 In the above grid, the largest plus sign can only be order 2. One of them is marked in bold.
Example 2:
Input: N = 2, mines = [] Output: 1 Explanation: There is no plus sign of order 2, but there is of order 1.
Example 3:
Input: N = 1, mines = [[0, 0]] Output: 0 Explanation: There is no plus sign, so return 0.
Note:
N
will be an integer in the range[1, 500]
.mines
will have length at most5000
.mines[i]
will be length 2 and consist of integers in the range[0, N-1]
.- (Additionally, programs submitted in C, C++, or C# will be judged with a slightly smaller time limit.)
How can we improve our bruteforce? One way is to try to speed up the inner loop involving
k
, the order of the candidate plus sign. If we knew the longest possible arm length in each direction from a center, we could know the order of a plus sign at that center. We could find these lengths separately using dynamic programming.
Algorithm
For each (cardinal) direction, and for each coordinate
(r, c)
let's compute the count
of that coordinate: the longest line of '1'
s starting from (r, c)
and going in that direction. With dynamic programming, it is either 0 if grid[r][c]
is zero, else it is 1
plus the count of the coordinate in the same direction. For example, if the direction is left and we have a row like 01110110
, the corresponding count values are 01230120
, and the integers are either 1 more than their successor, or 0. For each square, we want dp[r][c]
to end up being the minimum of the 4 possible counts. At the end, we take the maximum value in dp
.- Time Complexity: , as the work we do under two nested for loops is .
- Space Complexity: , the size of
dp
.
Set<Integer> banned = new HashSet();
int[][] dp = new int[N][N];
for (int[] mine: mines)
banned.add(mine[0] * N + mine[1]);
int ans = 0, count;
for (int r = 0; r < N; ++r) {
count = 0;
for (int c = 0; c < N; ++c) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = count;
}
count = 0;
for (int c = N-1; c >= 0; --c) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
}
}
for (int c = 0; c < N; ++c) {
count = 0;
for (int r = 0; r < N; ++r) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
}
count = 0;
for (int r = N-1; r >= 0; --r) {
count = banned.contains(r*N + c) ? 0 : count + 1;
dp[r][c] = Math.min(dp[r][c], count);
ans = Math.max(ans, dp[r][c]);
}
}
return ans;
}
https://leetcode.com/problems/largest-plus-sign/discuss/113314/JavaC++Python-O(N2)-solution-using-only-one-grid-matrix
Algorithms: For each position
(i, j)
of the grid
matrix, we try to extend in each of the four directions (left, right, up, down) as long as possible, then take the minimum length of 1
's out of the four directions as the order of the largest axis-aligned plus sign centered at position (i, j)
.
Optimizations: Normally we would need a total of five matrices to make the above idea work -- one matrix for the
grid
itself and four more matrices for each of the four directions. However, these five matrices can be combined into one using two simple tricks:- For each position
(i, j)
, we are only concerned with the minimum length of1
's out of the four directions. This implies we may combine the four matrices into one by only keeping tracking of the minimum length. - For each position
(i, j)
, the order of the largest axis-aligned plus sign centered at it will be0
if and only ifgrid[i][j] == 0
. This implies we may further combine thegrid
matrix with the one obtained above.
Implementations:
- Create an
N-by-N
matrixgrid
, with all elements initialized with valueN
. - Reset those elements to
0
whose positions are in themines
list. - For each position
(i, j)
, find the maximum length of1
's in each of the four directions and setgrid[i][j]
to the minimum of these four lengths. Note that there is a simple recurrence relation relating the maximum length of1
's at current position with previous position for each of the four directions (labeled asl
,r
,u
,d
). - Loop through the
grid
matrix and choose the maximum element which will be the largest axis-aligned plus sign of1
's contained in the grid.
Solutions: Here is a list of solutions for Java/C++/Python based on the above ideas. All solutions run at
O(N^2)
time with O(N^2)
extra space. Further optimizations are possible such as keeping track of the maximum plus sign currently available and terminating as early as possible if no larger plus sign can be found for current row/column.
Note: For those of you who got confused by the logic within the first nested for-loop, refer to andier's comment below for a more clear explanation.
public int orderOfLargestPlusSign(int N, int[][] mines) {
int[][] grid = new int[N][N];
for (int i = 0; i < N; i++) {
Arrays.fill(grid[i], N);
}
for (int[] m : mines) {
grid[m[0]][m[1]] = 0;
}
for (int i = 0; i < N; i++) {
for (int j = 0, k = N - 1, l = 0, r = 0, u = 0, d = 0; j < N; j++, k--) {
grid[i][j] = Math.min(grid[i][j], l = (grid[i][j] == 0 ? 0 : l + 1));
grid[i][k] = Math.min(grid[i][k], r = (grid[i][k] == 0 ? 0 : r + 1));
grid[j][i] = Math.min(grid[j][i], u = (grid[j][i] == 0 ? 0 : u + 1));
grid[k][i] = Math.min(grid[k][i], d = (grid[k][i] == 0 ? 0 : d + 1));
}
}
int res = 0;
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
res = Math.max(res, grid[i][j]);
}
}
return res;
}
X. Brute Force
Intuition and Algorithm
For each possible center, find the largest plus sign that could be placed by repeatedly expanding it. We expect this algorithm to be , and so take roughly operations. This is a little bit too big for us to expect it to run in time.
- Time Complexity: , as we perform two outer loops (), plus the inner loop involving
k
is . - Space Complexity: .
public int orderOfLargestPlusSign(int N, int[][] mines) {
Set<Integer> banned = new HashSet();
for (int[] mine: mines)
banned.add(mine[0] * N + mine[1]);
int ans = 0;
for (int r = 0; r < N; ++r) for (int c = 0; c < N; ++c) {
int k = 0;
while (k <= r && r < N-k && k <= c && c < N-k &&
!banned.contains((r-k)*N + c) &&
!banned.contains((r+k)*N + c) &&
!banned.contains(r*N + c-k) &&
!banned.contains(r*N + c+k))
k++;
ans = Math.max(ans, k);
}
return ans;
}