I new do dynamic programming and I am attempting the following problem on leetcode: 01 Matrix
Problem: Given an m x n binary matrix mat, return the distance of the nearest 0 for each cell. The distance between two adjacent cells is 1.
I have attempted the problem using top-down dynamic programming but cannot seem to get the right answer for all the test cases. In all most all cases, the matrix is almost optimized except for a few values. My algorithm entails finding a '1' and then doing a depth first search in all 4 directions and then taking the minimum of the values + 1 and saving that to the memoization table (ans[][]).
I have tried to search for a top down approach but all most all the solutions are bottom up. Can anyone please help me understand why taking the minimum of the steps in all 4 directions using memoization doesn't not yield an optimal solution or what my solution is missing?
class Solution {
public int[][] updateMatrix(int[][] mat) {
int m = mat.length;
int n = mat[0].length;
int[][] ans = new int[m][n];
for(int i = 0; i <m; i++){
for(int j = 0; j <n; j++){
if(mat[i][j] == 0){
ans[i][j] = 0;
}
else{
ans[i][j] = -1;
}
}
}
for(int i = 0; i < m; i++){
for(int j = 0; j < n; j++){
if(ans[i][j] == - 1){
boolean[][] visited = new boolean[m][n];
ans[i][j] = dfs(i, j, m, n, mat, ans, visited);
}
}
}
for(int i = 0; i < m; i++){
for(int j = 0; j < n; j++){
boolean[][] visited = new boolean[m][n];
ans[i][j] = Math.min(ans[i][j], dfs(i, j, m, n, mat, ans, visited));
}
}
return ans;
}
public int dfs(int i, int j, int m, int n, int[][]mat, int[][] ans, boolean[][] visited){
if(i >= m || i < 0 || j >=n || j < 0|| visited[i][j]){
return Integer.MAX_VALUE;
}
if(mat[i][j] == 0){
return 0;
}
if(ans[i][j] != -1){
return ans[i][j];
}
visited[i][j] = true;
int up = dfs(i - 1, j, m, n, mat, ans, visited);
int down = dfs(i + 1, j, m, n, mat, ans, visited);
int left = dfs( i, j - 1, m, n, mat, ans, visited);
int right = dfs(i, j + 1, m, n, mat, ans,visited);
visited[i][j] = false;
ans[i][j] = Math.min (up, Math.min(down, Math.min(left, right))) + 1;
return ans[i][j];
}
}
The 3 double-loops at the beginning could be reduced to one double-loop, the 3rd one seems to be completely superfluous. Dfs is not working here. E.g. you will go 4 fields up 1 left and 4 fields down and save 9 but actually that field could be reached in 1 step to the left, you just ignore it because you already have a result. Even if you would fix the problems, this is still O((nm)^2) instead of O(nm). For an O(nm) solution you can initialize the result with 0 where you have a 0, with -1 where there is no 0 next to it and 1 where you have a 1 with a 0 next to it, additionally add the position of this 1 to a list. Initialize c to 2. Go through the list and check the positions up, left, right and down, if it is a -1 in the result replace it by c and add this position to a new list. Replace the list by the new list, increment c and go through the list again and again until it is empty.
There is also a working DFS implementation that is easier to understand:
https://leetcode.com/problems/01-matrix/discuss/732601/DFS-O(1)-Space
class Solution {
public int[][] updateMatrix(int[][] dist) {
int m = dist.length;
int n = dist[0].length;
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
if (dist[i][j] != 0) {
dist[i][j] = Integer.MAX_VALUE;
}
}
}
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
if (dist[i][j] == 0) {
dfs(dist, i - 1, j, 1);
dfs(dist, i + 1, j, 1);
dfs(dist, i, j - 1, 1);
dfs(dist, i, j + 1, 1);
}
}
}
return dist;
}
private void dfs(int[][] dist, int i, int j, int value) {
if (i >= 0 && i < dist.length && j >= 0 && j < dist[0].length && value < dist[i][j]) {
dist[i][j] = value;
dfs(dist, i - 1, j, value + 1);
dfs(dist, i + 1, j, value + 1);
dfs(dist, i, j - 1, value + 1);
dfs(dist, i, j + 1, value + 1);
}
}
}
I have an engineering problem that requires me to partition a set of positive numbers to k parts, without changing the ordering of the numbers in the set, so that the sums of the parts are as equal as possible. I have understood that there are solutions to this 'linear partitioning problem'. In fact, I have successfully tried the dynamic programming solution already and it works.
Now the question: What if there is a maximum size limitation for the parts? ("No part can have more than m items") Can it be added e.g. to the DP solution or can some other technique be used for solving this.
Appreciate all comments, links and suggestions.
Edit: I added a crude sketch of the original algorithm below
// Partition number array seq to k parts with max number of m items in each part
public void LinearPartition(double[] seq, int k, int m)
{
double[,] table;
int[,] solution;
int n = seq.Length - 1;
linearPartitionTable(seq, k, m, out table, out solution);
k = k - 2;
while (k >= 0)
{
Console.WriteLine(solution[n - 1, k] + 1);
n = solution[n - 1, k];
k--;
}
}
private void linearPartitionTable(double[] seq, int k, int m,
out double[,] table, out int[,] solution)
{
int n = seq.Length;
table = new double[n, k];
solution = new int[n - 1, k - 1];
table[0, 0] = 0;
for (int i = 1; i < n; i++)
table[i, 0] = seq[i] + table[i - 1, 0];
for (int j = 0; j < k; j++)
table[0, j] = seq[0];
for (int i = 1; i < n; i++)
{
for (int j = 1; j < k; j++)
{
double currentMin = double.MaxValue;
int minX = 0;
for (int x = 0; x < i; x++)
{
double cost = Math.Max(table[x, j - 1], table[i, 0] - table[x, 0]);
if (cost < currentMin)
{
currentMin = cost;
minX = x;
}
}
table[i, j] = currentMin;
solution[i - 1, j - 1] = minX;
}
}
}
Find zeroes to be flipped so that number of consecutive 1’s is maximized.
Input: arr[] = {1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1}
m = 2
Output: 5 7
We are allowed to flip maximum 2 zeroes. If we flip
arr[5] and arr[7], we get 8 consecutive 1's which is
maximum possible under given constraints .
Now if we were to find just the maximum number of 1's that is possible, is it possible to solve using dynamic programming approach?
This problem can be solved in linear time O(N) and linear space O(N). Its not full fledged dynamic programming, but its similar to that as it uses precomputation.
Data Structures Used:
1.left: It is an integer array, of same length as given array. It is precomputed such that for every position i:
left[i] = Number of consecutive 1's to the left position i
2.right: It is an integer array, of same length as given array. It is precomputed such that for every position i:
right[i] = Number of consecutive 1's to the right position i
These can be computed in single traversal of the array.Assuming arr is the original array, following pseudocode does the job:
Pseudocode for populating left array
left()
{
int count = 0;
for(int i = 0;i < arr length; ++i)
{
if(i == 0)
{
left[i] = 0;
if(arr[i] == 1)
count++;
continue;
}
else
{
left[i] = count;
if(arr[i] == 1)
count++;
else count = 0;
}
}
}
Pseudocode for populating right array
right()
{
int count = 0;
for(int i = arr length - 1;i >= 0; --i)
{
if(i == arr length - 1)
{
right[i] = 0;
if(arr[i] == 1)
count++;
continue;
}
else
{
right[i] = count;
if(arr[i] == 1)
count++;
else count = 0;
}
}
}
Now the only thing we have to do is :check all pair of positions i and j (i < j) such that arr[i] = 0 and arr[j] = 0 and for no position between i and j arr[i] should be 0 and Keep track of the pair for which we get maximum value of the following:
left[i] + right[j] + right[l]
You could also use left[i] + right[j] + left[r].
left[i] tells the number of consecutive 1's to the left of position i and right[j] tells the number of consecutive 1's to the right of position j and the number of consecutive 1's between i and j can be counted be left[r] OR right[l], and therefore, we have two candidate expressions.
This can also be done in single traversal, using following pseudocode:
max_One()
{
max = 0;
l = -1, r = -1;
for(int i = 0;i < arr length; ++i)
{
if(arr[i] == 0)
{
if(l == -1)
l = i;
else
{
r = i;
if(left[l] + right[r] + right[l] > max)
{
max = left[l] + right[r] + right[l];
left_pos = l;
right_pos = r;
}
l = r;
}
}
}
}
You should use sliding window concept here - use start and end vars to store index of range. Whenever you encounter a 0, increment the counter of zeros received. Include it in current length.. If zeros encounter equals m+1, increment start till you encounter 0.
public static int[] zerosToFlip(int[] input, int m) {
if (m == 0) return new int[0];
int[] indices = new int[m];
int beginIndex = 0;
int endIndex = 0;
int maxBeginIndex=0;
int maxEndIndex=0;
int zerosIncluded = input[0] == 0 ? 1 : 0;
for (int i = 1; i < input.length; i++) {
if (input[i] == 0) {
if (zerosIncluded == m) {
if (endIndex - beginIndex > maxEndIndex - maxBeginIndex){
maxBeginIndex = beginIndex;
maxEndIndex = endIndex;
}
while (input[beginIndex] != 0) beginIndex++;
beginIndex++;
} else {
zerosIncluded++;
}
}
endIndex++;
}
if (endIndex - beginIndex > maxEndIndex - maxBeginIndex){
maxBeginIndex = beginIndex;
maxEndIndex = endIndex;
}
int j = 0;
for (int i = maxBeginIndex; i <= maxEndIndex; i++) {
if (input[i] == 0) {
indices[j] = i;
++j;
}
}
return indices;
}
This is programming puzzle. We have two arrays A and B. Both contains 0's and 1's only.
We have to two indices i, j such that
a[i] + a[i+1] + .... a[j] = b[i] + b[i+1] + ... b[j].
Also we have to maximize this difference between i and j. Looking for O(n) solution.
I found O(n^2) solution but not getting O(n).
Best solution is O(n)
First let c[i] = a[i] - b[i], then question become find i, j, which sum(c[i], c[i+1], ..., c[j]) = 0, and max j - i.
Second let d[0] = 0, d[i + 1] = d[i] + c[i], i >= 0, then question become find i, j, which d[j + 1] == d[i], and max j - i.
The value of d is in range [-n, n], so we can use following code to find the answer
answer = 0, answer_i = 0, answer_j = 0
sumHash[2n + 1] set to -1
for (x <- 0 to n) {
if (sumHash[d[x]] == -1) {
sumHash[d[x]] = x
} else {
y = sumHash[d[x]]
// find one answer (y, x), compare to current best
if (x - y > answer) {
answer = x - y
answer_i = y
answer_j = y
}
}
}
Here is an O(n) solution.
I use the fact that sum[i..j] = sum[j] - sum[i - 1].
I keep the leftmost position of each found sum.
int convertToPositiveIndex(int index) {
return index + N;
}
int mostLeft[2 * N + 1];
memset(mostLeft, -1, sizeof(mostLeft));
int bestLen = 0, bestStart = -1, bestEnd = -1;
int sumA = 0, sumB = 0;
for (int i = 0; i < N; i++) {
sumA += A[i];
sumB += B[i];
int diff = sumA - sumB;
int diffIndex = convertToPositiveIndex(diff);
if (mostLeft[diffIndex] != -1) {
//we have found the sequence mostLeft[diffIndex] + 1 ... i
//now just compare it with the best one found so far
int currentLen = i - mostLeft[diffIndex];
if (currentLen > bestLen) {
bestLen = currentLen;
bestStart = mostLeft[diffIndex] + 1;
bestEnd = i;
}
}
if (mostLeft[diffIndex] == -1) {
mostLeft[diffIndex] = i;
}
}
cout << bestStart << " " << bestEnd << " " << bestLen << endl;
P.S. mostLeft array is 2 * N + 1, because of the negatives.
This is a fairly straightforward O(N) solution:
let sa = [s1, s2, s3.. sn] where si = sum(a[0:i]) and similar for sb
then sum(a[i:j]) = sa[j]-sa[i]
and sum(b[i:j]) = sb[j] - sb[i]
Note that because the sums only increase by 1 each time, we know 0 <= sb[N], sa[N] <=N
difference_array = [d1, d2, .. dn] where di = sb[i] - sa[i] <= N
note if di = dj, then sb[i] - sa[i] = sb[j] - sa[j] which means they have the same sum (rearrange to get sum(b[i:j]) and sum(a[i:j]) from above).
Now for each difference we need its max position occurrence and min position occurrence
Now for each difference di, the difference between max - min, is an i-j section of equal sum. Find the maximum max-min value and you're done.
sample code that should work:
a = []
b = []
sa = [0]
sb = [0]
for i in a:
sa.append(sa[-1] + i)
for i in b:
sb.append(sb[-1] + i)
diff = [sai-sbi for sai, sbi in zip(sa, sb)]
min_diff_pos = {}
max_diff_pos = {}
for pos, d in enumerate(diff):
if d in min_diff_pos:
max_diff_pos[d] = pos
else:
min_diff_pos[d] = pos
ans = min(max_diff_pos[d] - min_diff_pos[d] for d in diff)
Basically, my solution goes like this.
Take a variable to take care of the difference since the beginning.
int current = 0;
for index from 0 to length
if a[i] == 0 && b[i] == 1
current--;
else if a[i] == 1 && b[i] == 0
current++;
else
// nothing;
Find the positions where the variable has the same value, which indicates that there are equal 1s and 0s in between.
Pseudo Code:
Here is my primary solution:
int length = min (a.length, b.length);
int start[] = {-1 ... -1}; // from -length to length
start[0] = -1;
int count[] = {0 ... 0}; // from -length to length
int current = 0;
for (int i = 0; i < length; i++) {
if (a[i] == 0 && b[i] == 1)
current--;
else if (a[i] == 1 && b[i] == 0)
current++;
else
; // nothing
if (start[current] == -1) // index can go negative here, take care
start[current] = current;
else
count[current] = i - start[current];
}
return max_in(count[]);
Input: A 2-dimensional array NxN - Matrix - with positive and negative elements.Output: A submatrix of any size such that its summation is the maximum among all possible submatrices.
Requirement: Algorithm complexity to be of O(N^3)
History: With the help of the Algorithmist, Larry and a modification of Kadane's Algorithm, i managed to solve the problem partly which is determining the summation only - below in Java.
Thanks to Ernesto who managed to solve the rest of the problem which is determining the boundaries of the matrix i.e. top-left, bottom-right corners - below in Ruby.
Here's an explanation to go with the posted code. There are two key tricks to make this work efficiently: (I) Kadane's algorithm and (II) using prefix sums. You also need to (III) apply the tricks to the matrix.
Part I: Kadane's algorithm
Kadane's algorithm is a way to find a contiguous subsequence with maximum sum. Let's start with a brute force approach for finding the max contiguous subsequence and then consider optimizing it to get Kadane's algorithm.
Suppose you have the sequence:
-1, 2, 3, -2
For the brute force approach, walk along the sequence generating all possible subsequences as shown below. Considering all possibilities, we can start, extend, or end a list with each step.
At index 0, we consider appending the -1
-1, 2, 3, -2
^
Possible subsequences:
-1 [sum -1]
At index 1, we consider appending the 2
-1, 2, 3, -2
^
Possible subsequences:
-1 (end) [sum -1]
-1, 2 [sum 1]
2 [sum 2]
At index 2, we consider appending the 3
-1, 2, 3, -2
^
Possible subsequences:
-1, (end) [sum -1]
-1, 2 (end) [sum -1]
2 (end) [sum 2]
-1, 2, 3 [sum 4]
2, 3 [sum 5]
3 [sum 3]
At index 3, we consider appending the -2
-1, 2, 3, -2
^
Possible subsequences:
-1, (end) [sum -1]
-1, 2 (end) [sum 1]
2 (end) [sum 2]
-1, 2 3 (end) [sum 4]
2, 3 (end) [sum 5]
3, (end) [sum 3]
-1, 2, 3, -2 [sum 2]
2, 3, -2 [sum 3]
3, -2 [sum 1]
-2 [sum -2]
For this brute force approach, we finally pick the list with the best sum, (2, 3), and that's the answer. However, to make this efficient, consider that you really don't need to keep every one of the lists. Out of the lists that have not ended, you only need to keep the best one, the others cannot do any better. Out of the lists that have ended, you only might need to keep the best one, and only if it's better than ones that have not ended.
So, you can keep track of what you need with just a position array and a sum array. The position array is defined like this: position[r] = s keeps track of the list which ends at r and starts at s. And, sum[r] gives a sum for the subsequence ending at index r. This is optimized approach is Kadane's algorithm.
Running through the example again keeping track of our progress this way:
At index 0, we consider appending the -1
-1, 2, 3, -2
^
We start a new subsequence for the first element.
position[0] = 0
sum[0] = -1
At index 1, we consider appending the 2
-1, 2, 3, -2
^
We choose to start a new subsequence because that gives a higher sum than extending.
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
At index 2, we consider appending the 3
-1, 2, 3, -2
^
We choose to extend a subsequence because that gives a higher sum than starting a new one.
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
position[2] = 1 sum[2] = 5
Again, we choose to extend because that gives a higher sum that starting a new one.
-1, 2, 3, -2
^
position[0] = 0 sum[0] = -1
position[1] = 1 sum[1] = 2
position[2] = 1 sum[2] = 5
positions[3] = 3 sum[3] = 3
Again, the best sum is 5 and the list is from index 1 to index 2, which is (2, 3).
Part II: Prefix sums
We want to have a way to compute the sum along a row, for any start point to any endpoint. I want to compute that sum in O(1) time rather than just adding, which takes O(m) time where m is the number of elements in the sum. With some precomputing, this can be achieved. Here's how. Suppose you have a matrix:
a d g
b e h
c f i
You can precompute this matrix:
a d g
a+b d+e g+h
a+b+c d+e+f g+h+i
Once that is done you can get the sum running along any column from any start to endpoint in the column just by subtracting two values.
Part III: Bringing tricks together to find the max submatrix
Assume that you know the top and bottom row of the max submatrix. You could do this:
Ignore rows above your top row and ignore rows below your bottom
row.
With what matrix remains, consider the using sum of each column to
form a sequence (sort of like a row that represents multiple rows).
(You can compute any element of this sequence rapidly with the prefix
sums approach.)
Use Kadane's approach to figure out best subsequence in this
sequence. The indexes you get will tell you the left and right
positions of the best submatrix.
Now, what about actually figuring out the top and bottom row? Just try all possibilities. Try putting the top anywhere you can and putting the bottom anywhere you can, and run the Kadane-base procedure described previously for every possibility. When you find a max, you keep track of the top and bottom position.
Finding the row and column takes O(M^2) where M is the number of rows. Finding the column takes O(N) time where N is the number of columns. So total time is O(M^2 * N). And, if M=N, the time required is O(N^3).
About recovering the actual submatrix, and not just the maximum sum, here's what I got. Sorry I do not have time to translate my code to your java version, so I'm posting my Ruby code with some comments in the key parts
def max_contiguous_submatrix_n3(m)
rows = m.count
cols = rows ? m.first.count : 0
vps = Array.new(rows)
for i in 0..rows
vps[i] = Array.new(cols, 0)
end
for j in 0...cols
vps[0][j] = m[0][j]
for i in 1...rows
vps[i][j] = vps[i-1][j] + m[i][j]
end
end
max = [m[0][0],0,0,0,0] # this is the result, stores [max,top,left,bottom,right]
# these arrays are used over Kadane
sum = Array.new(cols) # obvious sum array used in Kadane
pos = Array.new(cols) # keeps track of the beginning position for the max subseq ending in j
for i in 0...rows
for k in i...rows
# Kadane over all columns with the i..k rows
sum.fill(0) # clean both the sum and pos arrays for the upcoming Kadane
pos.fill(0)
local_max = 0 # we keep track of the position of the max value over each Kadane's execution
# notice that we do not keep track of the max value, but only its position
sum[0] = vps[k][0] - (i==0 ? 0 : vps[i-1][0])
for j in 1...cols
value = vps[k][j] - (i==0 ? 0 : vps[i-1][j])
if sum[j-1] > 0
sum[j] = sum[j-1] + value
pos[j] = pos[j-1]
else
sum[j] = value
pos[j] = j
end
if sum[j] > sum[local_max]
local_max = j
end
end
# Kadane ends here
# Here's the key thing
# If the max value obtained over the past Kadane's execution is larger than
# the current maximum, then update the max array with sum and bounds
if sum[local_max] > max[0]
# sum[local_max] is the new max value
# the corresponding submatrix goes from rows i..k.
# and from columns pos[local_max]..local_max
# the array below contains [max_sum,top,left,bottom,right]
max = [sum[local_max], i, pos[local_max], k, local_max]
end
end
end
return max # return the array with [max_sum,top,left,bottom,right]
end
Some notes for clarification:
I use an array to store all the values pertaining to the result for convenience. You can just use five standalone variables: max, top, left, bottom, right. It's just easier to assign in one line to the array and then the subroutine returns the array with all the needed information.
If you copy and paste this code in a text-highlight-enabled editor with Ruby support you'll obviously understand it better. Hope this helps!
There are already plenty of answers, but here is another Java implementation I wrote. It compares 3 solutions:
Naïve (brute force) - O(n^6) time
The obvious DP solution - O(n^4) time and O(n^3) space
The more clever DP solution based on Kadane's algorithm - O(n^3) time and O(n^2) space
There are sample runs for n = 10 thru n = 70 in increments of 10 with a nice output comparing run time and space requirements.
Code:
public class MaxSubarray2D {
static int LENGTH;
final static int MAX_VAL = 10;
public static void main(String[] args) {
for (int i = 10; i <= 70; i += 10) {
LENGTH = i;
int[][] a = new int[LENGTH][LENGTH];
for (int row = 0; row < LENGTH; row++) {
for (int col = 0; col < LENGTH; col++) {
a[row][col] = (int) (Math.random() * (MAX_VAL + 1));
if (Math.random() > 0.5D) {
a[row][col] = -a[row][col];
}
//System.out.printf("%4d", a[row][col]);
}
//System.out.println();
}
System.out.println("N = " + LENGTH);
System.out.println("-------");
long start, end;
start = System.currentTimeMillis();
naiveSolution(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms no auxiliary space requirements");
start = System.currentTimeMillis();
dynamicProgammingSolution(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms requires auxiliary space for "
+ ((int) Math.pow(LENGTH, 4)) + " integers");
start = System.currentTimeMillis();
kadane2D(a);
end = System.currentTimeMillis();
System.out.println(" run time: " + (end - start) + " ms requires auxiliary space for " +
+ ((int) Math.pow(LENGTH, 2)) + " integers");
System.out.println();
System.out.println();
}
}
// O(N^2) !!!
public static void kadane2D(int[][] a) {
int[][] s = new int[LENGTH + 1][LENGTH]; // [ending row][sum from row zero to ending row] (rows 1-indexed!)
for (int r = 0; r < LENGTH + 1; r++) {
for (int c = 0; c < LENGTH; c++) {
s[r][c] = 0;
}
}
for (int r = 1; r < LENGTH + 1; r++) {
for (int c = 0; c < LENGTH; c++) {
s[r][c] = s[r - 1][c] + a[r - 1][c];
}
}
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int r1 = 1; r1 < LENGTH + 1; r1++) { // rows 1-indexed!
for (int r2 = r1; r2 < LENGTH + 1; r2++) { // rows 1-indexed!
int[] s1 = new int[LENGTH];
for (int c = 0; c < LENGTH; c++) {
s1[c] = s[r2][c] - s[r1 - 1][c];
}
int max = 0;
int c1 = 0;
for (int c = 0; c < LENGTH; c++) {
max = s1[c] + max;
if (max <= 0) {
max = 0;
c1 = c + 1;
}
if (max > maxSum) {
maxSum = max;
maxRowStart = r1 - 1;
maxColStart = c1;
maxRowEnd = r2 - 1;
maxColEnd = c;
}
}
}
}
System.out.print("KADANE SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
// O(N^4) !!!
public static void dynamicProgammingSolution(int[][] a) {
int[][][][] dynTable = new int[LENGTH][LENGTH][LENGTH + 1][LENGTH + 1]; // [row][col][height][width]
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 0; h < LENGTH + 1; h++) {
for (int w = 0; w < LENGTH + 1; w++) {
dynTable[r][c][h][w] = 0;
}
}
}
}
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 1; h <= LENGTH - r; h++) {
int rowTotal = 0;
for (int w = 1; w <= LENGTH - c; w++) {
rowTotal += a[r + h - 1][c + w - 1];
dynTable[r][c][h][w] = rowTotal + dynTable[r][c][h - 1][w];
}
}
}
}
for (int r = 0; r < LENGTH; r++) {
for (int c = 0; c < LENGTH; c++) {
for (int h = 0; h < LENGTH + 1; h++) {
for (int w = 0; w < LENGTH + 1; w++) {
if (dynTable[r][c][h][w] > maxSum) {
maxSum = dynTable[r][c][h][w];
maxRowStart = r;
maxColStart = c;
maxRowEnd = r + h - 1;
maxColEnd = c + w - 1;
}
}
}
}
}
System.out.print(" DP SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
// O(N^6) !!!
public static void naiveSolution(int[][] a) {
int maxSum = Integer.MIN_VALUE;
int maxRowStart = -1;
int maxColStart = -1;
int maxRowEnd = -1;
int maxColEnd = -1;
for (int rowStart = 0; rowStart < LENGTH; rowStart++) {
for (int colStart = 0; colStart < LENGTH; colStart++) {
for (int rowEnd = 0; rowEnd < LENGTH; rowEnd++) {
for (int colEnd = 0; colEnd < LENGTH; colEnd++) {
int sum = 0;
for (int row = rowStart; row <= rowEnd; row++) {
for (int col = colStart; col <= colEnd; col++) {
sum += a[row][col];
}
}
if (sum > maxSum) {
maxSum = sum;
maxRowStart = rowStart;
maxColStart = colStart;
maxRowEnd = rowEnd;
maxColEnd = colEnd;
}
}
}
}
}
System.out.print(" NAIVE SOLUTION | Max sum: " + maxSum);
System.out.print(" Start: (" + maxRowStart + ", " + maxColStart +
") End: (" + maxRowEnd + ", " + maxColEnd + ")");
}
}
Here is a Java version of Ernesto implementation with some modifications:
public int[][] findMaximumSubMatrix(int[][] matrix){
int dim = matrix.length;
//computing the vertical prefix sum for columns
int[][] ps = new int[dim][dim];
for (int i = 0; i < dim; i++) {
for (int j = 0; j < dim; j++) {
if (j == 0) {
ps[j][i] = matrix[j][i];
} else {
ps[j][i] = matrix[j][i] + ps[j - 1][i];
}
}
}
int maxSum = matrix[0][0];
int top = 0, left = 0, bottom = 0, right = 0;
//Auxiliary variables
int[] sum = new int[dim];
int[] pos = new int[dim];
int localMax;
for (int i = 0; i < dim; i++) {
for (int k = i; k < dim; k++) {
// Kadane over all columns with the i..k rows
reset(sum);
reset(pos);
localMax = 0;
//we keep track of the position of the max value over each Kadane's execution
// notice that we do not keep track of the max value, but only its position
sum[0] = ps[k][0] - (i==0 ? 0 : ps[i-1][0]);
for (int j = 1; j < dim; j++) {
if (sum[j-1] > 0){
sum[j] = sum[j-1] + ps[k][j] - (i==0 ? 0 : ps[i-1][j]);
pos[j] = pos[j-1];
}else{
sum[j] = ps[k][j] - (i==0 ? 0 : ps[i-1][j]);
pos[j] = j;
}
if (sum[j] > sum[localMax]){
localMax = j;
}
}//Kadane ends here
if (sum[localMax] > maxSum){
/* sum[localMax] is the new max value
the corresponding submatrix goes from rows i..k.
and from columns pos[localMax]..localMax
*/
maxSum = sum[localMax];
top = i;
left = pos[localMax];
bottom = k;
right = localMax;
}
}
}
System.out.println("Max SubMatrix determinant = " + maxSum);
//composing the required matrix
int[][] output = new int[bottom - top + 1][right - left + 1];
for(int i = top, k = 0; i <= bottom; i++, k++){
for(int j = left, l = 0; j <= right ; j++, l++){
output[k][l] = matrix[i][j];
}
}
return output;
}
private void reset(int[] a) {
for (int index = 0; index < a.length; index++) {
a[index] = 0;
}
}
With the help of the Algorithmist and Larry and a modification of Kadane's Algorithm, here is my solution:
int dim = matrix.length;
//computing the vertical prefix sum for columns
int[][] ps = new int[dim][dim];
for (int i = 0; i < dim; i++) {
for (int j = 0; j < dim; j++) {
if (j == 0) {
ps[j][i] = matrix[j][i];
} else {
ps[j][i] = matrix[j][i] + ps[j - 1][i];
}
}
}
int maxSoFar = 0;
int min , subMatrix;
//iterate over the possible combinations applying Kadane's Alg.
for (int i = 0; i < dim; i++) {
for (int j = i; j < dim; j++) {
min = 0;
subMatrix = 0;
for (int k = 0; k < dim; k++) {
if (i == 0) {
subMatrix += ps[j][k];
} else {
subMatrix += ps[j][k] - ps[i - 1 ][k];
}
if(subMatrix < min){
min = subMatrix;
}
if((subMatrix - min) > maxSoFar){
maxSoFar = subMatrix - min;
}
}
}
}
The only thing left is to determine the submatrix elements, i.e: the top left and the bottom right corner of the submatrix. Anyone suggestion?
this is my implementation of 2D Kadane algorithm. I think it is more clear. The concept is based on just kadane algorithm. The first and second loop of the main part (that is in the bottom of the code) is to pick every combination of the rows and 3rd loop is to use 1D kadane algorithm by every following column sum (that can be computed in const time because of preprocessing of matrix by subtracting values from two picked (from combintation) rows). Here is the code:
int [][] m = {
{1,-5,-5},
{1,3,-5},
{1,3,-5}
};
int N = m.length;
// summing columns to be able to count sum between two rows in some column in const time
for (int i=0; i<N; ++i)
m[0][i] = m[0][i];
for (int j=1; j<N; ++j)
for (int i=0; i<N; ++i)
m[j][i] = m[j][i] + m[j-1][i];
int total_max = 0, sum;
for (int i=0; i<N; ++i) {
for (int k=i; k<N; ++k) { //for each combination of rows
sum = 0;
for (int j=0; j<N; j++) { //kadane algorithm for every column
sum += i==0 ? m[k][j] : m[k][j] - m[i-1][j]; //for first upper row is exception
total_max = Math.max(sum, total_max);
}
}
}
System.out.println(total_max);
I am going to post an answer here and can add actual c++ code if it is requested because I had recently worked through this. Some rumors of a divide and conqueror that can solve this in O(N^2) are out there but I haven't seen any code to support this. In my experience the following is what I have found.
O(i^3j^3) -- naive brute force method
o(i^2j^2) -- dynamic programming with memoization
O(i^2j) -- using max contiguous sub sequence for an array
if ( i == j )
O(n^6) -- naive
O(n^4) -- dynamic programming
O(n^3) -- max contiguous sub sequence
Have a look at JAMA package; I believe it will make your life easier.
Here is the C# solution. Ref: http://www.algorithmist.com/index.php/UVa_108
public static MaxSumMatrix FindMaxSumSubmatrix(int[,] inMtrx)
{
MaxSumMatrix maxSumMtrx = new MaxSumMatrix();
// Step 1. Create SumMatrix - do the cumulative columnar summation
// S[i,j] = S[i-1,j]+ inMtrx[i-1,j];
int m = inMtrx.GetUpperBound(0) + 2;
int n = inMtrx.GetUpperBound(1)+1;
int[,] sumMatrix = new int[m, n];
for (int i = 1; i < m; i++)
{
for (int j = 0; j < n; j++)
{
sumMatrix[i, j] = sumMatrix[i - 1, j] + inMtrx[i - 1, j];
}
}
PrintMatrix(sumMatrix);
// Step 2. Create rowSpans starting each rowIdx. For these row spans, create a 1-D array r_ij
for (int x = 0; x < n; x++)
{
for (int y = x; y < n; y++)
{
int[] r_ij = new int[n];
for (int k = 0; k < n; k++)
{
r_ij[k] = sumMatrix[y + 1,k] - sumMatrix[x, k];
}
// Step 3. Find MaxSubarray of this r_ij. If the sum is greater than the last recorded sum =>
// capture Sum, colStartIdx, ColEndIdx.
// capture current x as rowTopIdx, y as rowBottomIdx.
MaxSum currMaxSum = KadanesAlgo.FindMaxSumSubarray(r_ij);
if (currMaxSum.maxSum > maxSumMtrx.sum)
{
maxSumMtrx.sum = currMaxSum.maxSum;
maxSumMtrx.colStart = currMaxSum.maxStartIdx;
maxSumMtrx.colEnd = currMaxSum.maxEndIdx;
maxSumMtrx.rowStart = x;
maxSumMtrx.rowEnd = y;
}
}
}
return maxSumMtrx;
}
public static void PrintMatrix(int[,] matrix)
{
int endRow = matrix.GetUpperBound(0);
int endCol = matrix.GetUpperBound(1);
PrintMatrix(matrix, 0, endRow, 0, endCol);
}
public static void PrintMatrix(int[,] matrix, int startRow, int endRow, int startCol, int endCol)
{
StringBuilder sb = new StringBuilder();
for (int i = startRow; i <= endRow; i++)
{
sb.Append(Environment.NewLine);
for (int j = startCol; j <= endCol; j++)
{
sb.Append(string.Format("{0} ", matrix[i,j]));
}
}
Console.WriteLine(sb.ToString());
}
// Given an NxN matrix of positive and negative integers, write code to find the sub-matrix with the largest possible sum
public static MaxSum FindMaxSumSubarray(int[] inArr)
{
int currMax = 0;
int currStartIndex = 0;
// initialize maxSum to -infinity, maxStart and maxEnd idx to 0.
MaxSum mx = new MaxSum(int.MinValue, 0, 0);
// travers through the array
for (int currEndIndex = 0; currEndIndex < inArr.Length; currEndIndex++)
{
// add element value to the current max.
currMax += inArr[currEndIndex];
// if current max is more that the last maxSum calculated, set the maxSum and its idx
if (currMax > mx.maxSum)
{
mx.maxSum = currMax;
mx.maxStartIdx = currStartIndex;
mx.maxEndIdx = currEndIndex;
}
if (currMax < 0) // if currMax is -ve, change it back to 0
{
currMax = 0;
currStartIndex = currEndIndex + 1;
}
}
return mx;
}
struct MaxSum
{
public int maxSum;
public int maxStartIdx;
public int maxEndIdx;
public MaxSum(int mxSum, int mxStart, int mxEnd)
{
this.maxSum = mxSum;
this.maxStartIdx = mxStart;
this.maxEndIdx = mxEnd;
}
}
class MaxSumMatrix
{
public int sum = int.MinValue;
public int rowStart = -1;
public int rowEnd = -1;
public int colStart = -1;
public int colEnd = -1;
}
Here is my solution. It's O(n^3) in time and O(n^2) space.
https://gist.github.com/toliuweijing/6097144
// 0th O(n) on all candidate bottoms #B.
// 1th O(n) on candidate tops #T.
// 2th O(n) on finding the maximum #left/#right match.
int maxRect(vector<vector<int> >& mat) {
int n = mat.size();
vector<vector<int> >& colSum = mat;
for (int i = 1 ; i < n ; ++i)
for (int j = 0 ; j < n ; ++j)
colSum[i][j] += colSum[i-1][j];
int optrect = 0;
for (int b = 0 ; b < n ; ++b) {
for (int t = 0 ; t <= b ; ++t) {
int minLeft = 0;
int rowSum[n];
for (int i = 0 ; i < n ; ++i) {
int col = t == 0 ? colSum[b][i] : colSum[b][i] - colSum[t-1][i];
rowSum[i] = i == 0? col : col + rowSum[i-1];
optrect = max(optrect, rowSum[i] - minLeft);
minLeft = min(minLeft, rowSum[i]);
}
}
}
return optrect;
}
I would just parse the NxN array removing the -ves whatever remains is the highest sum of a sub matrix.
The question doesn't say you have to leave the original matrix intact or that the order matters.