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I know my title isn't very specific but that's because I have no idea where the problem comes from. I'm stuck with this problem since 2 or 3 hours and in theory everything should be working, but it's not.
This piece of code:
for ( int x = -1; x <= 1; x++ ) { //Iterate through the 8 neighbour cells plus the one indicated
for ( int y = -1; y <= 1; y++ ) {
neighbour = coords(locX + x, locY + y, width); //Get the cell index in the array
if (existsInOrtho(ortho, neighbour)) { //If the index exists in the array
if (ortho[neighbour] == 0) { //Cell is dead
cnt--; //Remove one from the number of alive neighbour cells
}
} else { //Cell is not in the zone
cnt--; //Remove one from the number of alive neighbour cells
}
}
}
Iterates through all the neighbour cells to get their value in the array (1 for alive, 0 for dead). The "coords" function, shown here:
int coords(int locX, int locY, int width)
{
int res = -1;
locX = locX - 1; //Remove one from both coordinates, since an index starts at 0 (and the zone starts at (1;1) )
locY = locY - 1;
res = locX * width + locY; //Small calculation to get the index of the pixel in the array
return res;
}
Gets the index of the cell in the array. But when I run the code, it doesn't work, the number of neighbour cells is not correct (it's like a cell is not counted every time there's some alive in the neighborhood). I tried decomposing everything manually, and it works, so I don't know what ruins everything in the final code... Here is the complete code. Sorry if I made any English mistake, it's not my native language.
This code ...
for ( int x = -1; x <= 1; x++ ) { //Iterate through the 8 neighbour cells plus the one indicated
for ( int y = -1; y <= 1; y++ ) {
Actually checks 9 cells. Perhaps you forgot that it checks (x,y) = (0,0). That would include the cell itself as well as its neighbours.
A simple fix is:
for ( int x = -1; x <= 1; x++ ) { //Iterate through the 8 neighbour cells plus the one indicated
for ( int y = -1; y <= 1; y++ ) {
if (x || y) {
Also, the simulate function (from your link) makes the common mistake of updating the value of the cell in the same array before processing state changes required for the cells beside it. The easiest fix is to keep two arrays -- two complete copies of the grid (two ortho arrays, in your code). When reading from orthoA, update orthoB. And then on the next generation, flip. Read from orthoB and write to orthoA.
Quick summary:
I create a cv::Mat by
cv::Mat m = cv::Mat::zeros(MAP_HEIGHT, MAP_WIDTH, CV_8UC1)
My approach after this is to see if i have any polygons in a list of polygons, and if i do, fill them in, and lastly i assign m to my public cv::Mat map (defined in the header-file).
What happens is basically:
cv::Mat m = cv::Mat::zeros(MAP_HEIGHT, MAP_WIDTH, CV_8UC1);
// possibly fill polygons with 1's. Nothing happens if there are no polygons
map = m;
The logic of my program is that position x,y is allowed if a 0 is occupying the cell. So no polygons => all map should be 'legit'.
I have defined this method to check whether a given x-y coordinate is allowed.
bool Map::isAllowed(bool res, int x, int y) {
unsigned char allowed = 0;
res = (map.ptr<unsigned char>(y)[x] == allowed);
}
Now the mystery begins.
cout << cv::countNonZero(map) << endl; // prints 0, meaning all cells are 0
for(int i = 0; i < MAP_HEIGHT; i++) {
unsigned char* c = map.ptr<unsigned char>(i);
for(int j = 0; j < MAP_WIDTH; j++) {
cout << c[j] << endl;
}
} // will print nothing, only outputs empty lines, followed by a newline.
If i print (c[j] == NULL) it prints 1.
If i print the entire Mat i see only 0's flashing over my screen, so they are clearly there.
Why does isAllowed(bool, x, y) return false for (0,0), when there is clearly a 0 there?
Let me know if any more information is needed, thanks!
Problem is solved now, here are my mistakes for future reference:
1: When printing, #Miki pointed out that unsigned characters -> ASCII value gets printed, not numerical representation.
2: in isAllowedPosition(bool res, int x, int y), res has a primitive type. Aka this is pushed on the stack and not a reference to a memorylocation. When writing to it, i write to the local copy and not to the one passed in as an argumet.
Two possible fixes, either pass in a pointer to a memorylocation and write to that, or simply return the result.
Since your data type is uchar (aka unsigned char), you're printing the ASCII value. Use
cout << int(c[j]) << endl;
to print the actual value.
Also map.ptr<unsigned char>(y)[x] can be rewritten simply as map.at<uchar>(y,x), or if you use Mat1b as map(y,x)
Given N boxes. How can i find the tallest tower made with them in the given order ? (Given order means that the first box must be at the base of the tower and so on). All boxes must be used to make a valid tower.
It is possible to rotate the box on any axis in a way that any of its 6 faces gets parallel to the ground, however the perimeter of such face must be completely restrained inside the perimeter of the superior face of the box below it. In the case of the first box it is possible to choose any face, because the ground is big enough.
To solve this problem i've tried the following:
- Firstly the code generates the rotations for each rectangle (just a permutation of the dimensions)
- secondly constructing a dynamic programming solution for each box and each possible rotation
- finally search for the highest tower made (in the dp table)
But my algorithm is taking wrong answer in unknown test cases. What is wrong with it ? Dynamic programming is the best approach to solve this problem ?
Here is my code:
#include <cstdio>
#include <vector>
#include <algorithm>
#include <cstdlib>
#include <cstring>
struct rectangle{
int coords[3];
rectangle(){ coords[0] = coords[1] = coords[2] = 0; }
rectangle(int a, int b, int c){coords[0] = a; coords[1] = b; coords[2] = c; }
};
bool canStack(rectangle ¤t_rectangle, rectangle &last_rectangle){
for (int i = 0; i < 2; ++i)
if(current_rectangle.coords[i] > last_rectangle.coords[i])
return false;
return true;
}
//six is the number of rotations for each rectangle
int dp(std::vector< std::vector<rectangle> > &v){
int memoization[6][v.size()];
memset(memoization, -1, sizeof(memoization));
//all rotations of the first rectangle can be used
for (int i = 0; i < 6; ++i) {
memoization[i][0] = v[0][i].coords[2];
}
//for each rectangle
for (int i = 1; i < v.size(); ++i) {
//for each possible permutation of the current rectangle
for (int j = 0; j < 6; ++j) {
//for each permutation of the previous rectangle
for (int k = 0; k < 6; ++k) {
rectangle &prev = v[i - 1][k];
rectangle &curr = v[i][j];
//is possible to put the current rectangle with the previous rectangle ?
if( canStack(curr, prev) ) {
memoization[j][i] = std::max(memoization[j][i], curr.coords[2] + memoization[k][i-1]);
}
}
}
}
//what is the best solution ?
int ret = -1;
for (int i = 0; i < 6; ++i) {
ret = std::max(memoization[i][v.size()-1], ret);
}
return ret;
}
int main ( void ) {
int n;
scanf("%d", &n);
std::vector< std::vector<rectangle> > v(n);
for (int i = 0; i < n; ++i) {
rectangle r;
scanf("%d %d %d", &r.coords[0], &r.coords[1], &r.coords[2]);
//generate all rotations with the given rectangle (all combinations of the coordinates)
for (int j = 0; j < 3; ++j)
for (int k = 0; k < 3; ++k)
if(j != k) //micro optimization disease
for (int l = 0; l < 3; ++l)
if(l != j && l != k)
v[i].push_back( rectangle(r.coords[j], r.coords[k], r.coords[l]) );
}
printf("%d\n", dp(v));
}
Input Description
A test case starts with an integer N, representing the number of boxes (1 ≤ N ≤ 10^5).
Following there will be N rows, each containing three integers, A, B and C, representing the dimensions of the boxes (1 ≤ A, B, C ≤ 10^4).
Output Description
Print one row containing one integer, representing the maximum height of the stack if it’s possible to pile all the N boxes, or -1 otherwise.
Sample Input
2
5 2 2
1 3 4
Sample Output
6
Sample image for the given input and output.
Usually you're given the test case that made you fail. Otherwise, finding the problem is a lot harder.
You can always approach it from a different angle! I'm going to leave out the boring parts that are easily replicated.
struct Box { unsigned int dim[3]; };
Box will store the dimensions of each... box. When it comes time to read the dimensions, it needs to be sorted so that dim[0] >= dim[1] >= dim[2].
The idea is to loop and read the next box each iteration. It then compares the second largest dimension of the new box with the second largest dimension of the last box, and same with the third largest. If in either case the newer box is larger, it adjusts the older box to compare the first largest and third largest dimension. If that fails too, then the first and second largest. This way, it always prefers using a larger dimension as the vertical one.
If it had to rotate a box, it goes to the next box down and checks that the rotation doesn't need to be adjusted there too. It continues until there are no more boxes or it didn't need to rotate the next box. If at any time, all three rotations for a box failed to make it large enough, it stops because there is no solution.
Once all the boxes are in place, it just sums up each one's vertical dimension.
int main()
{
unsigned int size; //num boxes
std::cin >> size;
std::vector<Box> boxes(size); //all boxes
std::vector<unsigned char> pos(size, 0); //index of vertical dimension
//gets the index of dimension that isn't vertical
//largest indicates if it should pick the larger or smaller one
auto get = [](unsigned char x, bool largest) { if (largest) return x == 0 ? 1 : 0; return x == 2 ? 1 : 2; };
//check will compare the dimensions of two boxes and return true if the smaller one is under the larger one
auto check = [&boxes, &pos, &get](unsigned int x, bool largest) { return boxes[x - 1].dim[get(pos[x - 1], largest)] < boxes[x].dim[get(pos[x], largest)]; };
unsigned int x = 0, y; //indexing variables
unsigned char change; //detects box rotation change
bool fail = false; //if it cannot be solved
for (x = 0; x < size && !fail; ++x)
{
//read in the next three dimensions
//make sure dim[0] >= dim[1] >= dim[2]
//simple enough to write
//mine was too ugly and I didn't want to be embarrassed
y = x;
while (y && !fail) //when y == 0, no more boxes to check
{
change = pos[y - 1];
while (check(y, true) || check(y, false)) //while invalid rotation
{
if (++pos[y - 1] == 3) //rotate, when pos == 3, no solution
{
fail = true;
break;
}
}
if (change != pos[y - 1]) //if rotated box
--y;
else
break;
}
}
if (fail)
{
std::cout << -1;
}
else
{
unsigned long long max = 0;
for (x = 0; x < size; ++x)
max += boxes[x].dim[pos[x]];
std::cout << max;
}
return 0;
}
It works for the test cases I've written, but given that I don't know what caused yours to fail, I can't tell you what mine does differently (assuming it also doesn't fail your test conditions).
If you are allowed, this problem might benefit from a tree data structure.
First, define the three possible cases of block:
1) Cube - there is only one possible option for orientation, since every orientation results in the same height (applied toward total height) and the same footprint (applied to the restriction that the footprint of each block is completely contained by the block below it).
2) Square Rectangle - there are three possible orientations for this rectangle with two equal dimensions (for examples, a 4x4x1 or a 4x4x7 would both fit this).
3) All Different Dimensions - there are six possible orientations for this shape, where each side is different from the rest.
For the first box, choose how many orientations its shape allows, and create corresponding nodes at the first level (a root node with zero height will allow using simple binary trees, rather than requiring a more complicated type of tree that allows multiple elements within each node). Then, for each orientation, choose how many orientations the next box allows but only create nodes for those that are valid for the given orientation of the current box. If no orientations are possible given the orientation of the current box, remove that entire unique branch of orientations (the first parent node with multiple valid orientations will have one orientation removed by this pruning, but that parent node and all of its ancestors will be preserved otherwise).
By doing this, you can check for sets of boxes that have no solution by checking whether there are any elements below the root node, since an empty tree indicates that all possible orientations have been pruned away by invalid combinations.
If the tree is not empty, then just walk the tree to find the highest sum of heights within each branch of the tree, recursively up the tree to the root - the sum value is your maximum height, such as the following pseudocode:
std::size_t maximum_height() const{
if(leftnode == nullptr || rightnode == nullptr)
return this_node_box_height;
else{
auto leftheight = leftnode->maximum_height() + this_node_box_height;
auto rightheight = rightnode->maximum_height() + this_node_box_height;
if(leftheight >= rightheight)
return leftheight;
else
return rightheight;
}
}
The benefits of using a tree data structure are
1) You will greatly reduce the number of possible combinations you have to store and check, because in a tree, the invalid orientations will be eliminated at the earliest possible point - for example, using your 2x2x5 first box, with three possible orientations (as a Square Rectangle), only two orientations are possible because there is no possible way to orient it on its 2x2 end and still fit the 4x3x1 block on it. If on average only two orientations are possible for each block, you will need a much smaller number of nodes than if you compute every possible orientation and then filter them as a second step.
2) Detecting sets of blocks where there is no solution is much easier, because the data structure will only contain valid combinations.
3) Working with the finished tree will be much easier - for example, to find the sequence of orientations of the highest, rather than just the actual height, you could pass an empty std::vector to a modified highest() implementation, and let it append the actual orientation of each highest node as it walks the tree, in addition to returning the height.
I am writing a program for K-means clustering, to find the clusters that each point should belong to. There are 8 points and 3 clusters for this code. Somehow in my code the 'entries' function is executing infinitely. I couldn't find where I have gone wrong. This is the logic that I'm following:
Hard coded input of the 8 points
Randomly generate 3 cluster centers
Calculate distance of each point from the 3 cluster centers and use arr1[][] to store the distances.
In cent_tally[][], store the number of the cluster that each point should belong to. eg. 0 for cluster 1, 1 for cluster 2 and 2 for cluster 3. (Also storing the same values in the 4th column of the 2-D array, 'arr1').
Calculate the mean centroids (cluster centers) by using the clusters nos. for each point.
Again call the 'entries' function to calculate the distances and the cluster no. to which each point should belong, but this time using the 2nd set of centroids.i.e. the mean centroids.
If the second set of cluster nos. for each point, (stored in the 2nd column of cent_tally[][]), tallies with the cluster nos. for each point using the randomly generated centroids(first column of cent_tally[][]), then print cent_tally[][], print arr1[][] and stop.
Here is the code:
#include<iostream>
#include<stdio.h>
#include<stdlib.h>
#include<time.h>
#include<math.h>
using namespace std;
class points
{
float x;
float y;
static int point_cnt;
static int flag;
int cent_tally[8][4];
int count2;
struct centroids
{
float cx;
float cy;
}c[3];
public:
points()
{
count2=0;
for(int i=0;i<3;i++)
{
c[i].cx=0;
c[i].cy=0;
}
for(int i=0;i<8;i++)
{
for(int j=0;j<4;j++)
{
cent_tally[i][j]=0;
}
}
}
void means(points * );
float dist(float a,float b,float c,float d);
int entries(float c11x,float c11y,float c22x,float c22y,float c33x,float c33y,float arr[8][4],points *p);
};
int points::point_cnt=8;
int points::flag=0;
int points::entries(float c11x,float c11y,float c22x,float c22y,float c33x,float c33y,float arr[8][4],points *p)
{
float sum1x,sum1y,sum2x,sum2y,sum3x,sum3y; //to calC mean centroids
sum1x=0;
sum1y=0;
sum2x=0;
sum2y=0;
sum3x=0;
sum3y=0;
int cnt1,cnt2,cnt3;
cnt1=0;
cnt2=0;
cnt3=0; //to calC mean centroids
//count2=0;
//in the first iteration of entries, count2=0
cout<<"count 2 value:"<<count2<<endl;
for(int k=0;k<8;k++) //0 to 7 for 8 points
{
arr[k][0]=dist(p[k].x,p[k].y,c11x,c11y);
arr[k][1]=dist(p[k].x,p[k].y,c22x,c22y);
arr[k][2]=dist(p[k].x,p[k].y,c33x,c33y);
float temp,min;
temp = (arr[k][0] < arr[k][1]) ? arr[k][0] : arr[k][1];
min = (arr[k][2] < temp) ? arr[k][2] : temp;
//cout<<"mins:"<<min<<endl;
for(int l=0;l<3;l++)
{
if(arr[k][l]==min)
{
arr[k][3]=l; //0 for c1, 1 for c2, 2 for c3 in 4th column of table
cent_tally[k][count2]=l;
if(l==0)
{
sum1x+=p[k].x;
sum1y+=p[k].y;
cnt1++;
}
else if (l==1)
{
sum2x+=p[k].x;
sum2y+=p[k].y;
cnt2++;
}
else if (l==2)
{ sum3x+=p[k].x;
sum3y+=p[k].y;
cnt3++;
}
else
{
cout<<"";
}
}
}
}
count2++;//for index into cent_tally
//finding mean centroid ...
//re entering values of mean centroid into the same structure created for 3 centroid coordinates ...
c[0].cx=sum1x/cnt1;
c[0].cy=sum1y/cnt1;
c[1].cx=sum2x/cnt2;
c[1].cy=sum2y/cnt2;
c[2].cx=sum3x/cnt3;
c[2].cy=sum3y/cnt3;
//now the struct contains mean centroids
for(int i=0;i<8;i++)
{ int temp=0;
temp=count2-1;
if(cent_tally[i][temp]==cent_tally[i][count2])
{
flag++;
}
else
{
break;
}
}
if(flag==8)
{
cout<<"centroids found: "<<endl;
for(int i=0;i<8;i++)
{
for(int j=0;j<4;j++)
{
cout<<" "<<cent_tally[i][j];
}
cout<<endl;
}
return 0;
}
else
{
return flag;
}
//while(flag!=8) //WHILE ALL 8 entries of latest 2 columns of cent_tally are not matching
//{
//entries(c[0].cx,c[0].cy,c[1].cx,c[1].cy,c[2].cx,c[2].cy,arr,&p[0]);
//}
}
float points::dist(float a,float b,float c,float d)
{
return (abs(a-c)+abs(b-d));
}
void points:: means(points * p)
{
float arr1[8][4]; //array to store dist b/w each point and cluster center and cluster values for each point after distance calculation
float arr2[8][4];
//let c1 c2 and c3 be initial cluster centers
//float c1x,c2x,c1y,c2y,c3x,c3y;
//Can take input from a file also...
p[0].x=2;
p[0].y=2;
p[1].x=1;
p[1].y=14;
p[2].x=10;
p[2].y=7;
p[3].x=1;
p[3].y=11;
p[4].x=3;
p[4].y=4;
p[5].x=11;
p[5].y=8;
p[6].x=4;
p[6].y=3;
p[7].x=12;
p[7].y=2;
srand ( time(NULL) );
for(int i=0;i<3;i++) //for 3 cluster centers, we need 3 centroids
{
int randIndex=1+rand()%(point_cnt-i-1);//where 8 is the no. of points
c[i].cx=p[randIndex].x;
c[i].cy=p[randIndex].y;
}
int val;
val=entries(c[0].cx,c[0].cy,c[1].cx,c[1].cy,c[2].cx,c[2].cy,arr1,&p[0]);
while(val!=8)
{
val=entries(c[0].cx,c[0].cy,c[1].cx,c[1].cy,c[2].cx,c[2].cy,arr1,&p[0]);
}
for(int i=0;i<8;i++)
{
for(int j=0;j<4;j++)
{
cout<<arr1[i][j]<<" ";
}
cout<<endl;
}
//displaying 1st table
//2.1 calculate mean centroid
//2.2 re enter new values in same table
//2.3 first 2 columns of cent_tally
//2.4 if not same repeat step 2.1
}
int main()
{
int c=8;
points p[8];
points obj;
obj.means(&p[0]);
return 0;
}
Another mistake I made, was not initialising flag=0 at the beginning of the 'entries' function!
Now my entries function is not running infinitely, but I have the following problems now:
Mean centroid(Second set of centroids onwards) is computed wrong after the first set of centroids are used
I'm trying to copy the fourth column of arr[][]into the first column and next columns eventually, of cent_tally[][] by using count2 as index , but the first column of cent-tally does not match the 4th column of arr[][]
I'm unable to figure where I have gone wrong.
Due to this logic in entries function
if(flag==8)
{
cout<<"centroids found: "<<endl;
for(int i=0;i<8;i++)
{
for(int j=0;j<4;j++)
{
cout<<" "<<cent_tally[i][j];
}
cout<<endl;
}
return 0;
}
else
{
return flag;
}
8 will never be returned from entries function.
On the other hand, this logic in means function
while(val!=8)
{
val=entries(c[0].cx,c[0].cy,c[1].cx,c[1].cy,c[2].cx,c[2].cy,arr1,&p[0]);
}
loops until 8 is returned from the entries function.
This seems to be the cause of the infinite loop. Consider adjusting the behavior of one of these two point.
Reasons for incorrect mean centroid computation:
Most Important: Inside the for loop where l runs from 0 to 2, if two values of distances are the same, the counts get incremented for two values of l, hence a flag can be used to ensure that only one minimum distance is taken into consideration, for deciding the centroid to which the point belongs to.
Abs takes integer values and hear we are dealing with float, so we need to define a function which handles float values .
Flag should be initialised to 0 in the beginning of 'entries' function.
If two randomly generated centroids are the same, you may not get the right answer.
I have m*n table which each entry have a value .
start position is at top left corner and I can go right or down until I reach lower right corner.
I want a path that if I multiply numbers on that path I get a number that have minimum number of zeros in it's right side .
example:
1 2 100
5 5 4
possible paths :
1*2*100*4=800
1*2*5*4= 40
1*5*5*4= 100
Solution : 1*2*5*4= 40 because 40 have 1 zero but other paths have 2 zero.
easiest way is using dfs and calculate all paths. but it's not efficient.
I'm looking for an optimal substructure for solving it using dynammic programming.
After thinking for a while I came up to this equation :
T(i,j) = CountZeros(T(i-1,j)*table[i,j]) < CountZeros(T(i,j-1)*table[i,j]) ?
T(i-1,j)*table[i,j] : T(i,j-1)*table[i,j]
Code :
#include <iostream>
#include <vector>
#include <algorithm>
#include <numeric>
using namespace std;
using Table = vector<vector<int>>;
const int rows = 2;
const int cols = 3;
Table memo(rows, vector<int>(cols, -1));
int CountZeros(int number)
{
if (number < 0)
return numeric_limits<int>::max();
int res = 0;
while (number != 0)
{
if (number % 10 == 0)
res++;
else break;
number /= 10;
}
return res;
}
int solve(int i, int j, const Table& table)
{
if (i < 0 || j < 0)
return -1;
if (memo[i][j] != -1)
return memo[i][j];
int up = solve(i - 1, j, table)*table[i][j];
int left = solve(i, j - 1, table)*table[i][j];
memo[i][j] = CountZeros(up) < CountZeros(left) ? up : left;
return memo[i][j];
}
int main()
{
Table table =
{
{ 1, 2, 100 },
{ 5, 5, 4 }
};
memo[0][0] = table[0][0];
cout << solve(1, 2, table);
}
(Run )
But it is not optimal (for example in above example it give 100 )
Any idea for better optimal sub-structure ? can I solve it with dynammic programming ?!
Let's reconsider the Bellman optimality equation for your task. I consider this as a systematic approach to such problems (whereas I often don't understand DP one-liners). My reference is the book of Sutton and Barto.
The state in which your system is can be described by a triple of integer numbers (i,j,r) (which is modeled as a std::array<int,3>). Here, i and j denote column and row in your rectangle M = m_{i,j}, whereas r denotes the multiplication result.
Your actions in state (i,j,r) are given by going right, with which you end in state (i, j+1, r*m_{i,j+1}) or by going down which leads to the state (i+1, j, r*m_{i+1,j}).
Then, the Bellman equation is given by
v(i,j,r) = min{ NullsIn(r*m_{i+1,j}) - NullsIn(r) + v_(i+1,j, r*m_{i+1,j})
NullsIn(r*m_{i,j+1}) - NullsIn(r) + v_(i,j+1, r*m_{i,j+1}) }
The rationale behind this equation is the following: NullsIn(r*m_{i+1,j}) - NullsIn(r) denotes the zeros you have to add when you take one of the two actions, i.e. the instant penalty. v_(i+1,j, r*m_{i+1,j}) denotes the zeros in the state you get to when you take this action. Now one wants to take the action which minimizes both contributions.
What you need further is only a function int NullsIn(int) which returns the nulls in a given integer. Here is my attempt:
int NullsIn(int r)
{
int ret=0;
for(int j=10; j<=r; j*=10)
{
if((r/j) * j == r)
++ret;
}
return ret;
}
For convenience I further defined a NullsDifference function:
int NullsDifference(int r, int m)
{
return NullsIn(r*m) - NullsIn(r);
}
Now, one has to do a backwards iteration starting from the initial state in the right bottom element of the matrix.
int backwardIteration(std::array<int,3> state, std::vector<std::vector<int> > const& m)
{
static std::map<std::array<int,3>, int> memoization;
auto it=memoization.find(state);
if(it!=memoization.end())
return it->second;
int i=state[0];
int j=state[1];
int r=state[2];
int ret=0;
if(i>0 && j>0)
{
int inew=i-1;
int jnew=j-1;
ret=std::min(NullsDifference(r, m[inew][j]) + backwardIteration({inew,j,r*m[inew][j]}, m),
NullsDifference(r, m[i][jnew]) + backwardIteration({i,jnew,r*m[i][jnew]}, m));
}
else if(i>0)
{
int inew=i-1;
ret= NullsDifference(r, m[inew][j]) + backwardIteration({inew,j,r*m[inew][j]}, m);
}
else if(j>0)
{
int jnew=j-1;
ret= NullsDifference(r, m[i][jnew]) + backwardIteration({i,jnew,r*m[i][jnew]}, m);
}
memoization[state]=ret;
return ret;
}
This routine is called via
int main()
{
int ncols=2;
int nrows=3;
std::vector<std::vector<int> > m={{1,2,100}, {5,5,4}};
std::array<int,3> initialState = {ncols-1, nrows -1, m[ncols-1][nrows - 1]};
std::cout<<"Minimum number of zeros: "backwardIteration(initialState, m)<<"\n"<<std::endl;
}
For your array, it prints out the desired 1 for the number of zeros.
Here is a live demo on Coliru.
EDIT
Here is an important thing: in production, you usually don't call backwardIteration as I did because it takes an exponentially increasing number of recursive calls. Rather, you start in the top left and call it, then store the result. Next you go left and down and each time call backwardIteration where you now use the previously stored result. And so on.
In order to do this, one needs a memoization concept within the function backwardIteration, which returns the already stored result instead of invoking another recursive call.
I've added memoization in the function call above. Now you can loop through the array from left top to right bottom in any way you like -- but prefereably take small steps, such as row-by-row, column-by-column, or rectangle-for-rectangle.
In fact, this and only this is the spirit of Dynamic Programming.