new question added

Author: arorarahul

README.md

@@ -178,6 +178,7 @@ nothing but fibonacci series.
 - In some algos involving DP you can start from n and in that case answer to n is dependent on n-1 and so on.
 - Most problems involving strings can be taken as S(i,j), either we compare last characters then if equal
 it gets converted to i-1,j-1 or we take min or max of i,j-1 and i-1,j
+- Somtimes if the problem is already a table, you need to reverse engineer it to find the min number of ways to do something. Example create a new table and start from the very bottom and construct the solution updwards. Answer in that case will be the cell 0,0
 
 # Topic0: Programming Questions
 
@@ -401,6 +402,7 @@ it gets converted to i-1,j-1 or we take min or max of i,j-1 and i-1,j
 - [Count the number of times string occured as the subsequence of the other string](/dynamic-programming/question25.c)
 - [Given an amount and some coints, find out in how many ways can this amount be formed](/dynamic-programming/question26.c)
 - [Given a 2xn board and tiles of size 2x1, count the number of ways to fill the board using 2x1 tiles](/dynamic-programming/question27.c)
+- [Given a cost matrix mxn having a cost at each cell. Find the min cost that it will take to reach cell (m-1,n-1) from top left corner cell (0,0) if the only allowed directions to move are right, down and diagnal down](/dynamic-programming/question28.c)
 
 ## Some important concepts to solve algos better
 

dynamic-programming/question28.c

@@ -0,0 +1,128 @@
+/*
+Given a cost matrix mxn having a cost at each cell. Find the min cost that it will take to 
+reach cell (m-1,n-1) from top left corner cell (0,0) if the only allowed directions to move are right, 
+down and diagnal down
+
+METHOD:
+In this case we cannot use greedy as each time we will try to find the min from the node and traverse that
+path, but we may end up with a node having max as the only option or min as the max number which may not
+be the shortest path
+
+Hence we use DP:
+Assume the matrix is something like this:
+
+10  3   4
+5   6   7
+13  4  11
+
+and we need to reach from 0,0 to n-1,m-1
+
+Therefore at each node I have three choices, recursive equation may look like this:
+
+C(0,0) = min{
+	10 + C(0,1),
+	10 + C(1,1),
+	10 + C(1,0)
+}
+
+Hence if we convert it into a tree, we can see repeating sub problems. The number of unique problems are clearly
+n*m and c(i,j)// i can take n values and j can take m values (entire row and columns)
+
+Now we make a matrix of size m*n and reverse engineer the solution. If we start from 0,0, it is the bigger
+problem we need to solve, but if we start from n-1,m-1, it is the smallest problem. Hence we try to construct
+the bottom most row, as 
+
+28 15 11
+
+as from cell in the middle it will take (4+11) 15 cost to reach 11 and similarly from cell 13 it will take(15+13)
+
+Now for the next rows, we keep taking the min of the three ways we have been given in the question and 0,0
+will be the answer.
+
+Time complexity: O(mn)
+Space complexity: O(mn)
+
+Here we can even construct the path by just finding min starting from 0,0 in this matrix
+We can also save the space complexity in case path is not required by just maintaing the values of two rows
+at a time at any given time.
+*/
+#include <stdio.h>
+#include <stdlib.h>
+
+int min(int a, int b, int c){
+	return (a<b?(a<c?a:c):(b<c?b:c));
+}
+
+int findCost(int rows, int columns,int arr[rows][columns]){
+
+	int cost[rows][columns];
+
+	int i,j;
+
+	cost[rows-1][columns-1] = arr[rows-1][columns-1];
+
+	for(i=rows-1;i>=0;i--){
+		for(j=columns-1;j>=0;j--){
+			if(i == rows-1){
+				cost[i][j] = arr[i][j] + cost[i][j+1];
+			}
+			else if(j == columns-1){
+				cost[i][j] = arr[i][j] + cost[i+1][j];
+			}else{
+				cost[i][j] = arr[i][j] + min(cost[i+1][j+1],cost[i][j+1],cost[i+1][j]);	
+			}
+		}
+	}
+
+
+	//printing the array
+	for(i=0;i<rows;i++){
+		for(j=0;j<columns;j++){
+			printf("%d ", cost[i][j]);
+		}
+		printf("\n");
+	}
+
+	printf("the path is: \n");
+	i=0,j=0;
+	while(i<=rows-1 && j<=columns-1){
+		printf("%d ", arr[i][j]);
+		if(cost[i+1][j+1] < cost[i+1][j]){
+			if(cost[i+1][j+1] < cost[i][j+1]){
+				i = i+1;
+				j = j+1;
+			}else{
+				i = i;
+				j = j+1;
+			}
+		}else{
+			if(cost[i+1][j] < cost[i][j+1]){
+				i = i+1;
+				j = j;
+			}else{
+				i = i;
+				j = j+1;
+			}
+		}
+	}
+
+	return cost[0][0];
+}
+
+int main(){
+
+	int cost[3][3] = {
+		{10,3,4},
+		{5,6,17},
+		{13,4,11},
+	};
+
+	int rows = sizeof(cost)/sizeof(cost[0]);
+	int columns = sizeof(cost[0])/sizeof(cost[0][0]);
+
+	int total = findCost(rows,columns,cost);
+	printf("total cost of reaching is %d\n", total);
+
+	return 0;
+}
+

nextquestions.md

@@ -2,6 +2,10 @@ GOOD QUESTIONS:
 - question18.c (arrays)
 
 
+C concepts
+- dynamic memory allocation for 2d arrays
+- fgets,scanf, getc etc difference and when to use what
+
 TODO: 
 - question14.c (general), 
 - question17.c (arrays), to be done after trees and linkedlist is done