SDE - 1

You have been given a matrix of ‘N’ rows and ‘M’ columns filled up with integers. Find the minimum sum that can be obtained from a path which from cell (x,y) and ends at the top left corner (1,1).

From any cell in a row, we can move to the right, down or the down right diagonal cell. So from a particular cell (row, col), we can move to the following three cells:

Down: (row+1,col)
Right: (row, col+1)
Down right diagonal: (row+1, col+1)

You are given an array 'arr' of length 'n', consisting of integers.

A subarray is a contiguous segment of an array. In other words, a subarray can be formed by removing 0 or more integers from the beginning and 0 or more integers from the end of an array.

Find the sum of the subarray (including empty subarray) having maximum sum among all subarrays.

The sum of an empty subarray is 0.

Example :

Input: 'arr' = [1, 2, 7, -4, 3, 2, -10, 9, 1]

Output: 11

Explanation: The subarray yielding the maximum sum is [1, 2, 7, -4, 3, 2].

You are given a real number, 'NUM'. You have to represent this number as an irreducible fraction of the form A/B, where 'A' and 'B' are the numerator and denominator respectively.

A fraction is called irreducible when the greatest common divisor (GCD/HCF) of the numerator and denominator is one.

Example :

Given 'NUM' : 1.75
Irreducible fraction  can be represented as 7/4.

Note that 14/8 = 1.75 as well, but 14/8 is not an irreducible fraction.

Note :

In order to preserve precision, the real number will be given to you in the form of two strings : the integer part, and the fractional part. 

The integer part will contain not more than 8 digits, whereas the fractional part will always contain 8 digits.

You have a binary tree of 'N' unique nodes and a Start node from where the tree will start to burn. Given that the Start node will always exist in the tree, your task is to print the time (in minutes) that it will take to burn the whole tree.

It is given that it takes 1 minute for the fire to travel from the burning node to its adjacent node and burn down the adjacent node.

For Example :

For the given binary tree: [1, 2, 3, -1, -1, 4, 5, -1, -1, -1, -1]
Start Node: 3

    1
   / \
  2   3
     / \
    4   5

Output: 2

Explanation :
In the zeroth minute, Node 3 will start to burn.

After one minute, Nodes (1, 4, 5) that are adjacent to 3 will burn completely.

After two minutes, the only remaining Node 2 will be burnt and there will be no nodes remaining in the binary tree. 

So, the whole tree will burn in 2 minutes.

You have been given an integer array/list 'ARR' of size 'N'. Your task is to return the total number of those subsequences of the array in which all the elements are equal.

A subsequence of a given array is an array generated by deleting some elements of the given array with the order of elements in the subsequence remaining the same as the order of elements in the array.

Note :

As this value might be large, print it modulo 10^9 + 7

You have been given an integer 'N'. Your task is to print the Magical Pattern(see examples) for the given 'N'.

Example :

For 'N' : 4
Pattern :
4 3 2 1 2 3 4                                                                   
3 3 2 1 2 3 3                                                                   
2 2 2 1 2 2 2                                                                   
1 1 1 1 1 1 1                                                                   
2 2 2 1 2 2 2                                                                   
3 3 2 1 2 3 3                                                                   
4 3 2 1 2 3 4