SDE - 1

Consider all numbers from 1 to ‘N’, your task is to find the sum of gcd of all pairs (i, j) such that 1 <= i < j <= N.

For example for N = 3, all pairs are { (1, 2), (1, 3), (2, 3) }.

Note :

Gcd of two numbers (X, Y) is defined as the largest integer that divides both ‘X’ and ‘Y’. 

Kevin has ‘N’ buckets each consisting of some fruits. Kevin wants to eat at least ‘M’ fruits and so, he decided to set a marker (integer) as maximum as possible such that if he eats “number of fruits in the i-th bucket - marker” fruits then it must be at least ‘M’.

Example:

Let’s assume, we have 5 buckets each having 10, 40, 30, and 20 fruits respectively. And, Kevin wants to eat at least 30 fruits. 

Now, if we set our marker at 20 then Kevin can eat: (10 - 20) = -10 => 0 (fruit’s count can’t be negative) fruits from 1st bucket, (40 - 20) = 20 fruits from 2nd bucket, (30 - 20) = 10 fruits from the third bucket, and (20 - 20) = 0 fruits from the last bucket.

So, he can eat 0 + 20 + 10 + 0 = 30 fruits in total.

Note:

Each bucket may not have the same amount of fruits. It is guaranteed that the sum of all fruits (including all the buckets) will be greater than M. Kevin can only eat fruits from a particular bucket if and only if the bucket has more fruits than the marker.

Given a string ‘S’ comprising of some brackets. You need to print the number of every bracket.

For Example:

If S = (pq)() 
Then the output will be 1 1 2 2. First pair of opening and closing brackets will get the same number and so does the 2nd pair.

You are given an arbitrary binary tree with N nodes, whose nodes have their values in the range of integers. You are given two nodes x, y from the tree. You have to print the least common ancestor of these nodes.

Least common ancestor of two nodes x, y in a tree or directed acyclic graph is the lowest node that has both nodes x, y as its descendants.

For example look at the tree below, the LCA of node 1 and 5 is 3.

Note :

You have to return the deepest node which has both x, y as its descendants.
There may be cases where one of u or v is not present in the tree. In those cases, the reference provided to u or v will be NULL.

You are given a directed graph having ‘N’ nodes. A matrix ‘EDGES’ of size M x 2 is given which represents the ‘M’ edges such that there is an edge directed from node EDGES[i][0] to node EDGES[i][1].

Find whether the graph contains a cycle or not, return true if a cycle is present in the given directed graph else return false.

For Example :

In the following directed graph has a cycle i.e. B->C->E->D->B.

Note :

1. The cycle must contain at least two nodes.
2. It is guaranteed that the given graph has no self-loops in the graph.
3. The graph may or may not be connected.
4. Nodes are numbered from 1 to N.
5. Your solution will run on multiple test cases. If you are using global variables make sure to clear them.

You are given two strings 'A' and 'B' where string 'A' is fixed. But you can perform left shift operations on string B any number of times.

Your task is to find out the minimum number of left-shift operations required in order to obtain the longest common prefix of string 'A' and 'B'.

Note:

Left shift is defined as a single circular rotation on the string after which the first character becomes the last character and all other characters are shifted one index to the left.

For Example:

If A = “an”, B = “can”.
After performing one left shift operation, string B becomes “anc”.
After performing two left shift operations, string B becomes “nca”.

Follow Up:

Can you solve this in linear time and space complexity?