Let G be a connected undirected weighted graph. Consider the following two statements.

S₁: There exists a minimum weight edge in G which is present in every minimum spanning tree of G.

S₂: If every edge in G has distinct weight, then G has a unique minimum spanning tree. Which one of the following options is correct?

For constants a ≥ 1 and b > 1, consider the following recurrence defined on the non-negative integers :

$$T\left( n \right) = a.T\left( {\frac{n}{b}} \right) + f\left( n \right)$$

Which one of the following options is correct about the recurrence T(n)?

Let H be a binary min-heap consisting of n elements implemented as an array. What is the worst case time complexity of an optimal algorithm to find the maximum element in H?

Consider the string abbccddeee. Each letter in the string must be assigned a binary code satisfying the following properties:

1. For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter.

2. For any two letters of the same frequency, the letter which occurs earlier in the dictionary order is assigned a code whose length is at most the length of the code assigned to the other letter.

Among the set of all binary code assignments which satisfy the above two properties, what is the minimum length of the encoded string?