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? 

Let G = (V, E) be a weighted undirected graph and let T be a Minimum Spanning Tree (MST) of G maintained using adjacency lists. Suppose a new weighted edge (u, v) $$ \in $$ V $$ \times $$ V is added to G. The worst case time complexity of determining if T is still an MST of the resultant graph is

Consider a graph G = (V, E), where V = {v₁, v₂, ...., v₁₀₀},
E = {(v_i, v_j) | 1 ≤ i < j ≤ 100}, and weight of the edge (v_i, v_j) is |i - j|. The weight of the minimum spanning tree of G is ______.

Consider the following undirected graph $$G: $$

Choose a value for $$x$$ that will maximize the number of minimum weight spanning trees $$(MWSTs)$$ of $$G.$$ The number of $$MWSTs$$ of $$G$$ for this value of $$x$$ is ______.