Improving Prim algorithm given some details and suggesting an algorithm for a connected graph

3. Suggest an improvement of Prim algorithm if the weight of every edge is either $1 or $2. What is the running time of the algorithm you propose ? 4. You are given a connected graph G(V, E) with weights assigned to its edges. Suggest an O(m log n) time algorithm to find the smallest value r* such that if we remove from G every edge with weight >r, then the graph is still connected. In other words, we want to remove from G all the expensive edges, and prefer to be as cheap as possible