(30 pts.) Minimum Spanning Trees Suppose you are building an electrical grid, and have to decide where to build power lines so that all of your customers are connected to the power plant. The customers and the power plant can be represented by vertices in an undirected graph G, where edges indicate possible locations of power lines and are weighted by the length of cable needed. Your main goal is to minimize the length of power lines you need to build, but there are also some edges in G which pass over unfavorable terrain and which you'd prefer to avoid, if possible. So if there are multiple layouts which would use the same amount of cable, you'd prefer the one with the least number of such bad edges. Describe an algorithm for choosing which edges of G to use, and give a short (few sentences) argument for why it is correct. (Hint: Think about how you can modify G so that an MST of the new graph is an MST of the original with as few bad edges as possible.) (30 pts.) Minimum Spanning Trees Suppose you are building an electrical grid, and have to decide where to build power lines so that all of your customers are connected to the power plant. The customers and the power plant can be represented by vertices in an undirected graph G, where edges indicate possible locations of power lines and are weighted by the length of cable needed. Your main goal is to minimize the length of power lines you need to build, but there are also some edges in G which pass over unfavorable terrain and which you'd prefer to avoid, if possible. So if there are multiple layouts which would use the same amount of cable, you'd prefer the one with the least number of such bad edges. Describe an algorithm for choosing which edges of G to use, and give a short (few sentences) argument for why it is correct. (Hint: Think about how you can modify G so that an MST of the new graph is an MST of the original with as few bad edges as possible.)