Problem 2 (20 points) 1. [15 points] Consider a DAG which has already been topologically sorted, so edges go from left to right. In the lecture notes, we have seen a Dynamic Programming algorithm that finds the shortest path in a DAG where edges have weights associated with them. Now consider the following greedy strategies. Show that all of them fail to find the shortest path from node 1 to node n by giving appropriate small examples. You should also explain what is the value of the optimal solution compared to the greedy one. Greedy strategy 1 Set i=1 While in Pick the edge (i, i) with the largest ; Set i-; and repeat Return the weight of the path constructed from node 1 to node n. Greedy strategy 2 Set in While iun, Pick the edge (i, j) with the smallest weight Set i-; and repeat Return the weight of the path constructed from node 1 to node n. Greedy strategy 3 Set i=1 While irn, Pick the edge (i, j) with the smallest ratio weight/(-i) Set i-; and repeat Return the weight of the path constructed from node 1 to node n. 2. 15 points] Construct the Huffman tree and show the encoding of each letter based on the frequencies below: Letter Frequency 25 10 13 6 20 14 5 3 4 If you have to encode a file consisting of 10' characters, how much space would the Huffman encoding require? What about the fixed size encoding