input:

17 22 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 31 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 9 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 9 0 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 20 0 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 20 0 1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 22 0 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5 0 1 2 3 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48

output:[17, 22]

Question 1 (Programming Assignment) [40 points]

Note

This assignment can be solved in either Java, Python or C++ (you should pick the language you are most comfortable with). Please make sure to look at the supporting documentation and files for the language of your choosing.

The Problem

In this problem, we will explore weighted graphs.

We are given a starting node ss and an ending node ee, for some undirected graph GG with nn nodes. Further, each node uu has its own weight, wuwu (0 <= wu <= 50). The graph is formatted as an adjacency list, with the leading number being the node weight, meaning that for each node, uu, we can access all of uu's neighbors, as well as uu's weight. The goal is to output the array of nodes in a shortest path from ss to ee. (The weight of a path P=(u1,,u)P=(u1,,u) is i=1wuii=1wui and the shortest path is the one with the minimum weight.)

Input

The input file is given with the first two lines being the start node and the end node, and the remainder of the file is an adjacency list with node weights for graph GG (we assume that the set of vertices is {0,1,,n1}{0,1,,n1}). The adjacency list assumes that the current node is the index of the line under consideration. For instance the third line of the input file has the list of all nodes adjacent to node 0 (the first two being the start and end nodes) and so on.

 s <- Staring node (some node between u0 and un-1). e <- Ending node (some node between u0 and un-1). w0 u1 u4 u6 <- All nodes that share edges with u0 w1 u3 uu <- All nodes that share edges with u1 w2 u0 <- All nodes that share edges with u2 . . . wn-1 u0 u4 u2 u7 <- All nodes that share edges with un-1 

Output

The output should be a list/ArrayList/vector (depending on your language of choice) where every 2 adjacent nodes have an edge between them. The 1st node in the output should be startNode and the last node in the output should be endNode.

For example, if startNode was 2 and endNode was 7, and the minimum path between them is 2-9-8-4-7 then you should output [2, 9, 8, 4, 7]. If there is no path between the start and end nodes, output an empty output.

 [s, n0 n1 n2... nm, e] <- Each entry is a node on the shortest path between 's' and 'e' 

Note

There is no guarantee that the graph is completely connected. In the situation where a node can not be reached from the starting node, you should return an empty array.

Note

For this problem there can be multiple shortest paths so it is possible that your solution (assuming it is correct) might not match the corresponding sample output.

Hint

The best possible algorithm for this problem that we are aware of runs in time O((m+n)logn)O((m+n)logn) where mm in the total number of edges and nn is the number of vertices.

Note

Both the input and output parsers in each of the three languages are already written for you. Note that you have to work with the input data structures provided (which will come pre-loaded with the data from an input file).

Addition is the only change you should make

Irrespective of what language you use, you will have to submit just one file. That file will come pre-populated with some stuff in it. You should not change any of those things because if you do you might break what the grader expects and end up with a zero on the question. You should of course add stuff to it (including helper functions and data structures as you see fit).