Write me a C code for an weighted, un-directed graph that performs dijkstra's algorithm on graph along with some basic functions as mentioned in description.

Problem Statement

• Input: A directed graph G = (V, E) with weight function w : E → N.

• Goal: Serve the following requests:

– Given u, v ∈ V , what is the weight of edge (u, v)?

– Given v ∈ V , run Dijkstra’s algorithm on G from vertex v.

– Given u, v ∈ V , output shortest path from a vertex u to v.

Input Format

General format:

A graph G = (V, E) is defined by first providing the number of vertices n = |V |.

Henceforth you shall assume that the vertex set is V = {1, 2, . . . , n}.

The edge set E and the weight function w are provided together.

Each neighbour v of a vertex u will be given in the adjacency list of u along with w(u, v).

Requests arrive only after the definition of G.

Format in detail:

Each line of the input looks like one of the following:

• ‘N’ followed by number of vertices n ∈ N.

• ‘E’ followed by a vertex and vertex-weight pairs that look like:

u, v1,w(u, v1), v2,w(u, v2) · · · , vk,w(u, vk)

This list gives the adjacency list of vertex u along with the respective edge weights.

The list is given as a space-separated list.

• ‘?’ followed by u, v ∈ [n] with a space separating them.

• ‘D’ followed by a u ∈ [n].

• ‘P’ followed by u, v ∈ [n] with a space separating them.

Each of the lines above ends with a ‘\n’ character. All lists are given as elements separated

by a space. No commas are used anywhere. All numbers used will fit inside an int. End of

input is indicated by EOF.

Output Format

• If input line started with ‘N’ or ‘E’, then no corresponding output.

• If input line was “? u v”: Output w(u, v) if (u, v) ∈ E, −1 otherwise.

• If input line was “D u”:

Let v1, v2, . . . , vn be the order of vertices visited on a run of of Dijkstra’s algorithm

from u (note that u = v1). Let δ(u, v) denote the shortest path from u to v. Output

a list of pairs: (v1, δ(u, v1)), . . . ,(vn, δ(u, vn)). If δ(u, v) = ∞ then output −1 in its

place. Use a space to separate v and δ(u, v) within a pair. Use a ‘\n’ between pairs.

• If input line was “P u v”:

If v is not reachable from u, output −1. Else output the shortest distance from u to

v followed by a shortest path from u to v as a space-separated list of vertices starting

with u.

All output lines have to end with a ‘\n’ character.

Implementation rules

• The input graph G = (V, E) is stored similar to Assignment 4. The edge weights are

stored in an additional variable inside the nodes in the adjacency list (see CLRS).

• Store each vertex’s adjacency list in the same order as provided in the input.

• Dijkstra’s algorithm requires a min-priority queue. Implement a min-priority queue by

using a min-heap.

Skills: Algorithm, C Programming, Mathematics

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