Daniel and Ria are taking a road trip from Somerville to Vancouver (that’s in Canada). Because it’s a 52-hour drive, Daniel and Ria decide to switch off driving at each rest stop they visit; however, because Ria has a better sense of direction than Daniel, she should be driving both when they depart and when they arrive (to navigate the city streets). Given a route map represented as a weighted undirected graph G = (V, E, w) with positive edge weights, where vertices represent rest stops and edges represent routes between rest stops, devise an efficient algorithm to find a route (if possible) of minimum distance between Somerville and Vancouver such that Daniel and Ria                  alternate edges and Ria drives the first and last edge. Specify the space and time complexity.

Consider a graph where vertices represent intersections and edges represent roads. The company Food To Go can place a maximum of one food cart at each intersection. A food cart placed at intersection X has an income equal to the number of roads at intersection X (ie, the degree of vertex X). If two food carts share a road, the income of that road is shared equally among the two. We are interested in calculating the total profit of all food carts placed, ie, the total income minus the cost, where each food cart costs 1 unit and each road provides an income of 1 unit. More formally, the profit for a graph G = (V, E) where the set SC V represents food cart locations, is calculated by the number of edges covered by the vertices in S minus the size of S (ie. |S). A D C E B In this example, placing a food cart on each of the vertices A and C will result in a profit of ISCA P (A D(CD) (CFU VACU-5

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Create a graph using networkx of CT towns that are within the red circle. For each town, output the shortest path to Fairfield. You can use networkx to get the shortest path. Output the number of cities using networkx Output the number of edges using networkx Submit python file or a jupyter notebook with all the code.

Consider a graph with five nodes labeled A, B, C, D, and E. Let's say we have the following edges with their weights: A to B with weight 3 A to C with weight 1 B to C with weight 2 B to D with weight 1 C to E with weight 4 D to E with weight 2 a. Find the shortest path from A to E using Dijkstra's algorithm. (Draw the finished shortest path) b. Use Prim to find the MST (Draw the finished MST) c. Use Kruskal to find the MST (Draw the finished MST) d. What's the difference between Prim and Kruskal algorithms? Do they always have the same result? Why or why not.

a. Build an adjacency matrix ? for this map. b. How many paths of length 2 from V5 to V1 exist? c. How many paths of length 3 from V5 to V1 exist?

(V, E) be a connected, undirected graph. Let A = V, B = V, and f(u) = neighbours of u. Select all that are true. Let G = a) f: AB is not a function Ob) f: A B is a function but we cannot always apply the Pigeonhole Principle with this A, B Odf: A B is a function but we cannot always apply the extended Pigeonhole Principle with this A, B d) none of the above

Consider the following graph: a Specify a simple cycle. Use only lower case letters. If the graph has any cycles which are not simple and which have no repeated edges, specify one. If there are none enter none Use only lower case letters.

Consider a graph with five nodes labeled A, B, C, D, and E. Let's say we have the following edges with their weights: A to B with weight 3 A to C with weight 1 B to C with weight 3 B to D with weight 1 C to E with weight 4 D to E with weight 2 a. Find the shortest path from A to E using Dijkstra's algorithm   (Would anything change if B to C weight was changed from 3 to 4? To 1? What about 5?)

An undirected graph G = (V,E) is said to be k-colorable if all the vertices of G can be colored using k different colors such that no two adjacent vertices have the same color. Design an algorithm that runs in time O(n + e) to color a graph with two colors or determine that the graph is not 2-colorable.