Prove: Proposition Q. ( Generic shortest-paths algorithm) Initialize distTo[s] to 0 and all other distTo[] values to infinity, and proceed as follows: Relax any edge in G, continuing until no edge is eligible. For all vertices w reachable from s, the value of distTo[w] after this computation is the length of a shortest path from s to w (and the value of edgeTo[] is the last edge on that path).

Algorithm : (1. Single-destination Shortest Path, 2. Bellman-Ford, 3. Negative-Weight Cycles), Dynamic Programming   Define and prove a recurrence for the following problem: Given a directed graph G = (V, E) with edge weight function w : E → R and a source vertex s ∈ V , find a shortest path from s to v for every vertex v ∈ V .

please explain fully

please answer both of the questions.   7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3
, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done..   8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree.    Please give proper explanation and typed answer only.