Dijkstra's algorithm for finding a shortest-path tree and Prim's algorithm for finding a minimum

Okay, this video we're going to be talking about a couple of statements about Dykstra's algorith

Hello everyone, in the first part of the question we have to use Prim's algorithm to find the mi

So our goal is to find the minimum weight spanning tree using Primps algorithm, which means that

The given sum can take values from from 0. All entries in the sum are 0, in the R0. 2 to D. All

So let's say we have given according to A part, G is connected and connected and regular. So fro

Question

Dijkstra's algorithm for finding a shortest-path tree and Prim's algorithm for finding a minimum spanning tree both effectively build the tree by adding low-weight edges, but they have different goals and need not produce exactly the same output. (a) Give an example of a weighted, undirected graph $G$ with a designated vertex $s$ such that Dijkstra's algorithm on $G$ starting from $s$ and Prim's algorithm on $G$ starting from $s$ produce different results. (Try to make your example as simple as possible.) (b) Suppose that after running Prim's algorithm on $G$ with starting vertex $s$ and producing a minimum spanning tree $T$, you would like to check whether $T$ is also the shortest path tree in $G$ starting from $s$. You could do this using Dijkstra's algorithm, but that would take time $O((m+n)\log n)$. Show (by giving an explicit algorithm) that in fact it is possible to determine whether $T$ is the shortest path tree in only $O(m+n)$ time. Your algorithm can take as input $G$ (including its edge weights), $s$, and $T$, where $T$ is represented by specifying the parent of each vertex. Prove both that your algorithm is correct and that its running time is $O(m+n)$. You can assume that all edge weights and shortest path lengths are distinct.

          Dijkstra's algorithm for finding a shortest-path tree and Prim's algorithm for finding a minimum
spanning tree both effectively build the tree by adding low-weight edges, but they have different
goals and need not produce exactly the same output.
(a) Give an example of a weighted, undirected graph $G$ with a designated vertex $s$ such that
Dijkstra's algorithm on $G$ starting from $s$ and Prim's algorithm on $G$ starting from $s$
produce different results. (Try to make your example as simple as possible.)
(b) Suppose that after running Prim's algorithm on $G$ with starting vertex $s$ and producing a
minimum spanning tree $T$, you would like to check whether $T$ is also the shortest path tree
in $G$ starting from $s$. You could do this using Dijkstra's algorithm, but that would take
time $O((m+n)\log n)$. Show (by giving an explicit algorithm) that in fact it is possible to
determine whether $T$ is the shortest path tree in only $O(m+n)$ time. Your algorithm can
take as input $G$ (including its edge weights), $s$, and $T$, where $T$ is represented by specifying
the parent of each vertex. Prove both that your algorithm is correct and that its running
time is $O(m+n)$. You can assume that all edge weights and shortest path lengths are
distinct.

Dijkstra's algorithm for finding a shortest-path tree and Prim's algorithm for finding a minimum
spanning tree both effectively build the tree by adding low-weight edges, but they have different
goals and need not produce exactly the same output.
(a) Give an example of a weighted, undirected graph G with a designated vertex s such that
Dijkstra's algorithm on G starting from s and Prim's algorithm on G starting from s
produce different results. (Try to make your example as simple as possible.)
(b) Suppose that after running Prim's algorithm on G with starting vertex s and producing a
minimum spanning tree T, you would like to check whether T is also the shortest path tree
in G starting from s. You could do this using Dijkstra's algorithm, but that would take
time O((m+n)log n). Show (by giving an explicit algorithm) that in fact it is possible to
determine whether T is the shortest path tree in only O(m+n) time. Your algorithm can
take as input G (including its edge weights), s, and T, where T is represented by specifying
the parent of each vertex. Prove both that your algorithm is correct and that its running
time is O(m+n). You can assume that all edge weights and shortest path lengths are
distinct.

Added by Jonathan A.

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