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Suppose we run Dijkstra's algorithm from a start node s in an weighted undirected graph. The distance to each reachable node t is given by distTo.get(t). What do we know about the shortest path between any two other nodes u and v-assuming neither is the start node? Explain your reasoning for all 4 answer choices. (It is sufficient to give a counterexample for incorrect answers.) There can only exist a single shortest path from u to v. A shortest path from u to v must include one or more edges in the shortest paths tree from s. If the shortest paths tree reaches u and v, then u and v are connected. The distance of a shortest path from u to v cannot be greater than distTo.get(u) + distTo.get(v).

          Suppose we run Dijkstra's algorithm from a start node s in an weighted undirected graph. The distance to each reachable node t is given by distTo.get(t).
What do we know about the shortest path between any two other nodes u and v-assuming neither is the start node? Explain your reasoning for all 4 answer choices. (It is sufficient to give a counterexample for incorrect answers.)
There can only exist a single shortest path from u to v.
A shortest path from u to v must include one or more edges in the shortest paths tree from s.
If the shortest paths tree reaches u and v, then u and v are connected.
The distance of a shortest path from u to v cannot be greater than distTo.get(u) + distTo.get(v).

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Suppose we run Dijkstra's algorithm from a start node s in an weighted undirected graph. The distance to each reachable node t is given by distTo.get(t).
What do we know about the shortest path between any two other nodes u and v-assuming neither is the start node? Explain your reasoning for all 4 answer choices. (It is sufficient to give a counterexample for incorrect answers.)
There can only exist a single shortest path from u to v.
A shortest path from u to v must include one or more edges in the shortest paths tree from s.
If the shortest paths tree reaches u and v, then u and v are connected.
The distance of a shortest path from u to v cannot be greater than distTo.get(u) + distTo.get(v).

Added by Danielle H.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

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Solved by Expert Christian Otero

06/10/2022

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There can only exist a single shortest path from u to U: This is not necessarily true. There can be multiple shortest paths between two nodes in a graph. For example, consider a graph with three nodes A, B, and C, where A is connected to B with weight 1, B is Show more…

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Suppose we run Dijkstra's algorithm from a start node s in a weighted undirected graph. The distance to each reachable node t is given by distTo.get(t). What do we know about the shortest path between any two other nodes u and v—assuming neither is the start node? Explain your reasoning for all 4 answer choices. (It is sufficient to give a counterexample for incorrect answers.) There can only exist a single shortest path from u to v. A shortest path from u to v must include one or more edges in the shortest paths tree from s. If the shortest paths tree reaches u and v, then u and v are connected. The distance of a shortest path from u to v cannot be greater than distTo.get(u) + distTo.get(v).

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Transcript

00:01 Okay, this video we're going to be talking about a couple of statements about dykstra's algorithm that we want to evaluate whether or not they are true or not.

00:12 So we're looking at dykstra, and they give us this notation where if you run it, we run diktra's on a weighted.

00:25 So we have a weighted undirected graph, and we run it from a start node s.

00:34 So start node is s and they give us this function or this method dist 2 .get and this gives us the shortest path from s to t in this graph and so we want to look at a couple of statements about this graph and dextras and one evaluate whether or not they're true so the first statement is there can only exist a single shortest path from u to v so in this case, u and v are two nodes in the graph, and none of them are start.

01:21 None of them are the start node.

01:22 So we have u and v, so uv, and the first statement is that there's only one single shortest path.

01:33 One single shortest path, u to v.

01:38 And this is false.

01:43 So this is false because there can exist multiple shortest paths.

01:48 In one graph.

01:50 So for example, let's say we have these two nodes, u and v.

01:55 We could have two different shortest paths.

01:59 So let's put these two nodes here.

02:01 It's called us x and y.

02:03 We can have these four edges...

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