Find a spanning tree with minimal total weight containing the edges {e, i} and {g, k} in the wei

Find a spanning tree with minimal total weight containing...

Key Concepts

Minimum Spanning Tree

A minimum spanning tree (MST) is a spanning tree with the smallest possible total edge weight. It is used in various optimization problems where the goal is to connect all points (vertices) with the least total cost, and common algorithms such as Kruskal's and Prim's are often employed to determine the MST.

Graph Algorithms for MST

Graph algorithms like Kruskal's and Prim's are designed to construct minimum spanning trees by iteratively selecting the smallest available edge that doesn't form a cycle for Kruskal's, or by growing the MST from a starting vertex for Prim's. These approaches rely on sorting edges or employing priority queues to ensure that at every step, the choice minimizes the total weight.

Forced Edge Inclusion in MST

In some MST problems, specific edges are required to be part of the spanning tree. This constraint forces the algorithm to begin with these edges, and the subsequent process must ensure that the remaining edges are selected in such a way as to complete the spanning tree while still aiming to minimize the overall weight. This often involves modifying the standard MST algorithms to account for pre-included edges.

Spanning Tree

A spanning tree is a subgraph that includes all the vertices of the original graph and is a tree (i.e., it is connected and has no cycles). It contains exactly (n - 1) edges for a graph with n vertices, ensuring that the entire graph is connected without any redundant connections.

Transcript

00:04 We're given a weighted graph and we're asked to find a spanning tree with minimal total weight that contains two edges from this weighted graph.

00:14 So the weighted graph is the one in figure 3 from the book in this section.

00:20 And we're asked to find a spanning tree with the minimal total weight contains the edges ei and gk.

00:30 So to do this, i'll use crucible algorithm.

00:34 So first of all, i'll construct a graph, which can consists only of the vertices of the weight graph.

00:44 So this is the vertices, abc, e, f, g, h, i, j, k, and l.

00:51 And start off, we'll first add the edges, ei, and gk, as we're required to do.

01:08 And these both have a weight of four.

01:23 And now we'll be applying kriscol's algorithm to determine the remaining edges.

01:27 So we have that the smallest weight, of the remaining edges of 1 occurs between b and f, c and d, and k and l.

01:40 So first if we add bf, we see that we don't create a circuit, so this is valid.

01:50 If we add cd, we don't create a circuit, so this is also valid.

01:56 And if we add kl, we don't create a circuit, so this is valid too.

02:06 Now in the remaining graph, the smallest weight, which is 2, is between the same.

02:12 Between a and b between c and g and between f and j.

02:18 So first, if we add a and b, we don't obtain a circuit...

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