3.3. Proof of Correctness of Prim's Algorithm

So, we are looking at the problem of constructing a minimum cost spanning tree in a weighted graph. We said there we have two basic strategies one could think of to do this. The first one leads to an algorithm called Prim's algorithm, and the second one leads to an algorithm called Kruskal's algorithm.

Prim's algorithm addresses the problem of creating a minimum-cost spanning tree in a weighted graph. In the context of graph theory, a spanning tree is a subset of the graph that connects all vertices without forming any cycles. Prim's algorithm is one of two main strategies used to find these spanning trees, the other being Kruskal's algorithm. The key aim is to ensure that the total weight of the edges in the tree is minimized.

Imagine you are planning a network of roads in a new town, and you want to connect all the important locations with the least amount of pavement. Prim's algorithm helps you build the most efficient network of roads, just as it finds the most efficient spanning tree in a graph.

We have a weighted undirected graph. So, V is the set of vertices, E is the set of edges and w is a weight function. We assumed that G is connected, because G is not connected and there is no way to build a spanning tree.

In graph theory, a weighted undirected graph consists of vertices (points) and edges (connections between points) that have assigned weights (values representing costs or distances). For a spanning tree to exist, the graph must be connected, meaning all vertices can be reached from any other vertex without isolated groups. If the graph is not connected, forming a spanning tree is impossible.

Think of a city where all locations (vertices) are connected by roads (edges). If there are some areas that cannot be reached from others (disconnected components), you cannot create a complete road network that allows travel between all locations, analogous to the inability to form a spanning tree in a disconnected graph.

So, this strategy in Prim's algorithm starts with the minimum cost edge, and keeps extending the tree with the smallest edge connected to the current tree.

Prim's algorithm begins by selecting the edge with the lowest weight from the graph. The tree grows iteratively by adding the smallest edge that connects a new vertex to the already built part of the tree. This process continues until all vertices are included, thereby forming the minimum cost spanning tree.

Imagine starting with a single road leading out from a central point in a city. Each time you choose the least expensive option to connect to another area, gradually expanding the road network while ensuring all parts of the city are connected at the lowest cost.

In order to prove Prim's algorithm correct and indeed, we will also use this to prove Kruskal's algorithm, correct later on. We prove a very useful remark called the minimum separator lemma.

The minimum separator lemma states that if a connected graph is divided into two non-empty disjoint sets of vertices, then the edge with the smallest weight crossing this partition must be included in every minimum cost spanning tree. This lemma is critical in justifying the correctness of Prim's algorithm, as it ensures the chosen edges during the process lead to an optimal tree.

Think of a bridge that connects two islands. If this bridge is the cheapest route between the islands, then any complete road network between the islands must include this bridge to minimize costs, paralleling how the minimum separator lemma ensures that the least expensive edge is included in the optimal spanning tree.

Suppose there must be some minimum cost spanning tree, because we know that the graph is connected. So there are many spanning trees unless we assume that the minimum cost spanning tree T which does not include this edge.

To establish the lemma, we consider a minimum cost spanning tree that does not include the smallest edge between two sets of vertices. If that is the case, there exists a path in the tree connecting vertices from one set to the other. By removing an edge from this tree and adding the smallest edge, we create a new tree with a lower total cost. This contradiction implies that the smallest edge must be part of any minimum spanning tree.

If you choose the longest, most expensive route to connect two parts of a town instead of the shortest bridge, you could easily discover that switching to that bridge reduces the overall travel distance, demonstrating that the cheapest connection is essential.

Once we had this lemma, the correctness of Prim's algorithm is very obvious. At every stage, Prim's algorithm, we have built this tree that consists of a few edges, and then we want to connect one of that.

By using the minimum separator lemma, we can confidently assert that each edge selected by Prim's algorithm is necessary for constructing the minimum spanning tree. Since at each step of the algorithm, the edge added is guaranteed to be the least weight edge connecting a vertex in the tree to one outside the tree, the overall structure remains optimal.

Imagine always investing in the cheapest links when building a water supply system so that every town gets water with minimum expenditure. This systematic choice mirrors how Prim's algorithm ensures the creation of a cost-effective network by continuously selecting the cheapest connection.

If I take any vertex, right and I look at all the edges going out of it, then I can take u to v the vertex itself and I can take w to be everything else. We set minus this vertex.

Prim's algorithm can be slightly modified to enhance flexibility. Instead of always starting with the minimum edge, you can begin from any vertex and look for the smallest edge connected to this vertex. The approach remains effective since the minimum separator lemma guarantees that this edge will also lead to an optimal spanning tree.

This is like starting from any city in a country and choosing the cheapest road to build an efficient national highway system, adjusting the starting point whenever necessary while still ensuring the overall cost remains minimized.

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