Suppose we want to find the minimum spanning tree of the...

Suppose we want to find the minimum spanning tree of the following graph. (a) Run Prim's algorithm; whenever there is a choice of nodes, always use alphabetic ordering (e.g., start from node $A$ ). Draw a table showing the intermediate values of the cost array. (b) Run Kruskal's algorithm on the same graph. Show how the disjoint-sets data structure looks at every intermediate stage (including the structure of the directed trees), assuming path compression is used.

Suppose we want to find the minimum spanning tree of the following graph.
(a) Run Prim's algorithm; whenever there is a choice of nodes, always use alphabetic ordering (e.g., start from node $A$ ). Draw a table showing the intermediate values of the cost array.
(b) Run Kruskal's algorithm on the same graph. Show how the disjoint-sets data structure looks at every intermediate stage (including the structure of the directed trees), assuming path compression is used.

Key Concepts

Path Compression

Path compression is a technique used in the disjoint-set data structure to flatten the structure of the trees representing the sets, thereby speeding up subsequent find operations. It optimizes the efficiency of union-find operations by making each node point directly to the root after a find operation, which is especially important when many operations are performed.

Disjoint-Set Data Structure

The disjoint-set data structure, also known as Union-Find, is used to efficiently manage a collection of non-overlapping sets. In the context of Kruskal's algorithm, it is employed to quickly determine whether two vertices are in the same set (which would create a cycle if connected) and to merge sets when a new edge is added to the MST.

Kruskal's Algorithm

Kruskal's algorithm constructs an MST by sorting all edges in increasing order of weight and adding them one by one, ensuring that no cycles form. It works by treating each vertex as a separate set and merging them as edges that do not form a cycle are added, which gradually builds the MST while maintaining minimal total cost.

Greedy Algorithm

Greedy algorithms build a solution piece by piece, always choosing the next piece that offers the most immediate benefit. In the context of MST, both Prim's and Kruskal's algorithms use greedy strategies to ensure that at each step, the smallest weight edge is chosen to extend the growing tree or forest, leading to an overall minimal spanning tree.

Minimum Spanning Tree

A minimum spanning tree (MST) is a subset of the edges of a connected, weighted graph that connects all the vertices together without any cycles and with the minimum possible total edge weight. It is a fundamental concept in graph theory used in network design, clustering, and many optimization problems.

Prim's Algorithm

Prim's algorithm finds an MST by starting at a chosen vertex and repeatedly adding the smallest edge that connects a vertex in the tree to a vertex outside the tree. Key to its process is maintaining a cost array (or priority queue) that tracks the smallest weights available to connect remaining vertices, updating it as new vertices are added to the tree.

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