Prim’s algorithm can be efficiently implemented using _____ for graphs with greater density.a.linear searchb.d-ary heapc.Fibonacci heapd.binary search

Break Down the Problem

1. Identify the context of the question: Prim's algorithm is used for finding the Minimum Spanning Tree (MST) in a graph.
2. Recognize that the question pertains to the efficient data structure implementation of Prim's algorithm.

Relevant Concepts

1. Prim's Algorithm: This algorithm is more efficient for dense graphs compared to sparse graphs, as it can directly benefit from data structures that provide efficient key updates and minimum retrievals.
2. Data Structures Options:
   • Linear Search: Not efficient for this problem.
   • D-ary Heap: A generalization of binary heaps but doesn't provide the best performance improvements over binary heaps for Prim's.
   • Fibonacci Heap: This data structure allows for very efficient decrease-key operations and can improve the performance of Prim's algorithm significantly for dense graphs.
   • Binary Search: Not applicable in the context of priority queues or graph algorithms.

Analysis and Detail

1. Efficiency of Fibonacci Heap: It allows operations such as insert, delete, and decrease key in O(1) amortized time, which is beneficial for the repeated key decrease that occurs in Prim's algorithm as new vertices are added to the MST. This is particularly useful for dense graphs where many edges can connect to various vertices.
2. Comparison to Others: While d-ary heaps may offer improvements over a binary heap in some cases, they do not yield as significant an improvement for Prim’s Algorithm as Fibonacci heaps do.

Verify and Summarize

1. Verify that Fibonacci heaps allow for efficient handling of graph nodes during Prim's execution.
2. Summary: For dense graphs, Prim's algorithm is best implemented with a Fibonacci heap due to its efficiency in handling frequently modified vertex keys.

Final Answer

c. Fibonacci heap