Computer science Consider a data processing system that needs to sort $25,000$ records using a merge sort algorithm. Explain the merge sort algorithm and determine its time complexity in both average and worst$-$case scenarios. Additionally, calculate the space complexity required for the auxiliary arrays used during the merge process.

In a binary search tree $($BST$)$ with $700$ nodes, where the tree is highly unbalanced, determine the maximum height of the tree and the number of nodes that can be present in the longest path from the root to a leaf.

For a graph with $200$ vertices and $600$ edges, apply the Floyd$-$Warshall algorithm to compute the shortest paths between all pairs of nodes. Calculate the time complexity of the Floyd$-$Warshall algorithm and discuss the size of the distance matrix used for storing shortest path distances.

Given a hash table with $9000$ slots and a load factor of $0\mathrm{.}65,$ estimate the total number of elements in the hash table. Compute the average number of elements per slot and evaluate the performance implications of this load factor on hash table operations such as insertion, deletion, and search.

A dynamic programming approach is used to solve the $0/1$ knapsack problem where the capacity of the knapsack is $1000,$ and there are $50$ items each with different weights and values. Determine the time complexity of the dynamic programming algorithm and the size of the DP table required for this problem.

For a network flow problem with $220$ nodes and $660$ edges, apply the Ford$-$Fulkerson method to find the maximum flow. Analyze the time complexity of the Ford$-$Fulkerson algorithm based on the number of nodes and edges, and discuss the impact of varying edge capacities on the algorithm's efficiency.

Implement a depth$-$first search $($DFS$)$ algorithm on a graph with $300$ nodes and $1000$ edges. Calculate the space complexity of the DFS algorithm considering both the recursion stack and the adjacency list representation.

Given an array of $25,000$ elements, use the radix sort algorithm to sort the array. Calculate the time complexity of radix sort, assuming that the maximum value in the array is $150,000,$ and explain the space complexity for the counting array used during the sorting process.

For a shortest path problem in a graph with $200$ nodes and $600$ edges, apply the Dijkstra$$s algorithm using a priority queue implemented as a binary heap. Determine the time complexity of the Dijkstra$$s algorithm and evaluate the number of heap operations required.

Apply Kruskal$$s algorithm to a graph with $190$ vertices and $570$ edges to find the minimum spanning tree. Calculate the time complexity of Kruskal$$s algorithm and the number of edges in the resulting minimum spanning tree, and discuss how edge weights influence the algorithm's performance.