Computer science Consider a database with $20,000$ records where each record has multiple fields, and you need to sort these records based on one of the fields using a radix sort algorithm. Explain the radix sort algorithm and calculate its time complexity, assuming that the key field being sorted is an integer with a maximum value of $100,000.$ Additionally, determine the space complexity for the counting array used during the sorting process.

In a binary search tree $($BST$)$ with $600$ nodes, where the tree is highly unbalanced, calculate the maximum height of the tree and the number of nodes that could be present in the longest path from the root to a leaf.

For a graph with $150$ vertices and $450$ edges, apply the Bellman$-$Ford algorithm to find the shortest paths from a single source. Calculate the time complexity of the Bellman$-$Ford algorithm and the maximum number of iterations required for this algorithm to complete.

Given a hash table with $8000$ slots and a load factor of $0\mathrm{.}7,$ estimate the number of elements currently stored in the hash table. Calculate the average number of elements per slot and discuss the implications of this load factor on the hash table$$s performance and collision resolution.

A dynamic programming approach is used to solve the longest common subsequence problem for two sequences, each of length $200.$ Determine the time complexity of the dynamic programming algorithm and the size of the DP table used in this solution.

For a network flow problem with $200$ nodes and $600$ edges, apply the Edmonds$-$Karp algorithm to find the maximum flow. Calculate the time complexity of the Edmonds$-$Karp algorithm in terms of the number of nodes and edges, and discuss how different network structures might affect its performance.

Implement a depth$-$first search $($DFS$)$ algorithm on a graph with $250$ nodes and $900$ edges. Compute the space complexity of the DFS algorithm considering both the recursion stack and the adjacency list representation used.

Given an array of $20,000$ elements, apply the heap sort algorithm. Calculate the number of comparisons performed by heap sort and determine the overall time complexity. Explain the impact of the initial ordering of the array on the performance of heap sort.

For a shortest path problem in a graph with $180$ nodes and $540$ edges, use the A$*$ search algorithm with a heuristic function that estimates the distance to the goal. Analyze the time complexity of the A$*$ algorithm and discuss how different heuristic functions affect the performance and efficiency of the search process.

Implement Kruskal$$s algorithm on a graph with $170$ vertices and $510$ edges to find the minimum spanning tree. Calculate the time complexity of Kruskal$$s algorithm and the number of edges in the resulting minimum spanning tree.