Computer science Given a data set with $15,000$ entries, where each entry is a record with multiple fields, implement a merge sort algorithm to sort the data based on a specific field. Analyze the time complexity of the merge sort algorithm and the space complexity for storing the temporary arrays used during the merge process. For a graph with $140$ vertices and $420$ edges, determine the number of distinct minimum spanning trees using Kruskal's algorithm and explain how the graph's structure affects this count. Given a hash table with $7000$ slots and a load factor of $0\mathrm{.}55,$ estimate the number of elements in the hash table and compute the expected number of collisions. Implement a binary search tree $($BST$)$ to maintain a dynamic set of $250$ integers, performing insertion, deletion, and search operations. Calculate the time complexity of each operation and discuss the impact of tree balance on performance. For a graph with $160$ vertices and $480$ edges, apply the Floyd$-$Warshall algorithm to find the shortest paths between all pairs of nodes. Determine the time complexity of the Floyd$-$Warshall algorithm and the size of the distance matrix used. A quicksort algorithm is used on an array of $15,000$ elements with a random pivot selection strategy. Calculate the expected number of comparisons made by the quicksort algorithm and its average$-$case time complexity. Given a set of $40$ integers and using a binary heap, implement heap sort to sort the integers. Calculate the number of comparisons performed during heap sort and the overall time complexity of the algorithm. For a network flow problem with $180$ nodes and $540$ edges, apply the Dinic's algorithm to find the maximum flow. Determine the time complexity of Dinic's algorithm with respect to the number of nodes and edges, and discuss the impact of different network topologies on the algorithm$$s performance. Using dynamic programming, solve the $0/1$ knapsack problem with a capacity of $500$ and items each with varying weights and values. Calculate the time complexity of the dynamic programming solution and the size of the DP table used. Implement a depth$-$first search $($DFS$)$ algorithm on a directed graph with $200$ nodes and $800$ edges. Compute the space complexity of the DFS algorithm considering both the recursion stack and the adjacency list representation. Finally, apply the A$*$ search algorithm to find the shortest path in a grid with $50$x$50$ cells, each cell having different traversal costs. Analyze the time complexity of the A$*$ algorithm and the effect of the heuristic function on its performance.