Given an array of integers with nnn elements, where nnn is $1000,$ apply a divide$-$and$-$conquer algorithm to sort the array. Analyze the time complexity of the sorting algorithm and calculate the number of comparisons required in the worst case. A binary search tree $($BST$)$ has $50$ nodes with a maximum height of $10.$ If the BST is balanced, determine the maximum number of nodes that could be in the left and right subtrees of the root. Consider a directed acyclic graph $($DAG$)$ with $20$ vertices and $40$ edges. Using Kahn$$s algorithm, calculate the total number of possible topological sorts for this graph. Given a graph with $100$ nodes and a maximum degree of $20,$ estimate the space complexity of representing this graph using an adjacency list. A hash table uses chaining to resolve collisions, with $200$ slots and an average chain length of $3$ elements. Determine the load factor of the hash table and the average number of comparisons required to find an element. Implement the Dijkstra$$s shortest path algorithm on a weighted graph with $100$ nodes and $500$ edges. Calculate the time complexity of the algorithm using a priority queue implemented as a binary heap. A quicksort algorithm is applied to an array of $500$ elements with a worst$-$case partitioning strategy. Determine the time complexity of the quicksort algorithm and the maximum number of recursive calls it will make. A dynamic programming algorithm is used to solve a knapsack problem with $50$ items and a maximum capacity of $200$ units. Calculate the time complexity of the algorithm and the maximum number of states in the DP table. Given a set of $15$ tasks with various start and end times, use the greedy algorithm to determine the maximum number of non$-$overlapping tasks that can be scheduled. A binary heap is used to implement a priority queue with $300$ elements. Calculate the time complexity of inserting a new element and deleting the minimum element from the heap. A breadth$-$first search $($BFS$)$ is performed on a graph with $150$ nodes and $600$ edges. Determine the space complexity of the BFS algorithm in terms of the number of nodes and edges. A graph coloring problem needs to be solved for a graph with $25$ vertices and $50$ edges using a greedy coloring algorithm. Estimate the number of colors required to color the graph such that no two adjacent vertices share the same color. Implement a depth$-$first search $($DFS$)$ on a graph with $40$ vertices and $80$ edges. Calculate the time complexity of the DFS algorithm and the maximum depth of recursion required.