$1.$ Consider an algorithm that selects the median of an array as the pivot for quicksort. What are the potential effects on the performance of the algorithm? A$)$ Improves the average$-$case time complexity to better than On log n$)$ B$)$ Ensures that the worst$-$case time complexity is O$($n log n$)$ C$)$ Can lead to degraded performance if all elements are identical D$)$ Eliminates the need for recursion in the sorting process $2.$ Which graph properties are utilized by the Boruvka's algorithm for finding a minimum spanning tree? A$)$ The graph is undirected. B$)$ The graph allows for cycles. C$)$ Edge weights can be negative. D$)$ Each component selects the cheapest outgoing edge. $3.$ Which conditions are necessary for successfully applying Dijkstra's algorithm to a graph? A$)$ The graph must not contain any cycles. B$)$ All edge weights must be non$-$negative. C$)$ The graph can be either directed or undirected. D$)$ The source vertex must have edges connecting to all other vertices. $4.$ In the context of dynamic programming, which statements are accurate for an algorithm solving the coin change problem $($minimum number of coins for a given value$)?$ A$)$ Can be solved using a greedy approach effectively for all sets of denominations B$)$ Utilizes a bottom$-$up approach to build the solution C$)$ The complexity necessarily improves with memoization D$)$ Is solvable in polynomial time for any set of denominations $5.$ In a directed graph, an algorithm is designed to find all the strongly connected components. Which of the following statements could be true for such an algorithm? A$)$ Utilizes depth$-$first search at least twice B$)$ Can be optimized using Dijkstra's algorithm C$)$ Has a best$-$case time complexity of O$($V $+$ E$)$ D$)$ Requires sorting of the vertices based on their in$-$degrees $6.$ In the context of branch and bound, which strategies are valid for improving efficiency? A$)$ Using a Least Cost Bound to prioritize nodes B$)$ Pruning branches that exceed a known upper bound C$)$ Incremental state space generation D$)$ Applying a greedy algorithm at each decision point

$7.$ What characteristics make backtracking a suitable approach for the n$-$Queens problem? A$)$ The solution space is discrete and finite. B$)$ Solutions can be built incrementally. C$)$ The problem can be solved without backtracking by using advanced matrix operations. D$)$ The state space shrinks as the solution progresses. $8.$ Which factors should be considered when choosing between backtracking and branch and bound for a given problem? A$)$ The availability of a good bounding function. B$)$ The problem's requirement for finding all solutions vs$.$ a single optimal solution. C$)$ The homogeneity of the search space. D$)$ The computational cost of maintaining branch states. $9.$ What is an effective method for the branch and bound algorithm when solving assignment problems? A$)$ Using a least$-$cost matrix reduction at each node. B$)$ Integrating backtracking to revert to previous solutions. C$)$ Employing a depth$-$first search to maximize memory efficiency. D$)$ Prioritizing branches based on historical data of similar problems. $10.$ When using backtracking to solve constraint satisfaction problems with multiple constraints, what techniques can prevent the combinatorial explosion of possibilities? A$)$ Applying constraint propagation to reduce the domains of future variables. B$)$ Limiting depth of search to a predefined threshold. C$)$ Using least constraining value heuristic to choose the next step. D$)$ Sequentially adding constraints to avoid early exhaustive searches. $11.$ Which of the following are valid optimizations that can be applied to the Ford$-$Fulkerson algorithm for improving performance in finding maximum flow in a network? A$)$ Using BFS instead of DFS to find augmenting paths, as in the Edmonds$-$Karp implementation. B$)$ Dynamically updating edge capacities using a scaling factor to handle high$-$capacity networks. C$)$ Applying a greedy algorithm to select the highest capacity paths first. D$)$ Using priority queues to manage the exploration of paths in terms of their current flow capacity. $12.$ Which techniques are crucial when adapting max flow algorithms like Ford$-$Fulkerson to real$-$world applications such as network traffic optimization or pipeline flow management? A$)$ Incremental flow adjustment to react to real$-$time demand changes. B$)$ Integration with cost optimization to not just maximize flow but also minimize operational costs. C$)$ Ensuring algorithmic stability and predictability in highly variable flow scenarios. D$)$ Utilizing feedback from flow measurements to adjust model parameters dynamically.

$13.$ What innovative applications of max flow algorithms could potentially enhance operations in urban traffic management systems? A$)$ Optimizing the flow of vehicles to reduce congestion. B$)$ Managing traffic signals dynamically based on real$-$time traffic data. C$)$ Routing emergency vehicles through the least congested paths. D$)$ Using predictive modeling to forecast traffic flow and adjust signals preemptively. $14.$ Which of the following strategies are typically used to approximate solutions to NP$-$hard problems? A$)$ Greedy algorithms that build a solution iteratively based on local optimal choices. B$)$ Dynamic programming to break down the problem into smaller, overlapping subproblems. C$)$ Heuristics that provide good$-$enough solutions in a reasonable time. D$)$ Exact algorithms that guarantee the optimal solution in polynomial time. $15.$ Which aspects are critical when designing approximation algorithms for NP$-$hard problems to ensure they are both efficient and effective? A$)$ The approximation ratio that guarantees how close the solution is to the optimal. B$)$ The worst$-$case time complexity to ensure the algorithm runs within acceptable limits$.$ C$)$ The adaptability of the algorithm to different problem instances and constraints. D$)$ The use of machine learning models to predict optimal solutions.