Algorithm Analysis and Design: Key Concepts Explained
Q1. Explain the significance of Asymptotic Analysis in algorithm design.
Asymptotic analysis helps in evaluating the efficiency of an algorithm by measuring its time and space requirements as the input size grows large. It allows comparison of algorithms independent of hardware, programming language, or machine implementation. By using notations like Big-O, Big-Ω, and Big-Θ, it focuses on growth rate rather than exact execution time, helping designers choose scalable and efficient algorithms.
Q2. Why is average-
case analysis of an algorithm often more useful than worst–
Case analysis?
Average-case analysis represents the expected performance of an algorithm for typical inputs, making it more realistic for practical use. Worst-case analysis considers rare scenarios that may not occur frequently, while average-case gives a balanced view of performance in real-world situations. Hence, average-case analysis helps in selecting algorithms that perform efficiently under normal conditions.
Q3. Explain the Time–Space Trade-off in algorithm design.
The time–space trade-off refers to the balance between an algorithm’s execution time and memory usage. An algorithm can run faster by using more memory, or it can save memory at the cost of increased execution time. For example, using hash tables improves speed but consumes more space. Algorithm designers choose an appropriate trade-off based on system requirements.
Q4. Explain the Insertion Sort algorithm and the role of the ‘key’ variable.
Insertion Sort works by dividing the array into sorted and unsorted parts. In each iteration, one element from the unsorted part is selected as the key and inserted into its correct position in the sorted part by shifting larger elements to the right. The key variable temporarily stores the current element being inserted and helps maintain the sorted order of the array.
Q5. Derive the worst-case time complexity of Insertion Sort using Big-O notation.
In the worst case, the input array is sorted in reverse order. For each element, all previous elements must be compared and shifted. The total number of comparisons is
1 + 2 + 3 + … + (n−1) = n(n−1)/2,
which is proportional to n². Therefore, the worst-case time complexity of Insertion Sort is O(n²).
Q6. Solve the recurrence relation T(n) = 2T(n/2) + n using the Master Theorem.
Here, a = 2, b = 2, and f(n) = n.
We calculate n^(log_b a) = n^(log₂2) = n.
Since f(n) = Θ(n), it matches Case 2 of the Master Theorem.
Therefore, the solution is T(n) = Θ(n log n).
Q7. Write the recurrence relation of Merge Sort and find its worst-case time complexity.
Merge Sort divides the array into two halves and recursively sorts them, then merges the results.
The recurrence relation is T(n) = 2T(n/2) + n.
Using the Master Theorem, the worst-case time complexity of Merge Sort is O(n log n).
Q8. Why is the worst-case time complexity of Quick Sort O(n²)? How can it be avoided?
The worst case of Quick Sort occurs when the pivot element is always the smallest or largest element, causing highly unbalanced partitions. This results in n recursive calls with n comparisons each, leading to O(n²) complexity. The worst case can be avoided by using randomized pivot selection or the median-of-three method, which ensures balanced partitions.
Q9. What is the 0/1 Knapsack problem?
Explain the Branch and Bound approach.
The 0/1 Knapsack problem involves selecting items with given weights and values to maximize profit without exceeding the knapsack capacity. The Branch and Bound approach explores all possible item combinations using a state-space tree while pruning branches that cannot produce better solutions than the current best. This reduces unnecessary computations and improves efficiency.
Q10. Differentiate between NP-Hard and NP-Complete problems with examples.
NP-Hard problems are at least as hard as the hardest problems in NP but may not belong to NP. NP-Complete problems are both in NP and NP-Hard, meaning their solutions can be verified in polynomial time.
Example of NP-Hard: Traveling Salesman Problem (optimization version).
Example of NP-Complete: Vertex Cover problem.