Manacher's Algorithm is able to achieve linear time complexity by exploiting the _______ of palindromes.
- Boundaries
- Linearity
- Reversibility
- Symmetry
Manacher's Algorithm exploits the symmetry of palindromes to achieve linear time complexity. It cleverly uses information from previously processed characters to avoid redundant computations, making it an efficient algorithm for finding palindromic substrings.
Discuss the time complexity of Dijkstra's algorithm and any potential optimizations to improve its performance.
- O((V + E) * log V) where V is vertices and E is edges
- O(V * E) where V is vertices and E is edges
- O(V log V + E log V) with Fibonacci heap
- O(V^2) with adjacency matrix, O(E + V log V) with heap
Dijkstra's algorithm has a time complexity of O((V + E) * log V) using a binary heap. Various optimizations can be applied, such as using a Fibonacci heap to achieve a time complexity of O(V log V + E log V). These optimizations aim to reduce the overall complexity, making Dijkstra's algorithm more efficient for large graphs.
How does the choice of compression algorithm impact the decompression process?
- Different algorithms may require different decompression techniques, impacting both speed and correctness.
- It does not impact decompression; all compression algorithms result in the same decompressed string.
- The choice of algorithm affects the speed of decompression but not the correctness.
- The choice of algorithm only impacts the compression ratio, not the decompression process.
The choice of compression algorithm can impact the decompression process as different algorithms may require different techniques to reconstruct the original string. The efficiency and correctness of decompression can vary based on the chosen algorithm.
Explain the concept of parenthesization in the context of Matrix Chain Multiplication.
- It is a technique used to factorize matrices.
- It is the placement of parentheses to determine the order of matrix multiplication.
- It is the removal of unnecessary parentheses in a mathematical expression.
- It refers to the process of adding parentheses to a mathematical expression.
Parenthesization in the context of Matrix Chain Multiplication refers to the placement of parentheses to determine the order in which matrices are multiplied. Dynamic programming helps find the optimal parenthesization to minimize the overall computational cost.
While topological sorting primarily applies to directed acyclic graphs (DAGs), certain algorithms can handle graphs with _______ edges by modifying the approach.
- Bidirectional
- Cyclic
- Undirected
- Weighted
While topological sorting primarily applies to directed acyclic graphs (DAGs), certain algorithms can handle graphs with cyclic edges by modifying the approach. Handling cycles requires additional considerations and modifications to traditional topological sorting algorithms.
How does Manacher's Algorithm achieve linear time complexity in finding the Longest Palindromic Substring?
- By cleverly exploiting the properties of previously processed palindromes to avoid unnecessary re-evaluations.
- By employing dynamic programming to optimize the computation of palindromic substrings.
- By using a brute-force approach to check all possible substrings for palindromicity.
- By utilizing a combination of hashing and greedy techniques to eliminate redundant computations.
Manacher's Algorithm achieves linear time complexity by intelligently utilizing information from previously processed palindromic substrings, avoiding redundant computations and optimizing the overall process.
The time complexity of BFS when implemented on an adjacency list representation of a graph is _______.
- O(E)
- O(V + E)
- O(V)
- O(log V)
The time complexity of BFS (Breadth-First Search) when implemented on an adjacency list representation of a graph is O(V + E), where V is the number of vertices and E is the number of edges. This is because each vertex and edge are examined once during the traversal.
Radix sort is often used to sort data represented in which numeric base?
- Binary
- Decimal
- Hexadecimal
- Octal
Radix sort is often used to sort data represented in the hexadecimal numeric base. It operates by processing digits from the least significant digit to the most significant digit.
The best-case time complexity of Insertion Sort is _______.
- O(1)
- O(n log n)
- O(n)
- O(n^2)
The best-case time complexity of Insertion Sort is O(1). This occurs when the input array is already sorted, and the algorithm needs only to check each element once.
How can you implement a queue using an array?
- Implement enqueue and dequeue at the middle of the array.
- Implement enqueue at the end and dequeue at the beginning, shifting elements accordingly.
- Use a single pointer for enqueue at the end and dequeue at the beginning.
- Use two pointers, one for enqueue and one for dequeue, and shift elements as needed.
A common way to implement a queue using an array is to use two pointers, one for enqueue at the end and one for dequeue at the beginning. Elements are shifted as needed to accommodate new elements and maintain the order of the queue.
Bubble sort's time complexity can be improved to _______ by implementing certain optimizations.
- O(n log n)
- O(n log^2 n)
- O(n)
- O(n^2)
Bubble sort's time complexity can be improved to O(n) by implementing certain optimizations. With optimized versions such as the flag-based check and other enhancements, the algorithm can achieve linear time complexity in scenarios where the array is already sorted or nearly sorted, making it more efficient in specific use cases.
In selection sort, how many comparisons are performed in the inner loop in each iteration?
- i
- n
- n - 1
- n - i
In each iteration of the inner loop in selection sort, where 'i' is the current iteration, n - i comparisons are performed. This is because the inner loop looks for the minimum element in the unsorted portion and places it at the beginning, reducing the number of comparisons in subsequent iterations.