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Questions and Answers
What is a heap?
What is a heap?
A specialized tree-based data structure that satisfies the heap property.
In a binary heap, the left child of a node at index i is found at index _____
In a binary heap, the left child of a node at index i is found at index _____
2i + 1
In a binary heap, the right child of a node at index i is found at index _____
In a binary heap, the right child of a node at index i is found at index _____
2i + 2
What is the time complexity of building a max-heap?
What is the time complexity of building a max-heap?
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Which algorithm is generally faster on average for sorting arrays?
Which algorithm is generally faster on average for sorting arrays?
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Radix Sort sorts numbers based on each digit starting from the most significant digit.
Radix Sort sorts numbers based on each digit starting from the most significant digit.
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What is the purpose of the Master Theorem?
What is the purpose of the Master Theorem?
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The time complexity of Counting Sort is _____
The time complexity of Counting Sort is _____
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What is the worst-case time complexity of QuickSort?
What is the worst-case time complexity of QuickSort?
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Which data structure allows for O(1) access time?
Which data structure allows for O(1) access time?
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Dijkstra's Algorithm can be used for graphs with negative weights.
Dijkstra's Algorithm can be used for graphs with negative weights.
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Define the term 'Big O' notation.
Define the term 'Big O' notation.
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In the Fractional Knapsack problem, the items are sorted by _____
In the Fractional Knapsack problem, the items are sorted by _____
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What is the time complexity of the 0-1 Knapsack problem using dynamic programming?
What is the time complexity of the 0-1 Knapsack problem using dynamic programming?
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Study Notes
Heaps and HeapSort
- A heap is a specialized tree structure that satisfies the heap property.
- In a max-heap, each parent node is greater than or equal to its children; in a min-heap, each parent is less.
- A binary heap is commonly implemented as an array.
- Parent index: floor((i−1)/2)
- Left child index: 2i + 1
- Right child index: 2i + 2
HeapSort
- Construct a max-heap from input data.
- Swap the root (maximum value) with the last item of the heap.
- Reduce heap size by one and heapify the root; repeat until the heap is empty.
- Time complexity for building the heap: O(n)
- Heapify operation time complexity: O(log n)
- Overall time complexity: O(n log n)
QuickSort
- Select a pivot element from the array.
- Partition the array into two halves based on the pivot values (less than and greater).
- Recursively sort the two halves.
- Best and average case time complexity: O(n log n)
- Worst-case time complexity: O(n^2), occurs with poor pivot choices on sorted arrays.
MergeSort
- Divide the array into two halves.
- Recursively sort each half.
- Merge the sorted halves back to form a sorted array.
- Time complexity is consistently O(n log n) for all cases.
Radix Sort
- Sort numbers based on each digit, from least significant to most significant.
- Utilize a stable sort (like Counting Sort) for sorting based on each digit.
- Time complexity: O(d(n + k)), where d is the number of digits and k is the range of digit values.
Counting Sort
- Count occurrences of each element.
- Compute positions for each element in the sorted array.
- Place elements in correct positions based on counts.
- Time complexity: O(n + k), where k is the range of input values.
Divide and Conquer Strategies
- Technique that breaks a problem into smaller subproblems, solves each independently, and combines their solutions.
- Examples include MergeSort, QuickSort, and Binary Search.
Greedy Algorithms
- Makes locally optimal choices at each step to solve optimization problems.
- Examples: Kruskal's Algorithm (MST), Prim's Algorithm (MST), Dijkstra's Algorithm (shortest paths).
- Suitable for problems with the greedy-choice property and optimal substructure.
Master Theorem
- A formula for solving recurrences of the form T(n)=aT(n/b)+f(n).
- Case when f(n)=O(n^c) and c > log_b(a): T(n)=O(f(n)).
Runtimes for Sorts
- Bubble Sort: O(n^2)
- Insertion Sort: O(n^2)
- Selection Sort: O(n^2)
- MergeSort: O(n log n)
- QuickSort: Average O(n log n), Worst O(n^2)
- HeapSort: O(n log n)
- Counting Sort: O(n + k)
- Radix Sort: O(d(n + k))
Runtimes for Data Structures
- Arrays: Access O(1), Search O(n), Insertion/Deletion O(n)
- Linked Lists: Access O(n), Search O(n), Insertion/Deletion O(1)
- Hash Tables: Access O(1), Search O(1), Insertion/Deletion O(1)
- Binary Search Trees: Access O(log n), Search O(log n), Insertion/Deletion O(log n)
- Heaps: Access O(1), Search O(n), Insertion/Deletion O(log n)
Fractional Knapsack
- Goal: Maximize total value by taking fractions of items with weights and values.
- Solution involves sorting items by value-to-weight ratio before selection.
- Time complexity is O(n log n) due to sorting.
0-1 Knapsack
- Goal: Maximize total value without breaking items.
- Dynamic programming builds a table (dp[i][w]) representing the maximum value for the first i items within weight limit w.
- Time complexity: O(nW), where n is number of items and W is capacity.
Dijkstra's Algorithm
- Aims to find shortest paths from a source vertex to all other vertices in a graph with non-negative weights.
- Uses a priority queue to select the vertex with the smallest distance iteratively.
- Time complexity: O(V^2) with an array, O((V + E) log V) with a priority queue.
Trees
- A tree is a connected acyclic graph, characterized by having n - 1 edges for n vertices.
- Unique path exists between any two vertices in a tree.
Big O Notation and Other Notations
- Big O (O): Defines the upper bound on time complexity.
- Big Omega (Ω): Indicates the lower bound on time complexity.
- Big Theta (Θ): Establishes a tight bound on time complexity.
- Little o (o): Indicates an upper bound that is not tight.
- Little omega (ω): Defines a lower bound that is not tight.
Asymptotic Analysis
- Used to analyze algorithm performance as input size increases.
- Considers worst-case, average-case, and best-case scenarios.
- Notations include Big O (upper bound) and Big Omega (lower bound).
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Description
Explore the fundamentals of heaps and the HeapSort algorithm. Understand the characteristics of max-heaps and min-heaps, and how binary heaps are implemented. This quiz will test your knowledge of tree-based data structures and their properties.
