Algorithm Time Complexities Quiz
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Questions and Answers

Which sorting algorithm is stable?

  • Heap sort
  • Quicksort
  • Selection sort
  • Merge sort (correct)

In a max-heap, A[Parent(i)] is less than or equal to A[i].

False (B)

What is the time complexity of the best case for Quicksort?

O(n log n)

A stack follows the _____ policy.

<p>Last in First Out</p> Signup and view all the answers

What are the three arrays used in Counting-Sort?

<p>Input array, output array, auxiliary array (C)</p> Signup and view all the answers

Match the functions with their corresponding data structure:

<p>Stack = Pop Queue = Dequeue</p> Signup and view all the answers

In a binary search tree, an in-order traversal visits nodes in ascending order.

<p>True (A)</p> Signup and view all the answers

What are the two main functions for manipulating elements in a Queue?

<p>Enqueue and Dequeue</p> Signup and view all the answers

A red-black tree satisfies _____ properties.

<p>five</p> Signup and view all the answers

What is the purpose of dynamic programming?

<p>To break the problem into overlapping subproblems (D)</p> Signup and view all the answers

Dynamic programming incurs a time-memory trade-off.

<p>True (A)</p> Signup and view all the answers

What does the C table store in the context of the Longest Common Subsequence (LCS) algorithm?

<p>The lengths of various subproblems of the LCS.</p> Signup and view all the answers

The __________ is used to represent directions for reconstructing the Longest Common Subsequence.

<p>B table</p> Signup and view all the answers

Match the following arrows with their meanings in the LCS algorithm:

<p>Diagonal Arrow = Indicates a match in characters Up Arrow = Indicates exclusion of first sequence character Left Arrow = Indicates exclusion of second sequence character</p> Signup and view all the answers

Which algorithm is used to calculate the minimum number of scalar products?

<p>Matrix-chain multiplication (C)</p> Signup and view all the answers

B-Trees are unbalanced search trees designed primarily for in-memory operations.

<p>False (B)</p> Signup and view all the answers

Name one method used for finding the longest common subsequence (LCS).

<p>Brute force or the optimal substructure property.</p> Signup and view all the answers

Which of the following algorithms has the worst case time complexity of O(n^2)?

<p>Insertion Sort (B)</p> Signup and view all the answers

The dynamic programming approach saves previously solved subproblems in a __________.

<p>table</p> Signup and view all the answers

What is indicated by a diagonal arrow in the context of the LCS algorithm?

<p>Characters match and are part of the LCS (C)</p> Signup and view all the answers

The time complexity of Quick Sort in the worst case is O(n log n).

<p>False (B)</p> Signup and view all the answers

What is in-place sorting? Provide an example of an algorithm that utilizes in-place sorting.

<p>In-place sorting is a method of sorting that requires no additional memory for transformation. An example is Insertion Sort.</p> Signup and view all the answers

The worst case time complexity for Balanced Binary Search Trees is O(logn) and O(______).

<p>h</p> Signup and view all the answers

Match the following algorithms with their respective sorting types:

<p>Quick Sort = Unstable Sort Merge Sort = Stable Sort Insertion Sort = Stable Sort Heap Sort = Unstable Sort</p> Signup and view all the answers

Which of the following algorithms uses the divide and conquer paradigm?

<p>Binary Search (B)</p> Signup and view all the answers

All linear data structures are arrays.

<p>False (B)</p> Signup and view all the answers

What is the worst case time complexity for Dijkstra's algorithm?

<p>O(V log V + E)</p> Signup and view all the answers

Counting Sort has a worst case time complexity of O(______ + n).

<p>k</p> Signup and view all the answers

Match the following data structures with their type:

<p>Array = Linear Graph = Nonlinear Tree = Nonlinear Stack = Linear</p> Signup and view all the answers

Flashcards

Divide and Conquer Paradigm

An algorithm that breaks a problem into smaller subproblems of the same type, solves the subproblems recursively, and combines their solutions to solve the original problem.

Not In-Place Sorting

Sorting algorithms that require additional memory space to store temporary data during the sorting process. They often achieve faster speeds than in-place sorts.

In-Place Sorting

Sorting algorithms that sort elements without requiring additional memory space apart from what is used to store the original input data. This means they modify the input array directly.

Stable Sorting

Sorting algorithms that maintain the relative order of elements with the same value. This is important when preserving certain properties of the data.

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Unstable Sorting

Sorting algorithms that do not guarantee the relative order of elements with the same value. The order of equal elements may change after sorting.

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Tree Data Structure

A data structure that arranges elements in a hierarchical manner. Each element, except the root, has a parent, and a parent can have multiple children.

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Graph Data Structure

A data structure that represents a set of interconnected nodes. Each node can be connected to multiple other nodes, forming edges.

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Hash Table

A data structure that allows you to store key-value pairs. The keys serve as unique identifiers for the corresponding values.

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Queue

A data structure that allows you to insert and retrieve elements in a FIFO (First-In, First-Out) manner. Think of a queue at a checkout counter.

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Stack

A data structure that allows you to insert and retrieve elements in a LIFO (Last-In, First-Out) manner. Think of a stack of plates.

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Stable Sorting Algorithm

A sorting algorithm where elements with the same value appear in the output array in the same order as they do in the input array.

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Unstable Sorting Algorithm

A sorting algorithm where the order of elements is not preserved during sorting and may appear in a different order in the output array.

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Max-heap Property

In a max-heap, the value of a parent node is always greater than or equal to the value of its children.

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Best Case Quicksort

The best case time complexity for Quicksort occurs when the pivot consistently divides the array in half, resulting in a balanced tree structure

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Auxiliary Array (C) in Counting Sort

It is an auxiliary array used in Counting Sort to store the frequency of each element in the input array.

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Input Array (A) in Counting Sort

It is the input array that contains the elements to be sorted in Counting Sort.

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Output Array (B) in Counting Sort

It is the output array that contains the sorted elements in Counting Sort.

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Stack Policy

The Last In First Out (LIFO) policy means that the most recently added element is the first to be removed.

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Queue Policy

The First In First Out (FIFO) policy means that the first element added is the first to be removed.

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Stack Functions

Push and Pop are functions for adding and removing elements from a stack, respectively.

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Hash Table Linked Lists

Elements with same hash values are stored in a linked list, which is attached to the corresponding slot in the hash table.

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Dynamic Programming Approach

Dynamic programming breaks down a problem into overlapping subproblems, solves them recursively only once and stores results in a table, and then combines these solutions to solve the original problem.

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Dynamic Programming Trade-Off

Dynamic programming uses additional memory to store intermediate results, which helps in avoiding redundant calculations and improves the overall time complexity.

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Dynamic Programming for Optimization

Dynamic Programming is often employed to find the optimal solution for optimization problems, where there are multiple possible solutions, and we aim to find the best.

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Matrix-chain Multiplication

The Matrix-chain Multiplication algorithm is a powerful technique used to find the minimum number of scalar multiplications required to compute a matrix-chain product.

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Longest Common Subsequence (LCS)

The Longest Common Subsequence (LCS) problem aims to find the longest sequence of characters common to two given input sequences.

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Brute Force LCS

The brute force approach for finding the LCS involves checking all possible subsequences of the input strings, which can be computationally expensive for large inputs.

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Optimal Substructure of LCS

The optimal substructure property of LCS ensures that the LCS of two strings is composed of the LCSs of their respective prefixes.

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LCS Algorithm Output

The LCS algorithm outputs both the length of the longest common subsequence and the actual sequence itself.

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C and B Tables in LCS

The C table stores the lengths of the longest common subsequences for all possible prefixes of the input strings, while the B table (directional table) guides in reconstructing the actual LCS.

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Study Notes

Algorithm Time Complexities

  • Insertion Sort: Worst case time complexity is O(n²).
  • Merge Sort: Worst case time complexity is O(n log n).
  • Heap Sort: Worst case time complexity is O(n log n).
  • Quick Sort: Worst case time complexity is O(n²). Best case is O(n log n).
  • Counting Sort: Worst case time complexity is O(k + n).
  • Searching Hash Table: Worst case time complexity is O(n).
  • Balanced Binary Search Tree: Worst case time complexity is O(log n).
  • Unbalanced Binary Search Tree: Worst case time complexity is O(n).
  • Inorder, Postorder, Preorder search: O(n).
  • Cut-Rod: O(2n).
  • Longest Common Subsequence: O(m + n).
  • Breadth-first search: O(|V| + |E|).
  • Depth-first search: O(|V| + |E|).
  • Dijkstra's algorithm: O(V log V + E).

Data Structures

  • Linear: Array, stack, queue, linked list
  • Non-linear: Tree, graph, hash table

Divide and Conquer Algorithms

  • Quicksort
  • Merge sort
  • Binary search

In-place Sorting

  • Insertion sort
    • Does not require additional space beyond the input array.

Not In-place Sorting

  • Merge sort
    • Requires additional space (memory) at least equal to the input, possibly more. It's often faster than in-place sorts.

Stable vs. Unstable Sorting

  • Stable sorting: Maintains the original order of equal elements in the output. Example: Merge sort
  • Unstable sorting: Does not guarantee the original order of equal elements in the output. Example: Quicksort.

Max-Heap Property

  • A[Parent(i)] ≥ A[i]

Best Case Time Complexity for Quicksort

  • Occurs when the pivot divides the array nearly in half.

Counting Sort Arrays

  • Input array (A): Holds the input values.
  • Output array (B): Holds the sorted output values.
  • Auxiliary array (C): Used for counting the occurrences of each input value.

Stack Policy

  • Last-In, First-Out (LIFO)

Queue Policy

  • First-In, First-Out (FIFO)

Dynamic Programming

  • Breaks down a problem into smaller overlapping subproblems.
  • Saves results of subproblems to avoid redundant calculations.
  • Used to solve optimization problems.

Trade-offs of Dynamic Programming

  • Increased memory usage for storing results of subproblems.
  • Often results in substantial time savings compared to recursive approaches.

Matrix-Chain Multiplication Algorithm

  • Used for calculating the minimum number of scalar multiplications.

LCS (Longest Common Subsequence) Algorithm

  • Finds the longest subsequence common to two input sequences.
  • Typically uses a 2D table (B and C) to build the solution from smaller subproblems.

B-Trees

  • Balanced search trees optimized for disk access.

Red-Black Tree Properties

  • Every node is either red or black.
  • The root is black.
  • Every leaf (NIL) is black.
  • If a node is red, both its children are black.
  • For each node, all simple paths from the node to descendant leaves contain the same number of black nodes.

Sentinel (in Merge Sort and Red-Black Trees)

  • Marker used as a boundary in arrays or other data structures to simplify processing.

Chaining in Hash Tables

  • Stores elements that hash to the same slot in a linked list.

Graph Traversal Algorithms

  • Depth-first search (DFS)
  • Breadth-first search (BFS)
  • Dijkstra's algorithm (shortest paths)

Strongly Connected Component

  • A group of vertices in a graph where, for every pair of vertices, there's a directed path from one to the other.

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Description

Test your knowledge on the time complexities of various algorithms, including sorting and searching algorithms. This quiz covers both worst-case scenarios and specific data structures. Challenge yourself with questions about linear and non-linear data structures as well as divide and conquer strategies.

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