Intractable Problems in Computer Science - Algorithms 1 Review
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Questions and Answers

What is an intractable problem in computer science?

  • A problem for which it is unlikely that a polynomial time algorithm solution exists (correct)
  • A problem with a solution that can be found by an exponential time algorithm
  • A problem for which a solution can be found in constant time
  • A problem with a solution that can be found by a polynomial time algorithm

    • What does it mean when an algorithm is said to be of polynomial time?

  • The algorithm cannot handle large inputs
  • The algorithm's running time is exponential
  • The algorithm always takes the same amount of time to execute
  • The algorithm's running time is upper bounded by a polynomial expression in the size of the input (correct)

    • Which of the following is an example of an intractable problem?

  • Traveling Salesman Problem (correct)
  • Linear Regression
  • Binary Search
  • Bubble Sort

    • What are some common examples of intractable problems?

    <p>Traveling Salesman Problem, Satisfiability (SAT), Clique</p> Signup and view all the answers

    What are some common data structures used to model intractable problems?

    <p>Graphs and Trees</p> Signup and view all the answers

    What does a graph illustrate?

    <p>The relationships between objects</p> Signup and view all the answers

    What is the formal definition of a graph G?

    <p>A set of vertices and a set of edges that relate the vertices</p> Signup and view all the answers

    What does a vertex represent in the context of this graph?

    <p>An airport</p> Signup and view all the answers

    Which type of graph has all its edges directed?

    <p>Directed graph</p> Signup and view all the answers

    What is the number of edges in a complete directed graph with n vertices?

    <p>$n \times (n-1)$</p> Signup and view all the answers

    In adjacency matrix representation, what does A(i,j) = 1 indicate?

    <p>(i,j) is an edge</p> Signup and view all the answers

    What is the space requirement for an adjacency matrix?

    <p>$O(n^2)$</p> Signup and view all the answers

    What does the adjacency list representation involve?

    <p>Generating a set of lists for each vertex and its neighbors</p> Signup and view all the answers

    What does null in an adjacency list represent?

    <p>The end of the list for the corresponding vertex</p> Signup and view all the answers

    What is the time complexity of the adjacency list representation?

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

    "What is a complete graph?"

    <p>&quot;A graph with n vertices and every vertex connected to every other vertex&quot;</p> Signup and view all the answers

    What are the two types of directed and undirected edges mentioned in the text?

    <p>Ordered pair and unordered pair</p> Signup and view all the answers

    How does a complete undirected graph differ from a complete directed graph in terms of its edges?

    <p>The number of edges in a complete undirected graph is $\frac{n(n-1)}{2}$</p> Signup and view all the answers

    Study Notes

    Intractable Problems

    • An intractable problem in computer science is a problem that cannot be solved in a reasonable amount of time (e.g., within seconds, minutes, or hours) due to its computational complexity.

    Algorithm Time Complexity

    • An algorithm is said to be of polynomial time if its running time increases polynomially (e.g., O(n), O(n^2), O(n^3)) with the size of the input.

    Examples of Intractable Problems

    • The traveling salesman problem is an example of an intractable problem.

    Graph Theory

    • A graph illustrates relationships between objects or nodes.
    • The formal definition of a graph G is G = (V, E), where V is the set of vertices and E is the set of edges.

    Graph Components

    • A vertex represents a node in the graph.
    • A graph with all its edges directed is called a directed graph.
    • The number of edges in a complete directed graph with n vertices is n(n-1).

    Graph Representations

    • In adjacency matrix representation, A(i,j) = 1 indicates that there is an edge from vertex i to vertex j.
    • The space requirement for an adjacency matrix is O(n^2).
    • Adjacency list representation involves storing edges as lists of vertices that are connected to each vertex.
    • Null in an adjacency list represents the absence of an edge.

    Graph Types

    • A complete graph is a graph in which every vertex is connected to every other vertex.

    Edge Types

    • The two types of edges are directed edges and undirected edges.
    • A complete undirected graph differs from a complete directed graph in that it has two edges between each pair of vertices (one in each direction), whereas a complete directed graph has only one edge between each pair of vertices (in one direction).

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    Description

    Test your understanding of intractable problems in computer science with this review quiz. Explore the classification of problems as tractable or intractable, and learn about polynomial time algorithms and their upper bound running time. This quiz is designed to help you reinforce your knowledge of algorithmic complexity.

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