Activity Selection Problem: Greedy Algorithms
40 Questions
1 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What does the shortest duration algorithm fail to provide in terms of the number of activities compared to an optimal solution?

  • It always selects the maximum number of activities possible.
  • It guarantees more activities than the optimal solution.
  • It produces exactly the same number of activities as the optimal solution.
  • It produces at least half the number of activities in the optimal solution. (correct)

    • In the worst-case scenario of selecting activities by earliest start time, how many activities can potentially be selected?

  • n-1 activities.
  • All activities can be selected.
  • n activities.
  • 1 activity. (correct)

    • According to the greedy choice property, includes which characteristic of an activity in an optimal solution?

  • The activity that starts latest.
  • The activity with the longest duration.
  • The activity with the latest finish time.
  • The activity with the earliest finish time. (correct)

    • What can be inferred if the optimal solution does not contain the activity with the earliest finish time?

    <p>The earliest finish time activity can replace a conflicting activity in the optimal solution.</p> Signup and view all the answers

    If the finish times of activities are sorted, what is the running time of the greedy algorithm?

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

    What does the activity selection problem aim to optimize?

    <p>The maximum number of non-conflicting activities.</p> Signup and view all the answers

    What is one criterion that makes the Greedy Algorithm for activity selection work?

    <p>Selecting the activity with the earliest finish time.</p> Signup and view all the answers

    What is the main issue with codewords in a prefix code?

    <p>One codeword can be a prefix of another.</p> Signup and view all the answers

    What is a prefix-free code?

    <p>A code where no codeword is a prefix of any other codeword.</p> Signup and view all the answers

    In a prefix code tree, what does each leaf node represent?

    <p>A character associated with a codeword.</p> Signup and view all the answers

    What does the Huffman algorithm aim to minimize in a prefix code scheme?

    <p>The sum of lengths of codewords weighted by their frequencies.</p> Signup and view all the answers

    What is the main goal of the greedy algorithm in the coin changing problem?

    <p>To make change using the fewest number of coins</p> Signup and view all the answers

    In the greedy algorithm for coin changing, what happens when x is greater than 0?

    <p>The largest coin less than or equal to x is found and added to the bag</p> Signup and view all the answers

    How do characters with the least frequencies relate in an optimal prefix code tree?

    <p>They share the same parent node.</p> Signup and view all the answers

    Why does the greedy algorithm not always work for certain coin denominations?

    <p>It can lead to a larger number of coins than necessary</p> Signup and view all the answers

    Which of the following describes the process of decoding a prefix code?

    <p>Recursively decoding after outputting a codeword.</p> Signup and view all the answers

    What property is crucial for determining whether a greedy algorithm will work correctly?

    <p>It must include the greedy choice in some optimal solution</p> Signup and view all the answers

    What is the significance of the lemma regarding characters with least frequencies in Huffman's algorithm?

    <p>Their optimal pairing impacts the overall structure of the tree.</p> Signup and view all the answers

    In the context of activity selection, what is the most effective criterion for choosing activities?

    <p>Selecting the activity with the earliest finish time</p> Signup and view all the answers

    What ensures that choosing the earliest finish time is the optimal strategy for activity selection?

    <p>It prevents any potential overlaps with other activities</p> Signup and view all the answers

    Which of the following best describes the general approach of a greedy algorithm?

    <p>It only focuses on the immediate benefit at each step</p> Signup and view all the answers

    How does the greedy algorithm handle subproblems in its solution?

    <p>It recursively solves subproblems while building up the solution</p> Signup and view all the answers

    What is the optimal substructure in the context of prefix codes?

    <p>Optimal prefix code trees can be constructed by merging leaves.</p> Signup and view all the answers

    What is the significance of merging leaves c and c' in tree T’?

    <p>It leads to the construction of an optimal prefix code for S’.</p> Signup and view all the answers

    Which of the following statements is true about the cost of tree T compared to tree T'?

    <p>Cost(T) is greater than cost(T').</p> Signup and view all the answers

    What is the running time of constructing a Huffman code tree?

    <p>O(n log n)</p> Signup and view all the answers

    In the Huffman coding scheme, what does the leaf node X represent?

    <p>The original nodes c and c' combined.</p> Signup and view all the answers

    What is the main implication of the lemma concerning tree T’ and optimal prefix code?

    <p>A contradiction arises if T is not optimal for S.</p> Signup and view all the answers

    Why must leaves a and b exist according to the prefix tree structure?

    <p>To fulfill the constraints of the prefix tree.</p> Signup and view all the answers

    What does the frequency value $f_X$ represent?

    <p>The combined frequency of c and c’.</p> Signup and view all the answers

    What is the goal of the Fractional Knapsack Problem?

    <p>To achieve the maximum total value without exceeding the weight limit.</p> Signup and view all the answers

    Which strategy is deemed correct for solving the Fractional Knapsack Problem?

    <p>Taking the densest item as much as possible.</p> Signup and view all the answers

    Why does the greedy algorithm fail for the 0-1 Knapsack Problem?

    <p>It does not evaluate multiple combinations of items.</p> Signup and view all the answers

    What characterizes variable-length encoding compared to fixed-length encoding?

    <p>More frequent characters are assigned shorter codes.</p> Signup and view all the answers

    How does fixed-length encoding compare in storage needs for a character set with unequal frequency?

    <p>It results in excessive storage use when frequencies vary greatly.</p> Signup and view all the answers

    Why is the strategy of taking fractions of items in the Fractional Knapsack Problem considered optimal?

    <p>It allows for flexibility in maximizing total value.</p> Signup and view all the answers

    What is the greedy-choice property?

    <p>A global optimal solution can be achieved through local choices.</p> Signup and view all the answers

    What is the implication of the optimal substructure property in algorithms?

    <p>An optimal solution can be constructed from optimal solutions to its subproblems.</p> Signup and view all the answers

    In the context of the 0-1 Knapsack Problem, which of the following items would be considered optimal?

    <p>The combination of items that maximizes value without exceeding weight.</p> Signup and view all the answers

    What flaw exists in this encoding scheme: a → 0, b → 1, c → 00, d → 01, e → 10, f → 11?

    <p>It can lead to ambiguity in decoding sequences.</p> Signup and view all the answers

    Study Notes

    Greedy Algorithm Overview

    • Greedy algorithms aim to make the locally optimal choice at each step, hoping to arrive at a globally optimal solution.
    • This approach is not always guaranteed to find the absolute best solution.
    • Sometimes, greedy algorithms produce near-optimal or optimal solutions for certain problems efficiently.

    Coin Changing Problem

    • Given a set of coin denominations, find the fewest number of coins to make a specific amount of money.
    • A greedy algorithm for this problem (selecting the largest coin denomination possible at each step) does not always produce the optimal solution in all cases (e.g., if denominations include $1, $4, $5, $10).

    Activity Selection Problem

    • Given a set of activities with start and finish times, select the maximum number of compatible activities.
    • The greedy approach of selecting the activity with the earliest finish time leads to an optimal solution.

    0-1 Knapsack Problem

    • The goal is to maximize the total value of items placed in a knapsack given a weight limit, where each item must be taken or left entirely.
    • There is no known greedy algorithm for this problem that will always provide an optimal solution.

    Fractional Knapsack Problem

    • This is variation of the knapsack problem; items can be taken in fractions.
    • The greedy strategy (take the item with the highest value-to-weight ratio first) yields an optimal solution.

    Encoding Characters

    • ASCII uses fixed-length encoding (e.g., 8 bits per character).
    • Variable-length encoding can reduce storage requirements by assigning shorter codes to more frequent characters and longer codes to less frequent ones.
    • Huffman coding is a variable-length prefix code that can be optimally computed using a greedy approach.

    Huffman Code

    • A variable-length prefix code where each character has a codeword that is not a prefix of any other codeword.
    • To construct an optimal Huffman code:
      • Find the two least frequent characters, combine them and assign new frequency.
      • Recursively build the Huffman coding tree in this way, following the greedy approach of combining the least frequent characters and expanding it step by step.
      • The running time is O(n log n) which is efficient for large input data.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Chapter 15 Greedy Algorithm PDF

    Description

    This quiz explores the concepts surrounding the Activity Selection Problem, particularly in the context of greedy algorithms. It covers important aspects such as the greedy choice property, the effect of choosing activities based on finish times, and the characteristics of prefix codes. Test your understanding of these algorithmic principles and their implications.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser