Design Ch.11
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Questions and Answers

What characterizes the choices made in the Greedy Technique?

  • They require backtracking to find the solution.
  • They can change based on the previous choices made.
  • They are always globally optimal.
  • They are feasible, locally optimal, and irrevocable. (correct)
  • Which algorithm is NOT typically associated with the Greedy Technique?

  • Prim’s Algorithm
  • A* Algorithm (correct)
  • Dijkstra’s Algorithm
  • Kruskal’s Algorithm
  • What is the primary function of Huffman Trees in algorithm design?

  • To ensure all nodes are connected with minimal weight.
  • To optimize data compression by assigning variable-length codes. (correct)
  • To sort data in ascending order.
  • To find the shortest path in a graph.
  • In which scenario does the Greedy Technique guarantee an optimal solution?

    <p>For certain problems where local optimization leads to global optimization.</p> Signup and view all the answers

    Which statement about the Greedy Technique is true?

    <p>It constructs a solution in a single pass without revisiting choices.</p> Signup and view all the answers

    What defines a minimum spanning tree in a weighted, connected graph?

    <p>A tree that includes all vertices and has the minimum total weight</p> Signup and view all the answers

    Why is exhaustive search not considered an efficient method for finding a minimum spanning tree?

    <p>It requires examining all possible spanning trees</p> Signup and view all the answers

    Which of the following properties do all minimum spanning trees share?

    <p>They may include different edges depending on the graph structure</p> Signup and view all the answers

    In the context of graphs, what is meant by a spanning tree?

    <p>A connected acyclic subgraph that contains all the vertices of the graph</p> Signup and view all the answers

    Which algorithm is commonly used to find minimum spanning trees?

    <p>Prim's algorithm</p> Signup and view all the answers

    What characterizes an undirected connected graph that contains a minimum spanning tree?

    <p>It includes all vertices without cycles</p> Signup and view all the answers

    In a weighted graph, how is the total weight of a spanning tree calculated?

    <p>By summing the weights of edges included in the spanning tree</p> Signup and view all the answers

    What is the main challenge of implementing Kruskal's algorithm?

    <p>Checking for cycles</p> Signup and view all the answers

    In Dijkstra’s algorithm, which expression is used to find the shortest path?

    <p>$d_v + w(v, u)$</p> Signup and view all the answers

    Which type of graphs is Dijkstra’s algorithm NOT applicable to?

    <p>Graphs with negative weights</p> Signup and view all the answers

    What is the efficiency of Dijkstra's algorithm using an adjacency list representation?

    <p>$O(|E| imes ext{log}|V|)$</p> Signup and view all the answers

    What does a cycle in a graph indicate when adding an edge in Kruskal's algorithm?

    <p>The edge connects vertices within the same connected component</p> Signup and view all the answers

    Which component is necessary for cycle detection in Kruskal's algorithm?

    <p>Union-find algorithms</p> Signup and view all the answers

    When is a vertex 'u' selected during Dijkstra's algorithm?

    <p>When it has the smallest sum of $d_v + w(v, u)$</p> Signup and view all the answers

    What does the variable $w(v, u)$ represent in Dijkstra's algorithm?

    <p>The weight of the edge from vertex v to vertex u</p> Signup and view all the answers

    What aspect of Kruskal's algorithm is generally easier compared to Prim's algorithm?

    <p>Finding the minimum spanning tree</p> Signup and view all the answers

    What is the primary criterion for selecting edges when constructing a minimum spanning tree?

    <p>They must be in nondecreasing order of their weights.</p> Signup and view all the answers

    What condition must be avoided when including an edge in the construction of a minimum spanning tree?

    <p>Creating a cycle.</p> Signup and view all the answers

    Which statement best describes the process of edge selection in minimum spanning tree construction?

    <p>Edges are selected in nondecreasing order of their weights.</p> Signup and view all the answers

    Why is it important to choose edges in nondecreasing order of their weights?

    <p>To ensure the minimum total weight for the spanning tree.</p> Signup and view all the answers

    What does the term 'minimum spanning tree' refer to?

    <p>A tree that minimizes the total weight of the edges.</p> Signup and view all the answers

    What is a cycle in the context of graph theory?

    <p>A sequence of edges that forms a closed loop.</p> Signup and view all the answers

    Which algorithm is commonly used to construct a minimum spanning tree by selecting edges wisely?

    <p>Kruskal’s Algorithm</p> Signup and view all the answers

    What is the effect of including an edge that creates a cycle in a minimum spanning tree construction?

    <p>It must be avoided to maintain a valid tree.</p> Signup and view all the answers

    In constructing a minimum spanning tree, adding edges must be based on what criteria?

    <p>They should not create a cycle.</p> Signup and view all the answers

    Study Notes

    Greedy Technique Overview

    • A greedy technique constructs solutions to optimization problems step by step, making locally optimal choices at each step.
    • The choices made in each step must be:
      • Feasible: The choice must be valid; it should satisfy the constraints of the problem.
      • Locally optimal: The choice must appear to be the best at that current step.
      • Irrevocable: The choice made cannot be changed later.
    • For certain types of problems, a greedy approach can yield optimal solutions for every case.
    • For most problems, however, it doesn't always produce optimal solutions but can be useful for creating approximations efficiently.

    Applications

    • Optimal solutions:
      • Coin change making (with typical denominations).
      • Minimum spanning trees (MST).
      • Single-source shortest paths.
      • Simple scheduling.
      • Huffman codes.
    • Approximations:
      • Traveling salesman problem (TSP).
      • Knapsack problem.
      • Other combinatorial optimization problems.

    Change-Making Problem

    • Given a set of denominations of coins (e.g. 25c, 10c, 5c, 1c), find the smallest number of coins required to make a specific amount of change.
    • A greedy approach to this problem may not always be optimal depending on the set of denominations.

    Minimum Spanning Tree (MST)

    • A spanning tree is a subset of edges of a connected graph that connects all vertices without forming cycles.
    • A minimum spanning tree (MST) is a spanning tree with the smallest possible total weight (sum of edge weights).

    Prim's Algorithm

    • Prim's algorithm is a greedy algorithm for constructing an MST.
    • It starts with a single vertex of the graph and adds the closest vertex to the existing subtree, repeatedly, until all vertices are in the tree.
    • Prim's algorithm requires a priority queue to efficiently locate the closest fringe vertex at each step.
    • Its complexity is O(n^2) for a matrix representation and O(E log V) for an adjacency list representation (using a min heap).

    Kruskal's Algorithm

    • Kruskal's algorithm is another greedy approach to finding an MST.
    • It sorts edges in increasing order of their weights.
    • It adds each edge (without cycles) to the minimum spanning tree.
    • It uses a concept called union–find algorithms to efficiently determine if adding an edge creates a cycle.

    Dijkstra's Algorithm

    • Dijkstra's algorithm finds the shortest paths from a single source vertex to all other vertices in a weighted graph.
    • It uses a priority queue to find the vertex with the smallest path cost from the source in each step.
    • Like Prim's algorithm, its efficiency depends on how the graph is implemented (weight matrix or adjacency list) - O(V^2) or O(E log V)
    • It does not work for graphs with negative weights.

    Huffman Codes

    • Huffman codes are a variable-length encoding method for efficiently encoding characters by assigning shorter codes to more frequent symbols and longer codes to less frequent ones.
    • The corresponding encoding algorithm is a greedy algorithm.
    • It aims to minimize the average codeword length.

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