Greedy Algorithms and Dijkstra's Algorithm Lecture

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What is the main difference between the Fractional Knapsack problem and the 0-1 Knapsack problem?

The Fractional Knapsack problem is used for optimization, while the 0-1 Knapsack problem is used for decision-making.

The Fractional Knapsack problem is a greedy algorithm, while the 0-1 Knapsack problem is a dynamic programming algorithm.

The Fractional Knapsack problem has a higher time complexity than the 0-1 Knapsack problem.

In the Fractional Knapsack problem, items can be taken in fractional amounts, while in the 0-1 Knapsack problem, items can only be taken in full. (correct)

What is the time complexity of the greedy approach to solving the Fractional Knapsack problem?

O(2^n)

O(n log n) (correct)

O(n^2)

O(n)

Which of the following statements about the Fractional Knapsack problem is not true?

The algorithm works by calculating the ratio of value to weight for each item and then adding the items with the highest ratios until the capacity is reached.

The problem can be solved using a brute-force approach, but this would be too time-consuming.

The greedy approach always gives the optimal solution.

The problem can be solved using dynamic programming, but this would be less efficient than the greedy approach. (correct)

In the Fractional Knapsack problem, if the capacity of the knapsack is 50 and the items are A (weight: 10, value: 60), B (weight: 20, value: 100), and C (weight: 30, value: 120), what is the best case solution?

Take all of A, B, and 2/3 of C. Signup and view all the answers

Which algorithm is typically used to solve the Fractional Knapsack problem?

The greedy approach Signup and view all the answers

In the 0-1 Knapsack problem, which of the following is true?

Each item can only be included either completely or not at all in the knapsack. Signup and view all the answers

What is the key difference between 0-1 Knapsack and Fractional Knapsack problems?

Fractional Knapsack allows breaking items into fractions, while 0-1 Knapsack does not. Signup and view all the answers

Which scenario would lead to an item being excluded in the 0-1 Knapsack problem?

Item weight is greater than the capacity of the knapsack. Signup and view all the answers

What does the term 'fractional' signify in Fractional Knapsack problem?

'Fractional' refers to breaking down items into smaller parts for inclusion. Signup and view all the answers

Which property distinguishes Dijkstra's algorithm from Greedy algorithms?

Dijkstra's algorithm guarantees finding the shortest path from one node to every other node. Signup and view all the answers

What characterizes an optimization problem?

Optimization problems involve maximizing or minimizing an objective function within constraints. Signup and view all the answers

What is the main characteristic of a greedy algorithm?

It makes locally optimal choices at each step Signup and view all the answers

In the 0-1 Knapsack problem, what does the integer array 'val' represent?

The values of n items to be placed in the knapsack Signup and view all the answers

What is the key difference between the Fractional Knapsack problem and the 0-1 Knapsack problem?

The Fractional Knapsack allows breaking items, while the 0-1 Knapsack does not Signup and view all the answers

Which problem-solving technique makes choices based on immediate benefit without considering future consequences?

Greedy algorithms Signup and view all the answers

What does Dijkstra's algorithm primarily aim to solve?

Find the shortest path in a graph Signup and view all the answers

Why might a greedy algorithm not always produce the optimal solution?

Because it makes locally optimal choices at each step Signup and view all the answers

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