Design Ch.10 1

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Questions and Answers

What type of problem does the knapsack problem represent?

Sorting problem

Graph traversal problem

Dynamic programming problem (correct)

Linear programming problem

What is the main objective when constructing a binary search tree (BST) given n keys and their search probabilities?

Minimize the average number of comparisons in successful search (correct)

Ensure all keys have equal spacing

Maximize the height of the tree

Maximize the frequency of search operations

If a binary search tree is constructed with keys in ascending order and equal probabilities, what is likely to happen?

The tree will be perfectly balanced

The performance of searches will be optimized

The tree will have a constant search time

The number of comparisons will be maximized (correct)

Which of the following is NOT an example of a dynamic programming algorithm?

Binary search algorithm Signup and view all the answers

Which property of the keys directly influences the construction of the optimal binary search tree?

The associated search probabilities Signup and view all the answers

In the context of the knapsack problem, what is meant by finding the most valuable subset of items?

Maximizing the total value of the items without exceeding weight capacity Signup and view all the answers

What is a defining characteristic of dynamic programming problems?

They have overlapping subproblems. Signup and view all the answers

How does dynamic programming typically compute solutions?

By solving subproblems in a bottom-up manner. Signup and view all the answers

Which algorithm would be used to solve the traveling salesman problem?

Dynamic programming Signup and view all the answers

When trying to construct a binary search tree with minimum average comparisons, what is considered a key factor in the decision process?

The distribution pattern of the probabilities Signup and view all the answers

What is the outcome of combining the solutions of subproblems in dynamic programming?

It yields the solution to the original problem. Signup and view all the answers

What is the expected outcome of a properly constructed binary search tree with respect to search operations?

It will minimize the number of comparisons based on key probabilities Signup and view all the answers

What does the variable 'W' represent in the context of the knapsack problem?

The maximum capacity of the knapsack Signup and view all the answers

Which statement is incorrect regarding dynamic programming?

Dynamic programming cannot solve problems that require memoization. Signup and view all the answers

What method is used to derive solutions from subproblems in dynamic programming?

An iterative approach that solves the simplest to the most complex. Signup and view all the answers

What is the mathematical formula used to calculate the total number of Binary Search Trees (BSTs) with n nodes?

$C(2n,n)/(n+1)$ Signup and view all the answers

Why is brute force not an efficient method to calculate the number of BSTs as n increases?

The total number of BSTs grows exponentially. Signup and view all the answers

What search probability is assigned to key C in the example of an optimal BST for A, B, C, and D?

0.4 Signup and view all the answers

Given the keys A, B, C, and D with respective search probabilities of 0.1, 0.2, 0.4, and 0.3, which key has the lowest search probability?

A Signup and view all the answers

In the context of optimal BSTs, which of the following is typically a consideration when organizing nodes?

Minimizing overall tree height. Signup and view all the answers

Study Notes

Dynamic Programming

Dynamic programming is a technique for solving problems with overlapping subproblems.
Subproblems often arise from a recurrence relating a solution to a given problem with solutions to its smaller subproblems.
Dynamic programming solves each smaller subproblem once and records the results in a table.
A solution to the original problem can then be obtained from the table.
Applicability to an optimization problem requires the principle of optimality.
This means an optimal solution to any instance is made up of optimal solutions to subinstances.

Examples of Dynamic Programming Algorithms

Knapsack Problem: This combinatorial optimization problem involves finding the most valuable subset of items that fit within a knapsack with a given capacity. A dynamic programming approach is used to solve this issue.
Optimal Binary Search Trees: Determining the optimal binary search tree for a given set of keys and search probabilities is solved using dynamic programming.
Warshall's Algorithm: This algorithm computes the transitive closure of a binary relation or the existence of all paths between vertices in a directed graph.
Floyd's Algorithm: This algorithm finds the shortest paths between all pairs of vertices in a weighted graph (with no negative-length cycle).

Optimal Binary Search Trees

Can be used to implement dictionaries (searching, insertion, deletion). Finding the BST with the minimum average number of comparisons in successful searches.
Using a brute force approach is not efficient as the number of possible BSTs increases exponentially with the number of nodes.
A dynamic programming solution utilizes a recurrence relation to determine the minimum average number of comparisons.

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Description

Explore the fundamentals of dynamic programming and its applications in solving problems with overlapping subproblems. This quiz covers key concepts such as the principle of optimality and examples like the Knapsack Problem and Optimal Binary Search Trees.