Fibonacci Sequence and Dynamic Programming

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Questions and Answers

What is the value of F(3) in the Fibonacci sequence?

2 (correct)

When calculating F(n) recursively, which of the following values is necessary to ensure the recursion has a base case?

F(0) = 0 (correct)

F(1) = 2

F(0) = 1

F(1) = 0

Which of the following recursive calls would calculate F(4) correctly?

F(3) + F(2) (correct)

F(4) + F(3)

F(3) + F(1)

F(4) + F(0)

How many recursive calls are made to compute F(4) using the Fibonacci definition?

7 Signup and view all the answers

What pattern can be observed in the Fibonacci sequence values as n increases?

Each term is always greater than the previous two. Signup and view all the answers

What is a key characteristic of dynamic programming problems?

They have overlapping subproblems. Signup and view all the answers

How does dynamic programming typically compute solutions?

Bottom-up fashion. Signup and view all the answers

What does dynamic programming do with the solutions of subproblems?

It combines them to form a solution to the original problem. Signup and view all the answers

Which of the following statements about dynamic programming is FALSE?

Dynamic programming involves re-solving subproblems multiple times. Signup and view all the answers

What is the typical advantage of using dynamic programming?

It reduces computation by storing solutions to subproblems. Signup and view all the answers

What is the first step in the dynamic programming approach?

Dividing the problem into subproblems Signup and view all the answers

In dynamic programming, how does the approach to solving problems differ from traditional recursion?

It avoids unnecessary recomputation of subproblems Signup and view all the answers

Why is dividing a problem into subproblems considered a beneficial strategy in dynamic programming?

It simplifies complex problems by breaking them down Signup and view all the answers

What characteristic is essential for problems suitable for dynamic programming?

They must have overlapping subproblems Signup and view all the answers

What is the primary goal of structuring a problem using dynamic programming?

To find the most efficient solution by reusing subproblem solutions Signup and view all the answers

What does Bellman’s principle of optimality state regarding optimization problems?

An optimal solution is made up of optimal solutions to its sub instances. Signup and view all the answers

Which of the following best describes the efficiency aspects considered in dynamic programming?

Both time and space efficiency are critical in dynamic programming. Signup and view all the answers

What is typically a result of applying dynamic programming to a problem?

Reduction of the number of subproblems needed to reach a solution. Signup and view all the answers

In the context of optimization problems, what does the term 'subinstances' refer to?

Parts of the main problem that need to be solved to find the overall solution. Signup and view all the answers

How does dynamic programming typically manage the results from subproblems?

It caches solutions of subproblems to avoid redundant calculations. Signup and view all the answers

What is a fundamental characteristic of problems addressed with dynamic programming?

They have overlapping subproblems and optimal substructure. Signup and view all the answers

What is the primary goal of Floyd’s Algorithm?

Find shortest paths between every pair of vertices Signup and view all the answers

During which iteration does Floyd's Algorithm update the shortest paths using vertex k as an intermediate?

D(k) Signup and view all the answers

What equation represents the update of the shortest path values in Floyd's Algorithm?

$D(k)[i,j] = min(D(k-1)[i,j], D(k-1)[i,k] + D(k-1)[k,j])$ Signup and view all the answers

Which statement correctly describes the structure of matrices in Floyd's Algorithm?

Each new matrix builds on the previous matrix to include additional vertices as intermediates. Signup and view all the answers

Why is it important to use a weighted graph in Floyd's Algorithm?

To accurately represent the cost of traversing edges Signup and view all the answers

What type of algorithm is Floyd's Algorithm considered to be?

Dynamic programming Signup and view all the answers

Which of the following best explains the role of intermediate vertices in Floyd's Algorithm?

They dictate the potential path options for vertex pairs. Signup and view all the answers

What is the time efficiency of Floyd's Algorithm?

Θ(n^3) Signup and view all the answers

In what scenario would Floyd's Algorithm not be suitable to use?

When needing to find the shortest path between a single pair of vertices Signup and view all the answers

Which of the following describes the space efficiency of Floyd's Algorithm?

Matrices can be stored over their predecessors Signup and view all the answers

What does the term D(k)[i,j] refer to in the context of Floyd's Algorithm?

The shortest path distance between vertices i and j using up to k vertices Signup and view all the answers

Which of the following statements is TRUE regarding shortest paths in Floyd's Algorithm?

Shortest paths can be derived from the matrix. Signup and view all the answers

What is a unique feature of the matrices generated in Floyd's Algorithm?

They indicate shortest paths between all pairs of nodes. Signup and view all the answers

In terms of algorithm design, how does Floyd's Algorithm approach pathfinding?

It iteratively updates distances based on predecessors. Signup and view all the answers

Which type of graph can Floyd's Algorithm be applied to?

Graphs with positive and negative weights Signup and view all the answers

What does Floyd's Algorithm compute from the distance matrices beyond shortest paths?

Intermediate nodes in the paths Signup and view all the answers

What can be inferred about the distance matrices when using Floyd's Algorithm?

They may include infinite values for unreachable nodes. Signup and view all the answers

Study Notes

Dynamic Programming

Dynamic programming is a technique for solving problems with overlapping subproblems.
Typically, subproblems arise from a recurrence relating a solution to a given problem to solutions of its smaller subproblems of the same type.
Dynamic programming suggests solving each smaller subproblem once and recording the results in a table from which a solution to the original problem can be obtained.
Applicability to an optimization problem requires the problem to satisfy the principle of optimality, where an optimal solution to any instance is made of optimal solutions to its sub-instances.

Knapsack Problem

A knapsack problem involves finding the most valuable subset of items that fit into a knapsack of integer capacity.
The technique to compute the solution is given, expressed with mathematical formulas.
The solution is derived using a table that stores solutions to smaller instances.

Warshall's Algorithm

Warshall's algorithm calculates the transitive closure of a relation, identifying the existence of all non-trivial paths in a digraph.
Transitive closure is a binary relation.
It's a dynamic programming algorithm.

Floyd's Algorithm

Floyd's algorithm determines the shortest paths between every pair of vertices in a weighted digraph.
It uses a series of matrices, incrementally increasing subsets of allowed intermediate vertices, to find shortest paths.
The algorithm constructs solutions in the bottom up manner constructing solutions through series of matrices (D(0),...,D(n)) .

Optimal Binary Search Trees

Optimal binary search trees are used to implement dictionaries, searching, insertion, and deleting of keys in a given set.
The goal is to find a BST with a minimum average number of comparisons in successful searches considering probabilities of searching.
Dynamic programming is used to find an optimal BST and the average number of comparisons needed to find one key.

Required Readings

Chapter 8 from Introduction to Design and Analysis of Algorithms by Anany Levitin (2012).
Chapter 12 from Algorithm Design and Applications by Michael T. Goodrich and Roberto Tamassia (2014).

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Description

This quiz covers key concepts related to the Fibonacci sequence, including recursive calculations, base cases, and the principles of dynamic programming. Answer questions about how these algorithms function and their efficiencies compared to traditional recursion.