Graph Theory and DFS Algorithm Quiz

A stack data structure is used.

The source node is added to the stack.

It is removed from the stack and its children are added to the stack.

<p>When the destination node is reached, a path from source to destination is confirmed.</p> Signup and view all the answers

<p>The weight is 12.</p> Signup and view all the answers

<p>Expanded nodes are added to a set to prevent re-expansion.</p> Signup and view all the answers

<p>The algorithm should not expand the destination node once reached.</p> Signup and view all the answers

<p>A dictionary provides an efficient way to access a node's children and their weights.</p> Signup and view all the answers

<p>Every node must have an even number of edges connected to it.</p> Signup and view all the answers

<p>An adjacency list is more memory efficient compared to an adjacency matrix.</p> Signup and view all the answers

<p>Each node is associated with a list of neighboring nodes using a dictionary.</p> Signup and view all the answers

<p>Adjacency lists are the primary data structure used in this course.</p> Signup and view all the answers

<p>It illustrates important concepts such as Eulerian paths and the conditions for traversing a graph.</p> Signup and view all the answers

<p>A stack is used in the DFS algorithm.</p> Signup and view all the answers

<p>The value of $M_{ij}$ would be 0.</p> Signup and view all the answers

<p>BFS uses a queue while DFS uses a stack.</p> Signup and view all the answers

<p>Identifiers can be numbers, strings, or other types of objects.</p> Signup and view all the answers

<p>Expanding a node means exploring its neighbors to continue the path search.</p> Signup and view all the answers

<p>Tracking the distance helps determine the shortest path from the source to the destination.</p> Signup and view all the answers

<p>Depth-First Search (DFS) and Breadth-First Search (BFS) are mentioned.</p> Signup and view all the answers

<p>The shortest path and distance are updated when the destination node is reached.</p> Signup and view all the answers

<p>The shortest path is initialized as an empty string and the shortest distance as infinity.</p> Signup and view all the answers

<p>The BFS algorithm terminates when the queue is empty.</p> Signup and view all the answers

<p>Their recorded distances might differ if the weights of edges along the path vary based on the algorithm used.</p> Signup and view all the answers

<p>The <code>add_node</code> method adds a new node to the graph's adjacency list if it does not already exist.</p> Signup and view all the answers

<p>The <code>add_edge</code> method adds a directed edge from a source node to a destination node with an associated weight.</p> Signup and view all the answers

<p>The output is: 1:[(3, 5), (2, 8)] 2:[(3, 14)] 3:[]</p> Signup and view all the answers

<p>A minimum cut is a set of edges whose removal partitions the graph into two subsets with no edges between them, minimizing the sum of the weights of the removed edges.</p> Signup and view all the answers

<p>DFS explores deeper paths first, traversing down a branch before backtracking, while BFS explores all neighboring nodes first, moving level by level.</p> Signup and view all the answers

<p>The <code>get_children</code> method returns the list of adjacent nodes (children) connected to the specified node.</p> Signup and view all the answers

<p>A maximum clique is the largest subset of nodes where every node is connected to every other node in that subset.</p> Signup and view all the answers

<p>Both the source and destination nodes must already exist in the graph's adjacency list.</p> Signup and view all the answers

<p>$O(V + E)$</p> Signup and view all the answers

<p>It has an optimal sub-structure and overlapping subproblems.</p> Signup and view all the answers

<p>Memoization is a top-down approach storing solved subproblems, while tabular is a bottom-up approach solving smaller subproblems first.</p> Signup and view all the answers

<p>The same subproblems are recalculated multiple times.</p> Signup and view all the answers

<p>A queue, which is a first-in-first-out data structure.</p> Signup and view all the answers

<p>It is a problem that benefits from finding optimal solutions through dynamic programming methods.</p> Signup and view all the answers

<p>It addresses the smallest subproblems first and combines their results progressively.</p> Signup and view all the answers

<p>The tabular approach requires more memory and a clear understanding of the problem structure, suitable for problems needing all subproblems to find the answer.</p> Signup and view all the answers

<p>Overlapping subproblems ensure that solutions can be reused rather than recalculated.</p> Signup and view all the answers

<p>Memoization is more intuitive as it uses modified recursive forms, requiring less computation of all subproblems and often utilizes advanced data structures like dictionaries.</p> Signup and view all the answers

<p>Results are retrieved from memory instead of re-computation.</p> Signup and view all the answers

<p>The time complexity is $ ext{O}(n)$ since each number from 0 to n requires a constant time calculation.</p> Signup and view all the answers

<p>Problems that exhibit optimal sub-structure and have overlapping subproblems.</p> Signup and view all the answers

<p>It can be expressed as $T(n) = T(n-1) + T(n-2) + C$, where C is a constant.</p> Signup and view all the answers

<p>The upper bound is $ ext{O}(2^{n/2})$, indicating exponential growth in computation time.</p> Signup and view all the answers

<p>As we substitute values, we can show that $T(n)$ will grow larger than various exponential forms, specifically exceeding $ ext{O}(2^{(n-1)/2})$.</p> Signup and view all the answers

<p>Memoization is preferred for these problems due to its intuitive approach and reduced computation of unnecessary subproblems.</p> Signup and view all the answers

<p>Understanding dictionaries is vital in memoization because they facilitate the storage and retrieval of previously computed values efficiently.</p> Signup and view all the answers

A graph representation where each node is represented by a unique identifier (e.g., a number, string, or object) and its connections (edges) are stored in a list associated with that identifier.

A way to represent a graph where rows and columns correspond to nodes, and a value of 1 indicates an edge between those nodes, while 0 indicates no edge.

A graph search algorithm that explores a graph by visiting nodes depth-first, meaning it explores all the children of a node before moving to its siblings.

A problem of determining whether a walk exists in a graph that crosses every edge exactly once.

A graph search algorithm that explores a graph by visiting nodes level by level, meaning it visits all nodes at a specific distance from the starting node before moving to the next level.

A technique for solving problems by breaking them down into smaller overlapping subproblems and storing the solutions to these subproblems to avoid redundant computations.

A type of problem that focuses on finding the optimal solution within a graph, often involving minimizing cost or maximizing profit.

A graph where each node represents a landmass and each edge represents a bridge. Used to solve problems like the Bridges of Königsberg.

A data structure used to represent relationships between entities (nodes). In a graph, nodes are connected by edges, which can be directed or undirected and may have weights associated with them.

A collection of nodes (vertices) connected by edges. In a directed graph, edges have a direction, meaning they flow from one node to another.

The numerical value associated with an edge in a graph. Edge weights often represent distance, cost, or any other relevant measure.

A list of all the nodes that a given node is connected to, along with their corresponding edge weights. This is a common way to represent a graph in Python.

The problem of finding the path with the smallest sum of edge weights between two nodes in a weighted graph. It is a common problem in graph theory with applications in routing, network optimization, and shortest path finding.

A set of edges in a graph whose removal partitions the graph into two disconnected subgraphs. The minimum cut is the set of edges with the minimum sum of weights that achieves this separation.

The total weight of the edges along a path in a graph.

The shortest path in a graph is the path with the minimum total weight.

A data structure used in DFS, which follows the last-in-first-out (LIFO) principle.

A data structure used in BFS, which follows the first-in-first-out (FIFO) principle.

The set of nodes that have already been explored in a search algorithm.

The algorithm uses a stack to explore the graph in depth-first manner, keeping track of the shortest path and distance.

The algorithm uses a queue to explore the graph in breadth-first manner, keeping track of the shortest path and distance.

Depth-First Search (DFS) is a graph traversal algorithm that explores as far as possible along each branch before backtracking. It uses a stack to keep track of which nodes to visit next.

The DFS algorithm starts by adding the source node to a stack. Then, it repeatedly takes a node from the stack and expands it if it hasn't been expanded before. This process continues until the stack is empty.

The DFS algorithm is often used to find paths between nodes in a graph, similar to how a maze solver works.

In a graph, a shortest path is the path between two nodes with the lowest total edge weights (if weights are assigned).

A graph can be represented using dictionaries, where each key represents a node and its value is a list of its neighboring nodes with their corresponding edge weights.

The Graph class is a Python class that represents a graph structure. It contains methods to retrieve neighbors and their weights for a given node.

The dfs(graph, source) function takes a graph and a source node as input and performs a Depth-First Search traversal from the source node. It prints the path taken during the search.

To adapt the DFS algorithm to find the shortest path, we need to keep track of the shortest path found so far. Whenever we reach the destination node, we update the shortest path if the current path is shorter. We also need to prevent the algorithm from expanding the destination node to avoid infinite loops.

A problem where the optimal solution can be constructed from optimal solutions of its subproblems (smaller parts).

A problem where overlapping subproblems arise, meaning some subproblems are solved multiple times during the problem solution process.

Dynamic Programming approach that saves the results of solved subproblems in a memo (structure) for future use, avoiding redundant calculations.

Dynamic Programming approach that solves the problem in a bottom-up manner, starting with the simplest subproblems and building up to the final solution.

The process of finding the optimal way to fill a knapsack with items, given a weight limit and item values, where items can either be fully included or excluded.

The sequence where each number is the sum of the two preceding numbers (e.g., 0, 1, 1, 2, 3, 5, 8, ...).

A mathematical notation used to describe the growth rate of an algorithm, ignoring constant factors and lower-order terms. For example, O(n) represents linear growth, while O(n^2) represents quadratic growth.

A method for solving problems where you calculate and store all possible subproblems beforehand, often using tables. It's better for problems where you need to solve all subproblems to get the final answer.

A method for solving problems where you break down the problem into smaller subproblems and calculate them as needed. It's more intuitive and uses recursive functions but requires more memory.

A function that calls itself within its definition. It's a powerful tool in recursion and dynamic programming.

A data structure that stores key-value pairs, allowing fast retrieval of values based on their keys. Commonly used in memoization for dynamic programming problems.

The time it takes to execute an algorithm. In dynamic programming, the goal is to optimize the time complexity of the algorithm.

The amount of memory an algorithm uses during execution. It's important to consider both time complexity and space complexity when designing algorithms.