Design Ch.11

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

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

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

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

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

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

<p>Prim's algorithm (A), Kruskal's algorithm (D)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A problem-solving method that builds an optimal solution by making locally optimal choices, one step at a time.

Choices that fit the rules or constraints of the problem.

Choices that are the best at the current step.

Choices that are not changeable.

Problems focused on finding the best or optimal solution from available options.

A graph where edges have no direction and every vertex is reachable from any other vertex.

A subgraph of a connected graph that includes all vertices and forms a tree (no cycles).

The spanning tree of a weighted graph with the smallest possible total edge weight.

The task of finding the MST for a given weighted connected graph.

Exhaustive search for MST involves trying all possible spanning trees, which grows exponentially with the number of vertices. This becomes computationally infeasible for large graphs.

A type of algorithm that makes locally optimal choices at each step to build the MST. It chooses the edge with the smallest weight that doesn't create a cycle.

A famous greedy algorithm for finding the MST. It sorts edges by weight and adds them one by one if they don't form a cycle.

An algorithm for finding the Minimum Spanning Tree (MST) of a graph, which connects all vertices with the lowest total edge weight.

A cycle is formed in Kruskal's algorithm when adding an edge connects two vertices already in the same connected component.

An algorithm used in Kruskal's algorithm to efficiently determine if two vertices belong to the same connected component.

A problem in graph theory to find the shortest paths from a single starting vertex (source) to all other vertices in the graph.

A greedy algorithm for finding the shortest paths from a single source vertex in a weighted graph, based on iteratively building a tree of shortest paths.

The numerical label assigned to a vertex in Dijkstra's algorithm, representing the length of the shortest path from the source to that vertex.

Dijkstra's algorithm doesn't work for graphs with negative edge weights, as it might lead to incorrect shortest paths.

The efficiency of Dijkstra's algorithm depends on the graph representation and the priority queue implementation.

Using an adjacency list to represent the graph in Dijkstra's algorithm leads to better time complexity (O(|E|log|V|)) compared to a weight matrix.

Building a minimum spanning tree by adding edges in order of increasing weight, avoiding cycles.

Adding an edge that creates a closed loop in the tree.

Choosing the locally best option at each step to build the MST.

Edges are considered in increasing weight order, from smallest to largest.

The best edge to add at the current step, even if it's not globally optimal.

The cost or value assigned to an edge in a weighted graph.

Connecting all vertices in a graph without creating cycles.

Cycles create redundant paths, increasing the total weight of the tree, making it non-minimal.