Weighted Graph

Preview

1. What is a weighted graph?
2. How are edge weights in a weighted graph represented?
3. What algorithms are commonly used to find the shortest path in a weighted graph?
4. What is the difference between a directed weighted graph and an undirected weighted graph?
5. Can a weighted graph have negative edge weights? If so, what are the implications for algorithms that use such graphs?

Explain

A weighted graph is a graph where each edge has a numerical value or weight assigned to it. The weight represents the cost, distance or some other relevant quantity associated with that edge.

For example, consider a city with multiple bus routes connecting different locations. A weighted graph can be used to represent the bus network where the nodes represent different stops or locations, and the edges represent the bus routes. The weight on each edge could be the time taken to travel between two stops or the fare for a trip. With a weighted graph, we can use algorithms like Dijkstra’s algorithm to find the shortest path between any two stops or locations based on cost, time or some other relevant criterion.

Keypoint

1. Weighted graph is a graph where each edge (or link) is associated with a numerical value, known as weight or cost.
2. Weighted graph used to represent networks where the edges have different costs, such as road networks with varying distances between nodes or computer networks with different bandwidths.
3. The weight of an edge can represent various things, such as the distance between nodes, the time it takes to travel between nodes, the cost of transmitting data between nodes, etc.
4. The weights on edges can be positive or negative, indicating a gain or loss of some sort.
5. A weighted graph can be represented using an adjacency matrix, where the elements of the matrix represent the weight of the edges between nodes.
6. Weighted graphs are used in several algorithms such as Dijkstra’s shortest path algorithm, Prim’s minimum spanning tree algorithm, and the Bellman-Ford algorithm.
7. The edges of a weighted graph may or may not have a direction (undirected or directed graphs).
8. The total weight of a path in a weighted graph is the sum of the weights of the edges on the path.
9. The goal in many applications involving weighted graphs is to find the shortest or minimum cost path from one node to another.

Review

1. What is a weighted graph?
   Answer: A weighted graph is a graph in which each edge is assigned a weight/value.

2. What is the purpose of assigning weights to edges in a graph?
   Answer: Assigning weights to edges in a graph allows the graph algorithms to calculate the shortest path, minimum spanning tree, and other useful calculations that are not feasible with unweighted graphs.

3. What is the difference between a weighted graph and a directed graph?
   Answer: A weighted graph can be either directed or undirected, but each edge has an assigned weight. In contrast, a directed graph (also known as a digraph) assigns a direction to each edge.

4. Can a weighted graph have negative edge weights?
   Answer: Yes, a weighted graph can have negative edge weights. Negative edge weights are usually used to represent scenarios where the weight of an edge represents a cost, and negative costs can occur in certain situations.

5. What is Dijkstra’s algorithm?
   Answer: Dijkstra’s algorithm is an algorithm used to find the shortest path between two nodes in a weighted graph. It works by maintaining a set of unvisited nodes and assigning tentative distances to each node in the graph, then selecting the node with the smallest tentative distance and updating its neighbors’ tentative distances. This process is repeated until the desired endpoint is reached, and the shortest path is calculated based on the distances assigned to each node.