What are weighted graphs?

Graphs with values assigned to edges.

How are shortest paths determined in weighted graphs?

By summing edge weights, not just number of edges.

Finding shortest path from one vertex to all others.

Useful for scenarios like optimizing routes.

Disrupt by allowing indefinite cost reduction.

Weighted graphs provide detailed information by assigning values to edges, such as road lengths or travel times, going beyond just showing connectivity.
Shortest paths in weighted graphs are determined by the sum of edge weights, not just the number of edges, influencing the path length, with negative cycles posing a challenge by allowing indefinite cost reduction and rendering shortest paths undefined.

What are weighted graphs?
Graphs with values assigned to edges.
How are shortest paths determined in weighted graphs?
By summing edge weights, not just number of edges.
What is the significance of single source shortest path problems?
Finding shortest path from one vertex to all others.
Why are negative edge weights in graphs practical?
Useful for scenarios like optimizing routes.
How do negative cycles impact shortest paths in graphs?
Disrupt by allowing indefinite cost reduction.

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Weighted graphs introduce values assigned to edges, like road lengths or travel times, providing more detailed information than just connectivity.
Weighted graphs consist of vertices, edges, and weights assigned by a function to each edge, typically positive but can be negative in some scenarios.
Representation of weighted graphs can be done using an adjacency matrix, where edge weights replace the usual 1s or 0s.
Shortest paths in weighted graphs are determined by the sum of edge weights, not just the number of edges.
Shortest paths in weighted graphs may not follow the minimum number of edges, as the weight of each edge influences the path length.
Single source shortest path problems involve finding the shortest path from one fixed vertex to all other vertices, crucial for applications like transportation logistics.
All pair shortest path problems aim to find the shortest distance between every pair of vertices in a graph, essential for various scenarios like booking sites.
Negative edge weights in graphs can be practical, such as in scenarios like taxi drivers optimizing routes based on earning potential.
Negative cycles in graphs, where a series of edges result in a negative total weight, can disrupt the concept of shortest paths by allowing indefinite cost reduction.
Graphs with negative cycles render shortest paths undefined, as one can continuously traverse the cycle to reduce costs infinitely, but graphs with only negative edges are still solvable.

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In weighted graphs, each edge is assigned a cost or weight, which is represented in the adjacency matrix by replacing the usual 1 with the specific cost. This allows for measuring the length of a path based on the total sum of the weights of the edges traversed, rather than just the number of edges, resulting in a new concept of shortest path that may differ from the shortest path in terms of the number of edges taken.
There are two main types of shortest path problems: the single source path, where the goal is to determine the fastest way to reach every other vertex from a fixed starting point (e.g., relevant for courier companies), and the all pairs problem, essential for scenarios like travel agencies needing to advise on the best routes between any two locations. Additionally, when dealing with negative edge weights, it is crucial to avoid negative cycles to ensure the possibility of finding shortest paths even with negative edge weights present.