Determine all shortest routes from A to B in the system of...

Determine all shortest routes from $A$ to $B$ in the system of intersections and streets (graph) in the following diagram. The numbers on the streets represent the lengths of the streets measured in terms of some unit.

Key Concepts

All Shortest Paths

In some cases, there might be multiple distinct routes between two nodes that all yield the same total minimal weight. Identifying all shortest paths involves not only finding a single optimal route but also enumerating every route that achieves the minimal cost, which can be useful in applications requiring redundancy or multiple options.

Graph Theory

Graph theory is the branch of mathematics that studies the properties of graphs, which consist of nodes (or vertices) connected by edges. In problems like this one, intersections can be modeled as nodes and streets as edges, helping to visualize and solve routing problems.

Weighted Graph

A weighted graph is a graph where each edge has an associated numerical value or weight, often representing a cost, distance, or time. These weights are used to quantify the cost of traveling from one node to another and are critical in determining the most efficient route.

Shortest Path Problem

The shortest path problem involves finding the path between two nodes in a weighted graph such that the sum of the weights along the path is minimized. This problem is central in many fields like transportation, computer networks, and logistics, where the goal is to optimize travel cost or time.

Dijkstra's Algorithm

Dijkstra's Algorithm is a popular method used to solve the shortest path problem on graphs with non-negative weights. It works by iteratively selecting the node with the smallest tentative distance from the source and updating the distances to its neighboring nodes, ensuring that the most efficient path is found.

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