Editing Shortest path problem (section)

== Applications ==
'''Network flows'''<ref>{{Cite book |last=Cormen |first=Thomas H. |title=Introduction to Algorithms |date=July 31, 2009 |publisher=MIT Press |isbn=9780262533058 |edition=3rd}}</ref> are a fundamental concept in graph theory and operations research, often used to model problems involving the transportation of goods, liquids, or information through a network. A network flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is the maximum amount that can flow through it. The goal is to find a feasible flow that maximizes the flow from a source node to a sink node.

'''Shortest Path Problems''' can be used to solve certain network flow problems, particularly when dealing with single-source, single-sink networks. In these scenarios, we can transform the network flow problem into a series of shortest path problems.

=== Transformation Steps ===
<ref>{{Cite book |last1=Kleinberg |first1=Jon |last2=Tardos |first2=Éva |title=Algorithm Design |publisher=Addison-Wesley |year=2005 |isbn=978-0321295354 |edition=1st |url=http://www.nytimes.com/2009/08/06/technology/06stats.html?_r=2&scp=1&sq=statistics&st=nyt}}</ref>
# '''Create a Residual Graph:'''
#* For each edge (u, v) in the original graph, create two edges in the residual graph:
#** (u, v) with capacity c(u, v)
#** (v, u) with capacity 0
#* The residual graph represents the remaining capacity available in the network.
# '''Find the Shortest Path:'''
#* Use a shortest path algorithm (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) to find the shortest path from the source node to the sink node in the residual graph.
# '''Augment the Flow:'''
#* Find the minimum capacity along the shortest path.
#* Increase the flow on the edges of the shortest path by this minimum capacity.
#* Decrease the capacity of the edges in the forward direction and increase the capacity of the edges in the backward direction.
# '''Update the Residual Graph:'''
#* Update the residual graph based on the augmented flow.
# '''Repeat:'''
#* Repeat steps 2-4 until no more paths can be found from the source to the sink.