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    <p>At a relatively <em>higher</em> C/C++ programmer level, how a graph/network is implemented depends quite heavily on what operations are performed on it. Being an OR person myself, I am probably biased in my response/example here. Some of the most efficient algorithms that can be implemented on graphs/networks are polynomial-time algorithms. Most, if not all, polynomial-time algorithms I can think of (Dijkstra's shortest path problem, transportation problem, max-flow problem, etc.) are special cases of the minimum-cost-flow (MCF) problem. Computationally, one of the most efficient ways of solving an MCF problem is via the network-simplex algorithm (which in itself is a specialization of the simplex algorithm to solve a general linear program). </p> <p>In the network-simplex-algorithm, a spanning tree (over the set of nodes) needs to be efficiently represented. To represent a spanning tree in a graph, a variety of data structures can be used. These include the following node-length</p> <pre><code>predecessor[], thread[] and depth[] arrays. </code></pre> <p>In the most efficient implementations of network-simplex-algorithms that I have come across, these are not represented as arrays but some sort of dynamically created block of memory via</p> <pre><code>calloc(number_of_nodes, sizeof(struct vertex)); </code></pre> <p>I am unsure (at a relatively <em>lower</em> level) internal to the compiler what/how this memory allocation is implemented - whether it is a list/set/map.</p> <p>So, in summary, the <em>best</em> way to implement a graph is highly dependent on the operations to be performed on it.</p> <p>Network simplex algorithm and the data structures needed to efficiently implement the same can be found in <a href="http://books.google.com/books?id=2DKKHvV_xVwC" rel="nofollow">this book</a>.</p>
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    1. POQuestion the way graphs are taught
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    1. COYeah, that's right. It depends on the operations you think you may need to perform. But, there are some certain facts that are common along all implementations. For example, if the graph can't have repeated elements (for example, people connected in facebook) and the dataset is big (for example, the 550M users of facebook), then if we use a simple list, the complexity increases linearly. Then, for insert a person, we should check 550M users first!! This, with other data structure can be performed easier. A HashTable (HashSet, too) is a good idea.
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