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<MIT's Introduction to Algorithms, Lectures 17, 18 and 19: Shortest Path Algorithms ⁄ >




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This is the twelfth post in an article series about MIT's lecture course "Introduction to Algorithms." In this post I will review a trilogy of lectures on graph and shortest path algorithms. They are lectures seventeen, eighteen and nineteen. They'll cover Dijkstra's Algorithm, Breadth-First Search Algorithm and Bellman-Ford Algorithm for finding single-source shortest paths as well as Floyd-Warshall Algorithm and Johnson's Algorithm for finding all-pairs shortest paths.

These algorithms require a thorough understanding of graphs. See the previous lecture for a good review of graphs.

Lecture seventeen focuses on the single-source shortest-paths problem: Given a graph G = (V, E), we want to find a shortest path from a given source vertex s &#8712; V to each vertex v &#8712; V. In this lecture the weights of edges are restricted to be positive which leads it to Dijkstra's algorithm and in a special case when all edges have unit weight to Breadth-first search algorithm.

Lecture eighteen also focuses on the same single-source shortest-paths problem, but allows edges to be negative. In this case a negative-weight cycles may exist and Dijkstra's algorithm would no longer work and would produce incorrect results. Bellman-Ford algorithm therefore is introduced that runs slower than Dijkstra's but detects negative cycles. As a corollary it is shown that Bellman-Ford solves Linear Programming problems with constraints in form xj - xi



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