A directed graph (or digraph) is a set of vertices and a collection of directed edges that each connects an ordered pair of vertices. We say that a directed edge points from the first vertex in the pair and points to the second vertex in the pair.
Given a weighted line-graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devise an algorithm that preprocesses the graph in linear time and can return the distance of the shortest path between any two vertices in constant time.
Given a connected graph, design a linear-time algorithm to find a vertex whose removal (deleting the vertex and all incident edges) does not disconnect the graph.
4.1 Undirected Graphs introduces the graph data type, including depth-first search and breadth-first search. 4.2 Directed Graphs introduces the digraph data type, including topological sort and strong components.
Given an undirected graph and two vertices s and t, find a simple cycle containing both s and t (or report that no such cycle exists). Hint: there is a directed cycle containing both s and t if and only if there are two (internally) vertex-disjoint paths between s and t.
If a graph is not connected, we can adapt our algorithms to compute the MSTs of each of its connected components, collectively known as a minimum spanning forest.
Physicists refer to it as the Hoshen–Kopelman algorithm although it is simply union–find on a grid graph with raster-scan order. Applications include modeling percolation and electrical conductance.
3.2 Binary Search Trees We examine a symbol-table implementation that combines the flexibility of insertion in linked lists with the efficiency of search in an ordered array. Specifically, using two links per node leads to an efficient symbol-table implementation based on the binary search tree data structure, which qualifies as one of the most fundamental algorithms in computer science ...
