So when Eric talked about depth first search and breadth first search in the last couple of lectures, we had directed graphs and undirected graphs. In particular, you have a much richer class of problems and applications if you allow for weights on graph edges.
Topological sorting Topological sorting is the algorithmic problem of finding a topological ordering of a given DAG. It can be solved in linear time. It maintains a list of vertices that have no incoming edges from other vertices that have not already been included in the partially constructed topological ordering; initially this list consists of the vertices with no incoming edges at all.
Then, it repeatedly adds one vertex from this list to the end of the partially constructed topological ordering, and checks whether its neighbors should be added to the list.
The algorithm terminates when all vertices have been processed in this way. The resulting orientation of the edges is called an acyclic orientation. Different total orders may lead to the same acyclic orientation, so an n-vertex graph can have fewer than n!
It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex. Any directed graph may be made into a DAG by removing a feedback vertex set or a feedback arc seta set of vertices or edges respectively that touches all cycles.
However, the smallest such set is NP-hard to find. Transitive closure and transitive reduction[ edit ] The transitive closure of a given DAG, with n vertices and m edges, may be constructed in time O mn by using either breadth-first search or depth-first search to test reachability from each vertex.
The transitive reduction consists of the edges that form length-one paths that are the only paths connecting their endpoints. Therefore, the transitive reduction can be constructed in the same asymptotic time bounds as the transitive closure. Closure problem The closure problem takes as input a directed acyclic graph with weights on its vertices and seeks the minimum or maximum weight of a closure, a set of vertices with no outgoing edges.
The problem may be formulated for directed graphs without the assumption of acyclicity, but with no greater generality, because in this case it is equivalent to the same problem on the condensation of the graph. It may be solved in polynomial time using a reduction to the maximum flow problem.
For example, it is possible to find shortest paths and longest paths from a given starting vertex in DAGs in linear time by processing the vertices in a topological order, and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges.
In this context, a dependency graph is a graph that has a vertex for each object to be updated, and an edge connecting two objects whenever one of them needs to be updated earlier than the other. A cycle in this graph is called a circular dependencyand is generally not allowed, because there would be no way to consistently schedule the tasks involved in the cycle.
Dependency graphs without circular dependencies form DAGs. For this problem, the tasks to be scheduled are the recalculations of the values of individual cells of the spreadsheet. Dependencies arise when an expression in one cell uses a value from another cell. In such a case, the value that is used must be recalculated earlier than the expression that uses it.
Topologically ordering the dependency graph, and using this topological order to schedule the cell updates, allows the whole spreadsheet to be updated with only a single evaluation per cell.
A somewhat different DAG-based formulation of scheduling constraints is used by the program evaluation and review technique PERTa method for management of large human projects that was one of the first applications of DAGs.
In this method, the vertices of a DAG represent milestones of a project rather than specific tasks to be performed. Instead, a task or activity is represented by an edge of a DAG, connecting two milestones that mark the beginning and completion of the task.
Each such edge is labeled with an estimate for the amount of time that it will take a team of workers to perform the task. The longest path in this DAG represents the critical path of the project, the one that controls the total time for the project. Individual milestones can be scheduled according to the lengths of the longest paths ending at their vertices.
In this representation, data enters a processing element through its incoming edges and leaves the element through its outgoing edges.
For instance, in electronic circuit design, static combinational logic blocks can be represented as an acyclic system of logic gates that computes a function of an input, where the input and output of the function are represented as individual bits.C++ Core Guidelines.
April 16, Editors: Bjarne Stroustrup; Herb Sutter; This is a living document under continuous improvement. Had it been an open-source (code) . Finding the longest common subsequence has applications in areas like biology. The longest subsequence (LCS) problem has an optimal substructure property.
We have discussed Knight’s tour and Rat in a Maze problems in Set 1 and Set 2 respectively. Let us discuss N Queen as another example problem that can be solved using Backtracking. Command Elements files indicates one or more filename or relative path filenames; see File Name Patterns for information on pattern matching path indicates a path on the local machine revision indicates a changeset which can be specified as a changeset revision number, a tag, or a unique substring of the changeset hash value.
The earliest instances of what might today be called genetic algorithms appeared in the late s and early s, programmed on computers by evolutionary biologists who were explicitly seeking to model aspects of natural evolution.
path. •Next shortest path is the shortest one edge extension of an already generated shortest path. Greedy Single Source All Destinations • Let d(i) (distanceFromSource(i)) be the length of a shortest one edge extension of an .