Hopcroft-Tarjan planarity test

Hopcroft-Tarjan planarity test is a well-known algorithm in the field of Computer Science, developed by John Hopcroft and Robert Tarjan, two prominent researchers from Cornell University and Stanford University. This test is used to determine whether a given graph is planar, meaning it can be drawn in a plane without any edge crossings, as studied by Leonhard Euler and Kazimierz Kuratowski. The Hopcroft-Tarjan planarity test has numerous applications in various fields, including Computer-aided design and Network topology, which were also explored by Donald Knuth and Andrew Yao.

Introduction

The Hopcroft-Tarjan planarity test is based on the concept of Graph embedding, which involves drawing a graph on a surface, such as a Sphere or a Torus, without any edge crossings, as discussed by William Tutte and Hassler Whitney. This test uses a combination of Depth-first search and Breadth-first search to traverse the graph and check for planarity, similar to the approaches used by Edsger W. Dijkstra and Robert Floyd. The algorithm is implemented using a data structure called a stack, which is also used in other algorithms, such as the Shunting-yard algorithm developed by Edsger W. Dijkstra. The Hopcroft-Tarjan planarity test has been widely used in various applications, including Computer networks and Database systems, which were also studied by Larry Ellison and Michael Stonebraker.

Algorithm

The Hopcroft-Tarjan planarity test algorithm works by first checking if the graph is connected, as defined by Paul Erdős and Alfréd Rényi. If the graph is not connected, the algorithm checks each connected component separately, using techniques similar to those developed by Frank Harary and Cassius Ionescu-Tulcea. The algorithm then performs a depth-first search on the graph to check for any edge crossings, using a similar approach to the one used by Niklaus Wirth and Donald Shell. If any edge crossings are found, the graph is not planar, as shown by Kurt Gödel and Stephen Smale. The algorithm also uses a technique called Planar separator theorem, which was developed by Richard Karp and Michael Garey, to find a planar separator in the graph.

Implementation

The implementation of the Hopcroft-Tarjan planarity test algorithm involves using a combination of data structures, such as stacks and queues, as discussed by Donald Knuth and Robert Sedgewick. The algorithm is typically implemented in a programming language, such as C or Java, using libraries and frameworks, such as Boost C++ Libraries and Apache Commons, developed by Bjarne Stroustrup and James Gosling. The implementation also involves using various algorithms, such as Depth-first search and Breadth-first search, which were developed by Edsger W. Dijkstra and Robert Floyd. The Hopcroft-Tarjan planarity test has been implemented in various software packages, including Graphviz and NetworkX, which were developed by John Ellson and Aric Hagberg.

Time Complexity

The time complexity of the Hopcroft-Tarjan planarity test algorithm is O(n), where n is the number of vertices in the graph, as shown by Michael R. Garey and David S. Johnson. This makes the algorithm very efficient for large graphs, as discussed by Richard Karp and Michael Garey. The algorithm's time complexity is also compared to other planarity testing algorithms, such as the Boyer-Myrvold planarity test, which was developed by John Boyer and Wendy Myrvold. The Hopcroft-Tarjan planarity test has a lower time complexity than many other planarity testing algorithms, making it a popular choice for many applications, including Computer-aided design and Network topology, which were also explored by Donald Knuth and Andrew Yao.

Applications

The Hopcroft-Tarjan planarity test has numerous applications in various fields, including Computer-aided design and Network topology, which were also explored by Donald Knuth and Andrew Yao. The algorithm is used in Computer networks to determine the planarity of network graphs, as discussed by Larry Ellison and Michael Stonebraker. The algorithm is also used in Database systems to optimize query performance, as shown by Edgar F. Codd and Christopher Date. The Hopcroft-Tarjan planarity test is also used in Graph drawing and visualization to draw planar graphs, as developed by William Tutte and Hassler Whitney. The algorithm has been used in various software packages, including Graphviz and NetworkX, which were developed by John Ellson and Aric Hagberg.

History

The Hopcroft-Tarjan planarity test was first developed in the 1970s by John Hopcroft and Robert Tarjan, two prominent researchers from Cornell University and Stanford University. The algorithm was first published in a paper titled "Efficient Planarity Testing" in the Journal of the ACM, which was also edited by Donald Knuth and Andrew Yao. The algorithm was later improved and optimized by other researchers, including Michael R. Garey and David S. Johnson, who developed the Planar separator theorem. The Hopcroft-Tarjan planarity test has since become a widely used algorithm in many fields, including Computer Science and Mathematics, as discussed by Stephen Cook and Richard Karp. The algorithm has been recognized as one of the most important algorithms in the field of Computer Science, and has been awarded several prizes, including the Turing Award, which was also awarded to Alan Turing and John von Neumann. Category:Graph algorithms