John E. Hopcroft

An

The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to

n^{5/2}

(m + n)\sqrt n

Algorithm for Maximum Matchings in Bipartite Graphs

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Efficient Planarity Testing

This paper describes an efficient algorithm to determine whether an arbitrary graph <italic>G</italic> can be embedded in the plane. The algorithm may be viewed as an iterative version of a method originally proposed by Auslander and Parter and correctly formulated by Goldstein. The algorithm used depth-first search and has <italic>O</italic>(<italic>V</italic>) time and space bounds, where <italic>V</italic> is the number of vertices in <italic>G</italic>. An ALGOL implementation of the algorithm succesfully tested graphs with as many as 900 vertices in less than 12 seconds.

Dividing a Graph into Triconnected Components

An algorithm for dividing a graph into triconnected components is presented. When implemented on a random access computer, the algorithm requires

The Directed Subgraph Homeomorphism Problem

O(V + E)

Algorithm 447: efficient algorithms for graph manipulation

time and space to analyze a graph with V vertices and E edges. The algorithm is both theoretically optimal to within a constant factor and efficient in practice.

An n log n algorithm for minimizing states in a finite automaton

The set of pattern graphs for which the directed subgraph homeomorphism problem is NP-complete is characterized. A polynomial time algorithm is given for the remaining cases. The restricted problem where the input graph is a directed acyclic graph is in polynomial time for all pattern graphs and an algorithm is given.

Routing, merging, and sorting on parallel models of computation

Efficient algorithms are presented for partitioning a graph into connected components, biconnected components and simple paths. The algorithm for partitioning of a graph into simple paths of iterative and each iteration produces a new path between two vertices already on paths. (The start vertex can be specified dynamically.) If V is the number of vertices and E is the number of edges, each algorithm requires time and space proportional to max ( V, E ) when executed on a random access computer.

On the complexity of motion planning for multiple independent objects: PSPACE-Hardness of the «Warehouseman's problem»

An algorithm is given for minimizing the number of states in a finite automaton or for determining if two finite automata are equivalent. The asymptotic running time of the algorithm is bounded by k n log n where k is some constant and n is the number of states. The constant k depends linearly on the size of the input alphabet.