## An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton (2010)

Venue: | THEOR. COMPUT. SCI |

Citations: | 7 - 4 self |

### BibTeX

@ARTICLE{Holzer10ann,

author = {Markus Holzer and Andreas Maletti},

title = { An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton},

journal = {THEOR. COMPUT. SCI},

year = {2010},

pages = {3404--3413}

}

### OpenURL

### Abstract

We improve a recent result [Badr: Hyper-minimization in O(n²). Int. J. Found. Comput. Sci. 20, 2009] for hyper-minimized finite automata. Namely, we present an O(n log n) algorithm that computes for a given deterministic finite automaton (dfa) an almostequivalent dfa that is as small as possible such an automaton is called hyper-minimal. Here two finite automata are almost-equivalent if and only if the symmetric difference of their languages is finite. In other words, two almost-equivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [Badr, Geffert, Shipman: Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43, 2009] and by Badr. Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O(n log n) is optimal for hyper-minimization. Independently, similar results were obtained in [Gawrychowski, Jez: Hyper-minimisation made efficient. Proc. MFCS,

### Citations

4092 |
Introduction to automata theory, languages, and computation. Menlo Park
- Hopcroft, Motwani, et al.
- 2000
(Show Context)
Citation Context ...ation of an equivalent minimal automaton is PSPACE-complete [4] and thus highly intractable, the corresponding problem for deterministic automata is known to be e ectively solvable in polynomial time =-=[5]-=-. An automaton is minimal if every other automaton with fewer states disagrees on acceptance for at least one input. Minimizing deterministic nite automata (dfa) is based on computing an equivalence r... |

246 |
Finite automata and their decision problems
- Rabin, Scott
- 1959
(Show Context)
Citation Context ...7-760. Preprint submitted to Theoretical Computer Science April 30, 20101. Introduction Early studies in automata theory revealed that nondeterministic and deterministic nite automata are equivalent =-=[1]-=-. However, nondeterministic automata can be exponentially more succinct [2, 3] (with respect to the number of states). In fact, nite automata are probably best known for being equivalent to right-line... |

68 |
Economy of description by automata, grammars, and formal systems
- Meyer, Fischer
- 1971
(Show Context)
Citation Context ... Introduction Early studies in automata theory revealed that nondeterministic and deterministic nite automata are equivalent [1]. However, nondeterministic automata can be exponentially more succinct =-=[2, 3]-=- (with respect to the number of states). In fact, nite automata are probably best known for being equivalent to right-linear context-free grammars, and thus, for capturing the lowest level of the Chom... |

44 |
Minimal NFA problems are hard
- Jiang, Ravikumar
- 1993
(Show Context)
Citation Context ...guages are used in many applications, and one may like to represent the languages succinctly. While for nondeterministic automata the computation of an equivalent minimal automaton is PSPACE-complete =-=[4]-=- and thus highly intractable, the corresponding problem for deterministic automata is known to be e ectively solvable in polynomial time [5]. An automaton is minimal if every other automaton with fewe... |

40 |
Depth- rst search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...ernel states. It is shown in [9, 10, 11], how to identify the kernel states in time O(mn). However, the kernel states can also be computed using a well-known algorithm (see Algorithm 3) due to Tarjan =-=[16]-=- in time O(m). Theorem 4. Ker(M ) can be computed in time O(m). Proof. With Tarjan's algorithm [16] (or equivalently the algorithms by Gabow [17, 18] or Kosaraju [19, 20]) we can identify the strongly... |

40 |
A strong-connectivity algorithm and its application in data flow analysis
- Sharir
- 1981
(Show Context)
Citation Context ...m (see Algorithm 3) due to Tarjan [16] in time O(m). Theorem 4. Ker(M ) can be computed in time O(m). Proof. With Tarjan's algorithm [16] (or equivalently the algorithms by Gabow [17, 18] or Kosaraju =-=[19, 20]-=-) we can identify the strongly connected components (strongly connected states) in time O(m + n). Algorithm 3 presents a simpli ed version of the general known algorithm because in our setting all sta... |

19 |
On the bounds of state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way automata
- Moore
- 1971
(Show Context)
Citation Context ... Introduction Early studies in automata theory revealed that nondeterministic and deterministic nite automata are equivalent [1]. However, nondeterministic automata can be exponentially more succinct =-=[2, 3]-=- (with respect to the number of states). In fact, nite automata are probably best known for being equivalent to right-linear context-free grammars, and thus, for capturing the lowest level of the Chom... |

18 | Algorithms for dense graphs and networks on the random access computer
- CHERIYAN, K
- 1996
(Show Context)
Citation Context ...a well-known algorithm (see Algorithm 3) due to Tarjan [16] in time O(m). Theorem 4. Ker(M ) can be computed in time O(m). Proof. With Tarjan's algorithm [16] (or equivalently the algorithms by Gabow =-=[17, 18]-=- or Kosaraju [19, 20]) we can identify the strongly connected components (strongly connected states) in time O(m + n). Algorithm 3 presents a simpli ed version of the general known algorithm because i... |

15 | On the complexity of Hopcroft’s state minimization algorithm
- Berstel, Carton
- 2004
(Show Context)
Citation Context ...abet of input symbols of the nite automaton. It is up to now the best known minimization algorithm for dfa in general. Recent developments have shown that this bound is tight for Hopcroft's algorithm =-=[7, 8]-=-. Thus, minimization can be seen as a form of lossless compression that can be done e ectively while preserving the accepted language exactly. Recently, a new form of minimization, namely hyper-minimi... |

8 |
Hyper-minimization in O(n 2
- Badr
- 2009
(Show Context)
Citation Context ...form of lossless compression that can be done e ectively while preserving the accepted language exactly. Recently, a new form of minimization, namely hyper-minimization, was studied in the literature =-=[9, 10, 11]-=-. There the minimization or compression is done while giving up the preservation of the semantics of nite automata; i.e., the accepted language. It is clear that the semantics cannot vary arbitrarily.... |

6 |
Hopcroft’s algorithm and cyclic automata
- Castiglione, Restivo, et al.
- 2008
(Show Context)
Citation Context ...abet of input symbols of the nite automaton. It is up to now the best known minimization algorithm for dfa in general. Recent developments have shown that this bound is tight for Hopcroft's algorithm =-=[7, 8]-=-. Thus, minimization can be seen as a form of lossless compression that can be done e ectively while preserving the accepted language exactly. Recently, a new form of minimization, namely hyper-minimi... |

5 |
I.: Hyper-minimizing minimized deterministic nite state automata
- Badr, ert, et al.
- 2009
(Show Context)
Citation Context ...form of lossless compression that can be done e ectively while preserving the accepted language exactly. Recently, a new form of minimization, namely hyper-minimization, was studied in the literature =-=[9, 10, 11]-=-. There the minimization or compression is done while giving up the preservation of the semantics of nite automata; i.e., the accepted language. It is clear that the semantics cannot vary arbitrarily.... |

5 |
A.: Hyper-minimisation made e cient
- Gawrychowski, Je»
- 2009
(Show Context)
Citation Context ...or hyper-minimization implies that (classical) minimization can be done within t(n). To this end, we linearly reduce minimization to hyper-minimization. Similar results were independently obtained in =-=[14]-=-. The results of this paper were rst reported in [15]. This version contains the full, detailed proofs of the claims, a more elaborate example, and a few minor corrections. The paper is organized as f... |

4 | An n log n Algorithm for Hyperminimizing States in a (Minimized) Deterministic Automaton
- Holzer, Maletti
- 2009
(Show Context)
Citation Context ...zation can be done within t(n). To this end, we linearly reduce minimization to hyper-minimization. Similar results were independently obtained in [14]. The results of this paper were rst reported in =-=[15]-=-. This version contains the full, detailed proofs of the claims, a more elaborate example, and a few minor corrections. The paper is organized as follows: In the next section we introduce the necessar... |

3 |
On Hopcroft’s minimization technique for DFA
- Paŭn, Paŭn, et al.
(Show Context)
Citation Context ...servation of the semantics of nite automata; i.e., the accepted language. It is clear that the semantics cannot vary arbitrarily. A related minimization method based on cover automata is presented in =-=[12, 13]-=-. Hyper-minimization [9, 10, 11] allows the accepted language to di er in acceptance on a nite number of inputs, which is called almost-equivalence. Thus, hyper-minimization aims to nd an almost-equiv... |

2 | Minimal cover-automata for nite languages, Theoret - Câmpeanu, Santean, et al. |

2 |
Path-based depth- rst search for strong and biconnected components
- Gabow
(Show Context)
Citation Context ...a well-known algorithm (see Algorithm 3) due to Tarjan [16] in time O(m). Theorem 4. Ker(M ) can be computed in time O(m). Proof. With Tarjan's algorithm [16] (or equivalently the algorithms by Gabow =-=[17, 18]-=- or Kosaraju [19, 20]) we can identify the strongly connected components (strongly connected states) in time O(m + n). Algorithm 3 presents a simpli ed version of the general known algorithm because i... |

2 |
Strong-connectivity algorithm, unpublished manuscript
- Kosaraju
- 1978
(Show Context)
Citation Context ...m (see Algorithm 3) due to Tarjan [16] in time O(m). Theorem 4. Ker(M ) can be computed in time O(m). Proof. With Tarjan's algorithm [16] (or equivalently the algorithms by Gabow [17, 18] or Kosaraju =-=[19, 20]-=-) we can identify the strongly connected components (strongly connected states) in time O(m + n). Algorithm 3 presents a simpli ed version of the general known algorithm because in our setting all sta... |