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Incremental Construction of Minimal Finite
State Automata
Jan Daciuk
Gdansk University of Technology
e-mail: jandac@eti.pg.gda.pl
Overview
• Incrementality and semi-incrementality
• Trie construction, minimization, synchronization
• Incremental algorithm for sorted data
• Unsorted data and confluence states
• Incremental algorithm for unsorted data
• Performance
Incrementality and automata
• Final automata
� ideal implementation of dictionaries
� very efficient once constructed
� traditional construction needs much memory
• Incremental and semi-incremental construction
requires less memory
• Moore’s law
Traditional construction
• Construct a trie
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• Minimize it
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Incremental vs. semi-incremental construction
algorithms
start
add word
minimize
start
add word
last?
minimize
last?
stop
stop
reduce
Incremental and semi-incremental algorithms
for acyclic automata
• Incremental algorithm for (lexicographically) sorted data (Daciuk,
Mihov, Ciura, Deorowicz)
• Incremental algorithm for unsorted data (Aoe, Morimoto, Hase,
Sgarbas, Fakotakis, Kokkinakis, Daciuk, Watson, Revuz. . . )
• Semi-incremental algorithm for data lexicographically sorted on
reversed strings (Revuz)
• Semi-incremental algorithm for data sorted on decreasing length
of strings (Watson)
Construction of the trie
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Construction of the trie
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Construction of the trie
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Construction of the trie
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What is actually minimization?
• M = (Q,Σ, δ, q0, F ), |M | = |Q|
• M is minimal iff ∀M ′:L(M ′)=L(M) |M | < |M ′|
• −→L (q) = {w : δ∗(q, w) ∈ F}, L(M) =−→L (q0)
• p ≡ q iff−→L (p) =
−→L (q).
• M is minimal iff ∀p,q∈Q p ≡ q ⇔ p = q
• −→L (q) =⋃a:δ(q,a) 6=⊥ a
−→L (δ(q, a)) ∪ ∅ q 6∈ F
ε q ∈ F
What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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What is minimization of a trie?
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The register
How does one check equivalence of two states?
• Use the recursive definition of−→L (q)
• Visit states using postorder, so that children have unique right
languages
• Keep pointers to states with unique−→L (q) in a sparse table
• Use hash function on finality and transitions
Result: Operations on the register are O(1)
Synchronization
• Add a word to the language of the automaton and minimize
the whole automaton again
� does not pose any restrictions on input data
� the same states have to processed over and over again – slow
• Add a word to the language of the automaton and minimize
the part that will not change in the future
� requires data to be sorted in some way
� faster due to procesing of states only when once
Synchronization – sorted data
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Synchronization – sorted data
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Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)←repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then repl or reg(δ(s, wi), w′i+1...|w′|); end if;
9: while i ≤ |w| do10: δ(s, wi)← new state; s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
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1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
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1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from sorted data –
examples
1: function sorted construction;2: w′ ← ε;3: while input not empty do4: s← q0; i← 1; w ← next word;5: while i ≤ |w| and δ(s, wi)! = ⊥ do6: s← δ(s, wi); i← i + 1;7: end while;8: if i ≤ |w′| then
repl or reg(δ(s, wi), w′i+1...|w′|);end if;
9: while i ≤ |w| do10: δ(s, wi)← new state;
s← δ(s, wi); i← i + 1;11: end while;12: F ← F ∪ {s}; w′ ← w13: end while;14: repl or reg(q0, w′);15: end function;16: function repl or reg(q, v);17: if v 6= ε then18: δ(q, v1)← repl or reg(δ(q, v1), v2...|v|);19: end if;20: if ∃r∈R r ≡ q then21: delete q; return r;22: else23: R← R ∪ {q}; return q;24: end if;25: end function;
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Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incremental construction from unsorted data1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥ and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi); push(s, P ); i← i + 1;
10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi); push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then R← R \ {top(P )}; u← u− 1; end if;18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s}; if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Incr. constr. from unsorted data – examples
1: function unsorted construction;2: while input not empty do3: s← q0; i← 0; w ← next word; push(s, P );4: while i ≤ |w| and δ(s, wi) 6= ⊥
and fanin(δ(s, wi)) ≤ 1 do5: s← δ(s, wi); push(s, P ); i← i + 1;6: end while;7: R← R \ {s}; u← i;8: while i ≤ |w| and δ(s, wi) 6= ⊥ do9: δ(s, wi)← clone(δ(s, wi)); s← δ(s, wi);
push(s, P ); i← i + 1;10: end while;11: while i ≤ |w| do12: s← new state; s← δ(s, wi);
push(s, P ); i← i + 1;13: end while;14: F ← F ∪ {s}; pop(P );15: while P not empty do16: if ∃r∈R r ≡ s then17: if i = u and i > 0 then
R← R \ {top(P )}; u← u− 1;end if;
18: delete s; δ(top(P ), wi)← r;19: else20: R← R ∪ {s};
if i = u then break; end if;21: end if;22: i← i− 1; s← pop(P );23: end while;24: reset P ;25: end while;26: end function;
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Complexity and performance
• Both algorithms keep intermediate automata (almost) minimal
• Both run in time proportional to input data size
• The algorithm for sorted data is faster but less flexible
• Traditional algorithms are slower and use much more memory
• There are extensions of both algorithms to the case off adding
words to a cyclic automaton
top related