diff --git a/src/bin/autfilt.cc b/src/bin/autfilt.cc
index 2fdb1ca31..14ca7b335 100644
--- a/src/bin/autfilt.cc
+++ b/src/bin/autfilt.cc
@@ -210,6 +210,7 @@ static int opt_seed = 0;
 static auto dict = spot::make_bdd_dict();
 static spot::tgba_digraph_ptr opt_product = nullptr;
 static bool opt_merge = false;
+static std::unique_ptr<spot::isomorphism_checker> isomorphism_checker = nullptr;
 static spot::tgba_digraph_ptr opt_are_isomorphic = nullptr;
 static bool opt_is_complete = false;
 static bool opt_is_deterministic = false;
@@ -574,10 +575,7 @@ namespace
       if (opt_is_deterministic)
 	matched &= is_deterministic(aut);
       if (opt_are_isomorphic)
-        {
-          spot::tgba_digraph_ptr tmp = make_tgba_digraph(aut);
-          matched &= (*spot::canonicalize(tmp) == *opt_are_isomorphic);
-        }
+        matched &= isomorphism_checker->is_isomorphic(aut);
       if (opt_is_empty)
 	matched &= aut->is_empty();
 
@@ -708,7 +706,8 @@ main(int argc, char** argv)
     {
       if (opt_merge)
         opt_are_isomorphic->merge_transitions();
-      spot::canonicalize(opt_are_isomorphic);
+      isomorphism_checker = std::unique_ptr<spot::isomorphism_checker>(
+                            new spot::isomorphism_checker(opt_are_isomorphic));
     }
 
 
diff --git a/src/tgbaalgos/are_isomorphic.cc b/src/tgbaalgos/are_isomorphic.cc
index acf54dba4..8e622c339 100644
--- a/src/tgbaalgos/are_isomorphic.cc
+++ b/src/tgbaalgos/are_isomorphic.cc
@@ -23,17 +23,127 @@
 #include "tgba/tgbagraph.hh"
 #include "tgbaalgos/are_isomorphic.hh"
 #include "tgbaalgos/canonicalize.hh"
+#include "tgbaalgos/isdet.hh"
+#include <vector>
+#include <queue>
+
+namespace
+{
+  typedef spot::tgba_digraph::graph_t::trans_storage_t tr_t;
+  bool
+  tr_t_less_than(const tr_t& t1, const tr_t& t2)
+  {
+    return t1.cond.id() < t2.cond.id();
+  }
+
+  bool
+  operator!=(const tr_t& t1, const tr_t& t2)
+  {
+    return t1.cond.id() != t2.cond.id();
+  }
+
+  bool
+  are_isomorphic_det(const spot::const_tgba_digraph_ptr aut1,
+                     const spot::const_tgba_digraph_ptr aut2)
+  {
+    typedef std::pair<unsigned, unsigned> state_pair_t;
+    std::queue<state_pair_t> workqueue;
+    workqueue.emplace(aut1->get_init_state_number(),
+                      aut2->get_init_state_number());
+    std::vector<unsigned> map(aut1->num_states(), -1U);
+    map[aut1->get_init_state_number()] = aut2->get_init_state_number();
+    std::vector<tr_t> trans1;
+    std::vector<tr_t> trans2;
+    state_pair_t current_state;
+    while (!workqueue.empty())
+      {
+        current_state = workqueue.front();
+        workqueue.pop();
+
+        for (auto& t : aut1->out(current_state.first))
+          trans1.emplace_back(t);
+        for (auto& t : aut2->out(current_state.second))
+          trans2.emplace_back(t);
+
+        if (trans1.size() != trans2.size())
+          return false;
+
+        std::sort(trans1.begin(), trans1.end(), tr_t_less_than);
+        std::sort(trans2.begin(), trans2.end(), tr_t_less_than);
+
+        for (auto t1 = trans1.begin(), t2 = trans2.begin();
+             t1 != trans1.end() && t2 != trans2.end();
+             ++t1, ++t2)
+          {
+            if (*t1 != *t2)
+              {
+                return false;
+              }
+            if (map[t1->dst] == -1U)
+              {
+                map[t1->dst] = t2->dst;
+                workqueue.emplace(t1->dst, t2->dst);
+              }
+            else if (map[t1->dst] != t2->dst)
+              {
+                return false;
+              }
+          }
+
+        trans1.clear();
+        trans2.clear();
+      }
+    return true;
+  }
+
+  bool
+  trivially_different(const spot::const_tgba_digraph_ptr aut1,
+                      const spot::const_tgba_digraph_ptr aut2)
+  {
+    return aut1->num_states() != aut2->num_states() ||
+      aut1->num_transitions() != aut2->num_transitions() ||
+      aut1->acc().num_sets() != aut2->acc().num_sets();
+  }
+}
 
 namespace spot
 {
-  bool
-  are_isomorphic(const const_tgba_digraph_ptr aut1,
-                 const const_tgba_digraph_ptr aut2)
+  isomorphism_checker::isomorphism_checker(const const_tgba_digraph_ptr ref)
   {
-    auto tmp1 = make_tgba_digraph(aut1);
-    auto tmp2 = make_tgba_digraph(aut2);
-    spot::canonicalize(tmp1);
-    spot::canonicalize(tmp2);
-    return *tmp1 == *tmp2;
+    ref_ = make_tgba_digraph(ref);
+    ref_deterministic_ = ref_->is_deterministic();
+    if (!ref_deterministic_)
+      {
+        nondet_states_ = spot::count_nondet_states(ref_);
+        ref_deterministic_ = (nondet_states_ == 0);
+      }
+    canonicalize(ref_);
+  }
+
+  bool
+  isomorphism_checker::is_isomorphic(const const_tgba_digraph_ptr aut)
+  {
+    if (trivially_different(ref_, aut))
+      return false;
+
+    if (ref_deterministic_)
+      {
+        if (aut->is_deterministic() || spot::is_deterministic(aut))
+          {
+            return are_isomorphic_det(ref_, aut);
+          }
+      }
+    else
+      {
+        if (aut->is_deterministic() ||
+            nondet_states_ != spot::count_nondet_states(aut))
+          {
+            return false;
+          }
+      }
+
+    auto tmp = make_tgba_digraph(aut);
+    spot::canonicalize(tmp);
+    return *tmp == *ref_;
   }
 }
diff --git a/src/tgbaalgos/are_isomorphic.hh b/src/tgbaalgos/are_isomorphic.hh
index 81ad0b138..5a1c2ac6d 100644
--- a/src/tgbaalgos/are_isomorphic.hh
+++ b/src/tgbaalgos/are_isomorphic.hh
@@ -25,19 +25,33 @@
 
 namespace spot
 {
-  /// \ingroup tgba_misc
-  /// \brief Check whether two automata are isomorphic.
-  ///
-  /// Two automata are considered isomorphic if there exists a bijection f
-  /// between the states of a1 and the states of a2 such that for any pair of
-  /// states (s1, s2) of a1, there is a transition from s1 to s2 with
-  /// condition c and acceptance set A iff there is a transition with
-  /// condition c and acceptance set A between f(s1) and f(s2) in a2.
-  /// This can be done simply by checking if
-  /// canonicalize(aut1) == canonicalize(aut2).
-  SPOT_API bool
-  are_isomorphic(const const_tgba_digraph_ptr aut1,
-		 const const_tgba_digraph_ptr aut2);
+  /// Check if two automata are isomorphic.
+  class SPOT_API isomorphism_checker
+  {
+  public:
+    isomorphism_checker(const const_tgba_digraph_ptr ref);
+
+    /// \ingroup tgba_misc
+    /// \brief Check whether an automaton is isomorphic to the one passed to
+    /// the constructor.
+    ///
+    /// Two automata are considered isomorphic if there exists a bijection f
+    /// between the states of a1 and the states of a2 such that for any pair of
+    /// states (s1, s2) of a1, there is a transition from s1 to s2 with
+    /// condition c and acceptance set A iff there is a transition with
+    /// condition c and acceptance set A between f(s1) and f(s2) in a2.
+    /// This can be done simply by checking if
+    /// canonicalize(aut1) == canonicalize(aut2), but is_isomorphic can do some
+    /// optimizations in some cases.
+    bool
+    is_isomorphic(const const_tgba_digraph_ptr aut);
+
+  private:
+    tgba_digraph_ptr ref_;
+    bool ref_deterministic_ = false;
+    unsigned nondet_states_ = 0;
+    std::vector<tgba_digraph::graph_t::trans_storage_t> reftrans_;
+  };
 }
 
 #endif
diff --git a/src/tgbatest/isomorph.test b/src/tgbatest/isomorph.test
index 34d5d6766..bb59a5170 100755
--- a/src/tgbatest/isomorph.test
+++ b/src/tgbatest/isomorph.test
@@ -22,13 +22,17 @@
 
 set -e
 
-for i in 0 1 2 3 4; do
+for i in 0 1 2; do
   ../../bin/randaut a b --seed=$i -S10 --hoa >iso$i
   ../../bin/autfilt iso$i --randomize --hoa >aut$i
 done
+for i in 3 4 5; do
+  ../../bin/randaut a b --seed=$i -S10 -D --hoa >iso$i
+  ../../bin/autfilt iso$i --randomize --hoa >aut$i
+done
 
-cat aut0 aut1 aut2 aut3 aut4 > all
-(for i in 0 1 2 3 4; do
+cat aut0 aut1 aut2 aut3 aut4 aut5 > all
+(for i in 0 1 2 3 4 5; do
   run 0 ../../bin/autfilt all --are-isomorphic iso$i --hoa
 done) > output
 diff all output