Hopcroft's minimization: Valmari and Lehtinen's variant for a partial transition function

include/mata/nfa/algorithms.hh

@@ -32,6 +32,16 @@ namespace mata::nfa::algorithms {
  */
 Nfa minimize_brzozowski(const Nfa& aut);
 
+/**
+ * Hopcroft minimization of automata. Based on the algorithm from the paper:
+ *  "Efficient Minimization of DFAs With Partial Transition Functions" by Antti Valmari and Petri Lehtinen.
+ *  The algorithm works in O(a*n*log(n)) time and O(m+n+a) space, where: n is the number of states, a is the size
+ *  of the alphabet, and m is the number of transitions. [https://dl.acm.org/doi/10.1016/j.ipl.2011.12.004]
+ * @param[in] dfa_trimmed Deterministic automaton without useless states. Perform trimming before calling this function.
+ * @return Minimized deterministic automaton.
+ */
+Nfa minimize_hopcroft(const Nfa& dfa_trimmed);
+
 /**
  * Complement implemented by determization, adding sink state and making automaton complete. Then it adds final states
  *  which were non final in the original automaton.

src/nfa/operations.cc

@@ -950,6 +950,359 @@ Nfa mata::nfa::minimize(
     return algo(aut);
 }
 
+// Anonymous namespace for the Hopcroft minimization algorithm.
+namespace {
+template <typename T>
+/**
+ * A class for partitionable sets of elements. All elements are stored in a contiguous vector.
+ * Elements from the same set are contiguous in the vector. Auxiliary data (indices, positions, etc.) are
+ * stored in separate vectors.
+ */
+class RefinablePartition {
+public:
+    static_assert(std::is_unsigned<T>::value, "T must be an unsigned type.");
+    static const T NO_MORE_ELEMENTS = std::numeric_limits<T>::max();
+    static const size_t NO_SPLIT = std::numeric_limits<size_t>::max();
+    size_t num_of_sets;            ///< The number of sets in the partition.
+    std::vector<size_t> set_idx;   ///< For each element, tells the index of the set it belongs to.
+
+private:
+    std::vector<T> elems;       ///< Contains all elements in an order such that the elements of the same set are contiguous.
+    std::vector<T> location;    ///< Contains the location of each element in the elements vector.
+    std::vector<size_t> first;  ///< Contains the index of the first element of each set in the elements vector.
+    std::vector<size_t> end;    ///< Contains the index after the last element of each set in the elements vector.
+    std::vector<size_t> mid;    ///< Contains the index after the last element of the first half of the set in the elements vector.
+
+public:
+    /**
+     * @brief Construct a new Refinable Partition for equivalence classes of states.
+     *
+     * @param n The number of states.
+     */
+    RefinablePartition(const size_t num_of_states)
+        : num_of_sets(1), set_idx(num_of_states), elems(num_of_states), location(num_of_states),
+          first(num_of_states), end(num_of_states), mid(num_of_states) {
+        // Initially, all states are in the same equivalence class.
+        first[0] = mid[0] = 0;
+        end[0] = num_of_states;
+        for (State e = 0; e < num_of_states; ++e) {
+            elems[e] = location[e] = e;
+            set_idx[e] = 0;
+        }
+    }
+
+    /**
+     * @brief Construct a new Refinable Partition for sets of splitters (transitions incoming to
+     *  equivalence classes under a common symbol). Initially, the partition has n sets, where n is the alphabet size.
+     *
+     * @param delta The transition function.
+     */
+    RefinablePartition(const Delta &delta)
+        : num_of_sets(0), set_idx(), elems(), location(), first(), end(), mid() {
+        size_t num_of_transitions = 0;
+        std::vector<size_t> counts;
+        std::unordered_map<Symbol, size_t> symbol_map;
+
+        // Transitions are grouped by symbols using counting sort in time O(m).
+        // Count the number of elements and the number of sets.
+        for (const auto &trans : delta.transitions()) {
+            const Symbol a = trans.symbol;
+            if (symbol_map.find(a) == symbol_map.end()) {
+                symbol_map[a] = num_of_sets++;
+                counts.push_back(1);
+            } else {
+                ++counts[symbol_map[a]];
+            }
+            ++num_of_transitions;
+        }
+
+        // Initialize data structures.
+        elems.resize(num_of_transitions);
+        location.resize(num_of_transitions);
+        set_idx.resize(num_of_transitions);
+        first.resize(num_of_transitions);
+        end.resize(num_of_transitions);
+        mid.resize(num_of_transitions);
+
+        // Compute set indices.
+        // Use (mid - 1) as an index for the insertion.
+        first[0] = 0;
+        end[0] = counts[0];
+        mid[0] = end[0];
+        for (size_t i = 1; i < num_of_sets; ++i) {
+            first[i] = end[i - 1];
+            end[i] = first[i] + counts[i];
+            mid[i] = end[i];
+        }
+
+        // Fill the sets from the back.
+        // Mid, decremented before use, is used as an index for the next element.
+        size_t trans_idx = 0;   // Index of the transition in the (flattened) delta.
+        for (const auto &trans : delta.transitions()) {
+            const Symbol a = trans.symbol;
+            const size_t a_idx = symbol_map[a];
+            const size_t trans_loc = mid[a_idx] - 1;
+            mid[a_idx] = trans_loc;
+            elems[trans_loc] = trans_idx;
+            location[trans_idx] = trans_loc;
+            set_idx[trans_idx] = a_idx;
+            ++trans_idx;
+        }
+    }
+
+    RefinablePartition(const RefinablePartition &other)
+        : elems(other.elems), set_idx(other.set_idx), location(other.location),
+          first(other.first), end(other.end), mid(other.mid), num_of_sets(other.num_of_sets) {}
+
+    RefinablePartition(RefinablePartition &&other) noexcept
+        : elems(std::move(other.elems)), set_idx(std::move(other.set_idx)), location(std::move(other.location)),
+          first(std::move(other.first)), end(std::move(other.end)), mid(std::move(other.mid)), num_of_sets(other.num_of_sets) {}
+
+    RefinablePartition& operator=(const RefinablePartition &other) {
+        if (this != &other) {
+            elems = other.elems;
+            set_idx = other.set_idx;
+            location = other.location;
+            first = other.first;
+            end = other.end;
+            mid = other.mid;
+            num_of_sets = other.num_of_sets;
+        }
+        return *this;
+    }
+
+    RefinablePartition& operator=(RefinablePartition &&other) noexcept {
+        if (this != &other) {
+            elems = std::move(other.elems);
+            set_idx = std::move(other.set_idx);
+            location = std::move(other.location);
+            first = std::move(other.first);
+            end = std::move(other.end);
+            mid = std::move(other.mid);
+            num_of_sets = other.num_of_sets;
+        }
+        return *this;
+    }
+
+    /**
+     * @brief Get the number of elements across all sets.
+     *
+     * @return The number of elements.
+     */
+    inline size_t size() const { return elems.size(); }
+
+    /**
+     * @brief Get the size of the set.
+     *
+     * @param s The set index.
+     * @return The size of the set.
+     */
+    inline size_t size_of_set(size_t s) const { return end[s] - first[s]; }
+
+    /**
+     * @brief Get the first element of the set.
+     *
+     * @param s The set index.
+     * @return The first element of the set.
+     */
+    inline T get_first(const size_t s) const { return elems[first[s]]; }
+
+    /**
+     * @brief Get the next element of the set.
+     *
+     * @param e The element.
+     * @return The next element of the set or NO_MORE_ELEMENTS if there is no next element.
+     */
+    inline T get_next(const T e) const {
+        if (location[e] + 1 >= end[set_idx[e]]) { return NO_MORE_ELEMENTS; }
+        return elems[location[e] + 1];
+    }
+
+    /**
+     * @brief Mark the element and move it to the first half of the set.
+     *
+     * @param e The element.
+     */
+    void mark(const T e) {
+        const size_t e_set = set_idx[e];
+        const size_t e_loc = location[e];
+        const size_t e_set_mid = mid[e_set];
+        if (e_loc >= e_set_mid) {
+            elems[e_loc] = elems[e_set_mid];
+            location[elems[e_loc]] = e_loc;
+            elems[e_set_mid] = e;
+            location[e] = e_set_mid;
+            mid[e_set] = e_set_mid + 1;
+        }
+    }
+
+    /**
+     * @brief Test if the set has no marked elements.
+     *
+     * @param s The set index.
+     * @return True if the set has no marked elements, false otherwise.
+     */
+    inline bool has_no_marks(const size_t s) const { return mid[s] == first[s]; }
+
+    /**
+     * @brief Split the set into two sets according to the marked elements (the mid).
+     * The first set name remains the same.
+     *
+     * @param s The set index.
+     * @return The new set index.
+     */
+    size_t split(const size_t s) {
+        if (mid[s] == end[s]) {
+            // If no elements were marked, move the mid to the end (no split needed).
+            mid[s] = first[s];
+        }
+        if (mid[s] == first[s]) {
+            // If all or no elements were marked, there is no need to split the set.
+            return NO_SPLIT;
+        } else {
+            // Split the set according to the mid.
+            first[num_of_sets] = first[s];
+            mid[num_of_sets] = first[s];
+            end[num_of_sets] = mid[s];
+            first[s] = mid[s];
+            for (size_t l = first[num_of_sets]; l < end[num_of_sets]; ++l) {
+                set_idx[elems[l]] = num_of_sets;
+            }
+            return num_of_sets++;
+        }
+    }
+};
+} // namespace
+
+
+Nfa mata::nfa::algorithms::minimize_hopcroft(const Nfa& dfa_trimmed) {
+    if (dfa_trimmed.delta.num_of_transitions() == 0) {
+        // The automaton is trivially minimal.
+        return Nfa{ dfa_trimmed };
+    }
+    assert(dfa_trimmed.is_deterministic());
+    assert(dfa_trimmed.get_useful_states().size() == dfa_trimmed.num_of_states());
+
+    // Initialize equivalence classes. The initial partition is {Q}.
+    RefinablePartition<State> brp(dfa_trimmed.num_of_states());
+    // Initialize splitters. A splitter is a set of transitions
+    // over a common symbol incoming to an equivalence class. Initially,
+    // the partition has m splitters, where m is the alphabet size.
+    RefinablePartition<size_t> trp(dfa_trimmed.delta);
+
+    // Initialize vector of incoming transitions for each state. Transitions
+    // are represented only by their indices in the (flattened) delta.
+    // Initialize mapping from transition index to its source state.
+    std::vector<State> trans_source_map(trp.size());
+    std::vector<std::vector<size_t>> incomming_trans_idxs(brp.size(), std::vector<size_t>());
+    size_t trans_idx = 0;
+    for (const auto &trans : dfa_trimmed.delta.transitions()) {
+        trans_source_map[trans_idx] = trans.source;
+        incomming_trans_idxs[trans.target].push_back(trans_idx);
+        ++trans_idx;
+    }
+
+    // Worklists for the Hopcroft algorithm.
+    std::stack<size_t> unready_spls;    // Splitters that will be used in the backpropagation.
+    std::stack<size_t> touched_blocks;  // Blocks (equivalence classes) touched during backpropagation.
+    std::stack<size_t> touched_spls;    // Splitters touched (in the split_block function) as a result of backpropagation.
+
+    /**
+     * @brief Split the block (equivalence class) according to the marked states.
+     *
+     * @param b The block index.
+     */
+    auto split_block = [&](size_t b) {
+        // touched_spls has been moved to a higher scope to avoid multiple constructions/destructions.
+        assert(touched_spls.empty());
+        size_t b_prime = brp.split(b);  // One block will keep the old name 'b'.
+        if (b_prime == RefinablePartition<size_t>::NO_SPLIT) {
+            // All or no states in the block were marked (touched) during the backpropagation.
+            return;
+        }
+        // According to the Hopcroft's algorithm, it is sufficient to work only with the smaller block.
+        if (brp.size_of_set(b) < brp.size_of_set(b_prime)) {
+            b_prime = b;
+        }
+        // Split the transitions of the splitters according to the new partitioning.
+        // Transitions in one splitter must have the same symbol and go to the same block.
+        for (State q = brp.get_first(b_prime); q != RefinablePartition<State>::NO_MORE_ELEMENTS; q = brp.get_next(q)) {
+            for (const size_t trans_idx : incomming_trans_idxs[q]) {
+                const size_t splitter_idx = trp.set_idx[trans_idx];
+                if (trp.has_no_marks(splitter_idx)) {
+                    touched_spls.push(splitter_idx);
+                }
+                // Mark the transition in the splitter and move it to the first half of the set.
+                trp.mark(trans_idx);
+            }
+        }
+        // Refine all splitters where some transitions were marked.
+        while (!touched_spls.empty()) {
+            const size_t splitter_idx = touched_spls.top();
+            touched_spls.pop();
+            const size_t splitter_pime = trp.split(splitter_idx);
+            if (splitter_pime != RefinablePartition<size_t>::NO_SPLIT) {
+                // Use the new splitter for further refinement of the equivalence classes.
+                unready_spls.push(splitter_pime);
+            }
+        }
+    };
+
+    // Use all splitters for the initial refinement.
+    for (size_t splitter_idx = 0; splitter_idx < trp.num_of_sets; ++splitter_idx) {
+        unready_spls.push(splitter_idx);
+    }
+
+    // In the first refinement, we split the equivalence classes according to the final states.
+    for (const State q : dfa_trimmed.final) {
+        brp.mark(q);
+    }
+    split_block(0);
+
+    // Main loop of the Hopcroft's algorithm.
+    while (!unready_spls.empty()) {
+        const size_t splitter_idx = unready_spls.top();
+        unready_spls.pop();
+        // Backpropagation.
+        // Fire back all transitions of the splitter. (Transitions over the same
+        // symbol that go to the same block.) Mark the source states of these transitions.
+        for (size_t trans_idx = trp.get_first(splitter_idx); trans_idx != RefinablePartition<size_t>::NO_MORE_ELEMENTS; trans_idx = trp.get_next(trans_idx)) {
+            const State q = trans_source_map[trans_idx];
+            const size_t b_prime = brp.set_idx[q];
+            if (brp.has_no_marks(b_prime)) {
+                touched_blocks.push(b_prime);
+            }
+            brp.mark(q);
+        }
+        // Try to split the blocks touched during the backpropagation.
+        // The block will be split only if some states (not all) were touched (marked).
+        while (!touched_blocks.empty()) {
+            const size_t b = touched_blocks.top();
+            touched_blocks.pop();
+            split_block(b);
+        }
+    }
+
+    // Construct the minimized automaton using equivalence classes (BRP).
+    assert(dfa_trimmed.initial.size() == 1);
+    Nfa result(brp.num_of_sets, StateSet{ brp.set_idx[*dfa_trimmed.initial.begin()] }, StateSet{});
+    for (size_t block_idx = 0; block_idx < brp.num_of_sets; ++block_idx) {
+        const State q = brp.get_first(block_idx);
+        if (dfa_trimmed.final.contains(q)) {
+            result.final.insert(block_idx);
+        }
+        StatePost &mut_state_post = result.delta.mutable_state_post(block_idx);
+        for (const SymbolPost &symbol_post : dfa_trimmed.delta[q]) {
+            assert(symbol_post.targets.size() == 1);
+            const State target = brp.set_idx[*symbol_post.targets.begin()];
+            mut_state_post.push_back(SymbolPost{ symbol_post.symbol, StateSet{ target } });
+        }
+    }
+
+    return result;
+}
+
+
 Nfa mata::nfa::intersection(const Nfa& lhs, const Nfa& rhs, const Symbol first_epsilon, std::unordered_map<std::pair<State, State>, State>  *prod_map) {
 
     auto both_final = [&](const State lhs_state,const State rhs_state) {