DiscreteMorseSandwich.h

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#pragma once
 
#include <DiscreteGradient.h>
 
#include <algorithm>
#include <numeric>
 
namespace ttk {
  class DiscreteMorseSandwich : virtual public Debug {
  public:
    DiscreteMorseSandwich();
 
    struct PersistencePair {
      SimplexId birth;
      SimplexId death;
      int type;
 
      PersistencePair(SimplexId b, SimplexId d, int t)
        : birth{b}, death{d}, type{t} {
      }
    };
 
    inline void preconditionTriangulation(AbstractTriangulation *const data) {
      this->dg_.preconditionTriangulation(data);
    }
 
    inline void setInputOffsets(const SimplexId *const offsets) {
      this->dg_.setInputOffsets(offsets);
    }
 
    inline void setComputeMinSad(const bool data) {
      this->ComputeMinSad = data;
    }
    inline void setComputeSadSad(const bool data) {
      this->ComputeSadSad = data;
    }
    inline void setComputeSadMax(const bool data) {
      this->ComputeSadMax = data;
    }
 
    template <typename triangulationType>
    inline int buildGradient(const void *const scalars,
                             const size_t scalarsMTime,
                             const SimplexId *const offsets,
                             const triangulationType &triangulation) {
      this->dg_.setDebugLevel(this->debugLevel_);
      this->dg_.setThreadNumber(this->threadNumber_);
      this->dg_.setInputOffsets(offsets);
      this->dg_.setInputScalarField(scalars, scalarsMTime);
      return this->dg_.buildGradient(triangulation);
    }
 
    inline void setGradient(ttk::dcg::DiscreteGradient &&dg) {
      this->dg_ = std::move(dg);
      // reset gradient pointer to local storage
      this->dg_.setLocalGradient();
    }
    inline ttk::dcg::DiscreteGradient &&getGradient() {
      return std::move(this->dg_);
    }
 
    template <typename triangulationType>
    inline SimplexId
      getCellGreaterVertex(const dcg::Cell &c,
                           const triangulationType &triangulation) {
      return this->dg_.getCellGreaterVertex(c, triangulation);
    }
 
    inline const std::vector<std::vector<SimplexId>> &
      get2SaddlesChildren() const {
      return this->s2Children_;
    }
 
    template <typename triangulationType>
    int computePersistencePairs(std::vector<PersistencePair> &pairs,
                                const SimplexId *const offsets,
                                const triangulationType &triangulation,
                                const bool ignoreBoundary,
                                const bool compute2SaddlesChildren = false);
 
    struct GeneratorType {
      std::vector<SimplexId> boundary;
      SimplexId critTriangleId;
      std::array<SimplexId, 2> critVertsIds;
    };
 
  protected:
    template <typename triangulationType>
    std::vector<std::vector<SimplexId>>
      getSaddle1ToMinima(const std::vector<SimplexId> &criticalEdges,
                         const triangulationType &triangulation) const;
 
    template <typename triangulationType,
              typename GFS,
              typename GFSN,
              typename OB>
    std::vector<std::vector<SimplexId>>
      getSaddle2ToMaxima(const std::vector<SimplexId> &criticalCells,
                         const GFS &getFaceStar,
                         const GFSN &getFaceStarNumber,
                         const OB &isOnBoundary,
                         const triangulationType &triangulation) const;
 
    template <typename triangulationType>
    void getMinSaddlePairs(std::vector<PersistencePair> &pairs,
                           std::vector<bool> &pairedMinima,
                           std::vector<bool> &paired1Saddles,
                           const std::vector<SimplexId> &criticalEdges,
                           const std::vector<SimplexId> &critEdgesOrder,
                           const SimplexId *const offsets,
                           const triangulationType &triangulation) const;
 
    template <typename triangulationType>
    void getMaxSaddlePairs(std::vector<PersistencePair> &pairs,
                           std::vector<bool> &pairedMaxima,
                           std::vector<bool> &pairedSaddles,
                           const std::vector<SimplexId> &criticalSaddles,
                           const std::vector<SimplexId> &critSaddlesOrder,
                           const std::vector<SimplexId> &critMaxsOrder,
                           const triangulationType &triangulation) const;
 
    template <typename triangulationType>
    void getSaddleSaddlePairs(std::vector<PersistencePair> &pairs,
                              std::vector<bool> &paired1Saddles,
                              std::vector<bool> &paired2Saddles,
                              const bool exportBoundaries,
                              std::vector<GeneratorType> &boundaries,
                              const std::vector<SimplexId> &critical1Saddles,
                              const std::vector<SimplexId> &critical2Saddles,
                              const std::vector<SimplexId> &crit1SaddlesOrder,
                              const triangulationType &triangulation) const;
 
    template <typename triangulationType>
    void extractCriticalCells(
      std::array<std::vector<SimplexId>, 4> &criticalCellsByDim,
      std::array<std::vector<SimplexId>, 4> &critCellsOrder,
      const SimplexId *const offsets,
      const triangulationType &triangulation,
      const bool sortEdges) const;
 
    void displayStats(
      const std::vector<PersistencePair> &pairs,
      const std::array<std::vector<SimplexId>, 4> &criticalCellsByDim,
      const std::vector<bool> &pairedMinima,
      const std::vector<bool> &paired1Saddles,
      const std::vector<bool> &paired2Saddles,
      const std::vector<bool> &pairedMaxima) const;
 
    using tripletType = std::array<SimplexId, 3>;
 
    void tripletsToPersistencePairs(std::vector<PersistencePair> &pairs,
                                    std::vector<bool> &pairedExtrema,
                                    std::vector<bool> &pairedSaddles,
                                    std::vector<SimplexId> &reps,
                                    std::vector<tripletType> &triplets,
                                    const SimplexId *const saddlesOrder,
                                    const SimplexId *const extremaOrder,
                                    const SimplexId pairDim) const;
 
    template <typename triangulationType, typename Container>
    SimplexId
      eliminateBoundariesSandwich(const SimplexId s2,
                                  std::vector<bool> &onBoundary,
                                  std::vector<Container> &s2Boundaries,
                                  const std::vector<SimplexId> &s2Mapping,
                                  const std::vector<SimplexId> &s1Mapping,
                                  std::vector<SimplexId> &partners,
                                  std::vector<Lock> &s1Locks,
                                  std::vector<Lock> &s2Locks,
                                  const triangulationType &triangulation) const;
 
    template <size_t n>
    struct Simplex {
      SimplexId id_{};
      std::array<SimplexId, n> vertsOrder_{};
      friend bool operator<(const Simplex<n> &lhs, const Simplex<n> &rhs) {
        return lhs.vertsOrder_ < rhs.vertsOrder_;
      }
    };
 
    struct EdgeSimplex : Simplex<2> {
      template <typename triangulationType>
      void fillEdge(const SimplexId id,
                    const SimplexId *const offsets,
                    const triangulationType &triangulation) {
        this->id_ = id;
        triangulation.getEdgeVertex(id, 0, this->vertsOrder_[0]);
        triangulation.getEdgeVertex(id, 1, this->vertsOrder_[1]);
        this->vertsOrder_[0] = offsets[this->vertsOrder_[0]];
        this->vertsOrder_[1] = offsets[this->vertsOrder_[1]];
        // sort vertices in decreasing order
        std::sort(this->vertsOrder_.rbegin(), this->vertsOrder_.rend());
      }
    };
 
    struct TriangleSimplex : Simplex<3> {
      template <typename triangulationType>
      void fillTriangle(const SimplexId id,
                        const SimplexId *const offsets,
                        const triangulationType &triangulation) {
        this->id_ = id;
        triangulation.getTriangleVertex(id, 0, this->vertsOrder_[0]);
        triangulation.getTriangleVertex(id, 1, this->vertsOrder_[1]);
        triangulation.getTriangleVertex(id, 2, this->vertsOrder_[2]);
        this->vertsOrder_[0] = offsets[this->vertsOrder_[0]];
        this->vertsOrder_[1] = offsets[this->vertsOrder_[1]];
        this->vertsOrder_[2] = offsets[this->vertsOrder_[2]];
        // sort vertices in decreasing order
        std::sort(this->vertsOrder_.rbegin(), this->vertsOrder_.rend());
      }
    };
 
    struct TetraSimplex : Simplex<4> {
      template <typename triangulationType>
      void fillTetra(const SimplexId id,
                     const SimplexId *const offsets,
                     const triangulationType &triangulation) {
        this->id_ = id;
        triangulation.getCellVertex(id, 0, this->vertsOrder_[0]);
        triangulation.getCellVertex(id, 1, this->vertsOrder_[1]);
        triangulation.getCellVertex(id, 2, this->vertsOrder_[2]);
        triangulation.getCellVertex(id, 3, this->vertsOrder_[3]);
        this->vertsOrder_[0] = offsets[this->vertsOrder_[0]];
        this->vertsOrder_[1] = offsets[this->vertsOrder_[1]];
        this->vertsOrder_[2] = offsets[this->vertsOrder_[2]];
        this->vertsOrder_[3] = offsets[this->vertsOrder_[3]];
        // sort vertices in decreasing order
        std::sort(this->vertsOrder_.rbegin(), this->vertsOrder_.rend());
      }
    };
 
    template <typename triangulationType>
    void alloc(const triangulationType &triangulation) {
      Timer tm{};
      const auto dim{this->dg_.getDimensionality()};
      if(dim > 3 || dim < 1) {
        return;
      }
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel master num_threads(threadNumber_)
#endif
      {
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif // TTK_ENABLE_OPENMP
        this->firstRepMin_.resize(triangulation.getNumberOfVertices());
        if(dim > 1) {
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->firstRepMax_.resize(triangulation.getNumberOfCells());
        }
        if(dim > 2) {
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->critEdges_.resize(triangulation.getNumberOfEdges());
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->edgeTrianglePartner_.resize(
            triangulation.getNumberOfEdges(), -1);
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->onBoundary_.resize(triangulation.getNumberOfEdges(), false);
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->s2Mapping_.resize(triangulation.getNumberOfTriangles(), -1);
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->s1Mapping_.resize(triangulation.getNumberOfEdges(), -1);
        }
        for(int i = 0; i < dim + 1; ++i) {
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->pairedCritCells_[i].resize(
            this->dg_.getNumberOfCells(i, triangulation), false);
        }
        for(int i = 1; i < dim + 1; ++i) {
#ifdef TTK_ENABLE_OPENMP
#pragma omp task
#endif
          this->critCellsOrder_[i].resize(
            this->dg_.getNumberOfCells(i, triangulation), -1);
        }
      }
      this->printMsg("Memory allocations", 1.0, tm.getElapsedTime(), 1,
                     debug::LineMode::NEW, debug::Priority::DETAIL);
    }
 
    void clear() {
      Timer tm{};
      this->firstRepMin_ = {};
      this->firstRepMax_ = {};
      this->edgeTrianglePartner_ = {};
      this->s2Mapping_ = {};
      this->s1Mapping_ = {};
      this->critEdges_ = {};
      this->pairedCritCells_ = {};
      this->onBoundary_ = {};
      this->critCellsOrder_ = {};
      this->printMsg("Memory cleanup", 1.0, tm.getElapsedTime(), 1,
                     debug::LineMode::NEW, debug::Priority::DETAIL);
    }
 
    dcg::DiscreteGradient dg_{};
 
    // factor memory allocations outside computation loops
    mutable std::vector<SimplexId> firstRepMin_{}, firstRepMax_{},
      edgeTrianglePartner_{}, s2Mapping_{}, s1Mapping_{};
    mutable std::vector<EdgeSimplex> critEdges_{};
    mutable std::array<std::vector<bool>, 4> pairedCritCells_{};
    mutable std::vector<bool> onBoundary_{};
    mutable std::array<std::vector<SimplexId>, 4> critCellsOrder_{};
    mutable std::vector<std::vector<SimplexId>> s2Children_{};
 
    bool ComputeMinSad{true};
    bool ComputeSadSad{true};
    bool ComputeSadMax{true};
    bool Compute2SaddlesChildren{false};
  };
} // namespace ttk
 
template <typename triangulationType>
std::vector<std::vector<SimplexId>>
  ttk::DiscreteMorseSandwich::getSaddle1ToMinima(
    const std::vector<SimplexId> &criticalEdges,
    const triangulationType &triangulation) const {
 
  Timer tm{};
 
  std::vector<std::vector<SimplexId>> res(criticalEdges.size());
 
  // follow vpaths from 1-saddles to minima
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel for num_threads(threadNumber_)
#endif
  for(size_t i = 0; i < criticalEdges.size(); ++i) {
    auto &mins = res[i];
 
    const auto followVPath = [this, &mins, &triangulation](const SimplexId v) {
      std::vector<Cell> vpath{};
      this->dg_.getDescendingPath(Cell{0, v}, vpath, triangulation);
      const Cell &lastCell = vpath.back();
      if(lastCell.dim_ == 0 && this->dg_.isCellCritical(lastCell)) {
        mins.emplace_back(lastCell.id_);
      }
    };
 
    // critical edge vertices
    SimplexId v0{}, v1{};
    triangulation.getEdgeVertex(criticalEdges[i], 0, v0);
    triangulation.getEdgeVertex(criticalEdges[i], 1, v1);
 
    // follow vpath from each vertex of the critical edge
    followVPath(v0);
    followVPath(v1);
  }
 
  this->printMsg("Computed the descending 1-separatrices", 1.0,
                 tm.getElapsedTime(), this->threadNumber_, debug::LineMode::NEW,
                 debug::Priority::DETAIL);
 
  return res;
}
 
template <typename triangulationType, typename GFS, typename GFSN, typename OB>
std::vector<std::vector<SimplexId>>
  ttk::DiscreteMorseSandwich::getSaddle2ToMaxima(
    const std::vector<SimplexId> &criticalCells,
    const GFS &getFaceStar,
    const GFSN &getFaceStarNumber,
    const OB &isOnBoundary,
    const triangulationType &triangulation) const {
 
  Timer tm{};
 
  const auto dim = this->dg_.getDimensionality();
  std::vector<std::vector<SimplexId>> res(criticalCells.size());
 
  // follow vpaths from 2-saddles to maxima
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel for num_threads(threadNumber_)
#endif
  for(size_t i = 0; i < criticalCells.size(); ++i) {
    const auto sid = criticalCells[i];
    auto &maxs = res[i];
 
    const auto followVPath
      = [this, dim, &maxs, &triangulation](const SimplexId v) {
          std::vector<Cell> vpath{};
          this->dg_.getAscendingPath(Cell{dim, v}, vpath, triangulation);
          const Cell &lastCell = vpath.back();
          if(lastCell.dim_ == dim && this->dg_.isCellCritical(lastCell)) {
            maxs.emplace_back(lastCell.id_);
          } else if(lastCell.dim_ == dim - 1) {
            maxs.emplace_back(-1);
          }
        };
 
    const auto starNumber = getFaceStarNumber(sid);
 
    for(SimplexId j = 0; j < starNumber; ++j) {
      SimplexId cellId{};
      getFaceStar(sid, j, cellId);
      followVPath(cellId);
    }
 
    if(isOnBoundary(sid)) {
      // critical saddle is on boundary
      maxs.emplace_back(-1);
    }
  }
 
  this->printMsg("Computed the ascending 1-separatrices", 1.0,
                 tm.getElapsedTime(), this->threadNumber_, debug::LineMode::NEW,
                 debug::Priority::DETAIL);
 
  return res;
}
 
template <typename triangulationType>
void ttk::DiscreteMorseSandwich::getMinSaddlePairs(
  std::vector<PersistencePair> &pairs,
  std::vector<bool> &pairedMinima,
  std::vector<bool> &paired1Saddles,
  const std::vector<SimplexId> &criticalEdges,
  const std::vector<SimplexId> &critEdgesOrder,
  const SimplexId *const offsets,
  const triangulationType &triangulation) const {
 
  Timer tm{};
 
  auto saddle1ToMinima = getSaddle1ToMinima(criticalEdges, triangulation);
 
  Timer tmseq{};
 
  auto &firstRep{this->firstRepMin_};
  std::iota(firstRep.begin(), firstRep.end(), 0);
  std::vector<tripletType> sadMinTriplets{};
 
  for(size_t i = 0; i < saddle1ToMinima.size(); ++i) {
    auto &mins = saddle1ToMinima[i];
    const auto s1 = criticalEdges[i];
    // remove duplicates
    TTK_PSORT(this->threadNumber_, mins.begin(), mins.end());
    const auto last = std::unique(mins.begin(), mins.end());
    mins.erase(last, mins.end());
    if(mins.size() != 2) {
      continue;
    }
    sadMinTriplets.emplace_back(tripletType{s1, mins[0], mins[1]});
  }
 
  tripletsToPersistencePairs(pairs, pairedMinima, paired1Saddles, firstRep,
                             sadMinTriplets, critEdgesOrder.data(), offsets, 0);
 
  const auto nMinSadPairs = pairs.size();
 
  this->printMsg(
    "Computed " + std::to_string(nMinSadPairs) + " min-saddle pairs", 1.0,
    tm.getElapsedTime(), this->threadNumber_);
 
  this->printMsg("min-saddle pairs sequential part", 1.0,
                 tmseq.getElapsedTime(), 1, debug::LineMode::NEW,
                 debug::Priority::VERBOSE);
}
 
template <typename triangulationType>
void ttk::DiscreteMorseSandwich::getMaxSaddlePairs(
  std::vector<PersistencePair> &pairs,
  std::vector<bool> &pairedMaxima,
  std::vector<bool> &pairedSaddles,
  const std::vector<SimplexId> &criticalSaddles,
  const std::vector<SimplexId> &critSaddlesOrder,
  const std::vector<SimplexId> &critMaxsOrder,
  const triangulationType &triangulation) const {
 
  Timer tm{};
 
  const auto dim = this->dg_.getDimensionality();
 
  auto saddle2ToMaxima
    = dim == 3
        ? getSaddle2ToMaxima(
          criticalSaddles,
          [&triangulation](const SimplexId a, const SimplexId i, SimplexId &r) {
            return triangulation.getTriangleStar(a, i, r);
          },
          [&triangulation](const SimplexId a) {
            return triangulation.getTriangleStarNumber(a);
          },
          [&triangulation](const SimplexId a) {
            return triangulation.isTriangleOnBoundary(a);
          },
          triangulation)
        : getSaddle2ToMaxima(
          criticalSaddles,
          [&triangulation](const SimplexId a, const SimplexId i, SimplexId &r) {
            return triangulation.getEdgeStar(a, i, r);
          },
          [&triangulation](const SimplexId a) {
            return triangulation.getEdgeStarNumber(a);
          },
          [&triangulation](const SimplexId a) {
            return triangulation.isEdgeOnBoundary(a);
          },
          triangulation);
 
  Timer tmseq{};
 
  auto &firstRep{this->firstRepMax_};
  std::iota(firstRep.begin(), firstRep.end(), 0);
  std::vector<tripletType> sadMaxTriplets{};
 
  for(size_t i = 0; i < saddle2ToMaxima.size(); ++i) {
    auto &maxs = saddle2ToMaxima[i];
    // remove duplicates
    TTK_PSORT(this->threadNumber_, maxs.begin(), maxs.end(),
              [](const SimplexId a, const SimplexId b) {
                // positive values (actual maxima) before negative ones
                // (boundary component id)
                if(a * b >= 0) {
                  return a < b;
                } else {
                  return a > b;
                }
              });
    const auto last = std::unique(maxs.begin(), maxs.end());
    maxs.erase(last, maxs.end());
 
    // remove "doughnut" configurations: two ascending separatrices
    // leading to the same maximum/boundary component
    if(maxs.size() != 2) {
      continue;
    }
 
    const auto s2 = criticalSaddles[i];
    if(!pairedSaddles[s2]) {
      sadMaxTriplets.emplace_back(tripletType{s2, maxs[0], maxs[1]});
    }
  }
 
  const auto nMinSadPairs = pairs.size();
 
  tripletsToPersistencePairs(pairs, pairedMaxima, pairedSaddles, firstRep,
                             sadMaxTriplets, critSaddlesOrder.data(),
                             critMaxsOrder.data(), dim - 1);
 
  const auto nSadMaxPairs = pairs.size() - nMinSadPairs;
 
  this->printMsg(
    "Computed " + std::to_string(nSadMaxPairs) + " saddle-max pairs", 1.0,
    tm.getElapsedTime(), this->threadNumber_);
 
  this->printMsg("saddle-max pairs sequential part", 1.0,
                 tmseq.getElapsedTime(), 1, debug::LineMode::NEW,
                 debug::Priority::VERBOSE);
}
 
template <typename triangulationType, typename Container>
SimplexId ttk::DiscreteMorseSandwich::eliminateBoundariesSandwich(
  const SimplexId s2,
  std::vector<bool> &onBoundary,
  std::vector<Container> &s2Boundaries,
  const std::vector<SimplexId> &s2Mapping,
  const std::vector<SimplexId> &s1Mapping,
  std::vector<SimplexId> &partners,
  std::vector<Lock> &s1Locks,
  std::vector<Lock> &s2Locks,
  const triangulationType &triangulation) const {
 
  auto &boundaryIds{s2Boundaries[s2Mapping[s2]]};
 
  const auto addBoundary = [&boundaryIds, &onBoundary](const SimplexId e) {
    // add edge e to boundaryIds/onBoundary modulo 2
    if(!onBoundary[e]) {
      boundaryIds.emplace(e);
      onBoundary[e] = true;
    } else {
      const auto it = boundaryIds.find(e);
      boundaryIds.erase(it);
      onBoundary[e] = false;
    }
  };
 
  const auto clearOnBoundary = [&boundaryIds, &onBoundary]() {
    // clear the onBoundary vector (set everything to false)
    for(const auto e : boundaryIds) {
      onBoundary[e] = false;
    }
  };
 
  if(!boundaryIds.empty()) {
    // restore previously computed s2 boundary
    for(const auto e : boundaryIds) {
      onBoundary[e] = true;
    }
  } else {
    // init cascade with s2 triangle boundary (3 edges)
    for(SimplexId i = 0; i < 3; ++i) {
      SimplexId e{};
      triangulation.getTriangleEdge(s2, i, e);
      addBoundary(e);
    }
  }
 
  // lock the 2-saddle to ensure that only one thread can perform the
  // boundary expansion
  s2Locks[s2Mapping[s2]].lock();
 
  while(!boundaryIds.empty()) {
    // tau: youngest edge on boundary
    const auto tau{*boundaryIds.begin()};
    // use the Discrete Gradient to find a triangle paired to tau
    auto pTau{this->dg_.getPairedCell(Cell{1, tau}, triangulation)};
    bool critical{false};
    if(pTau == -1) {
      // maybe tau is critical and paired to a critical triangle
      do {
#ifdef TTK_ENABLE_OPENMP
#pragma omp atomic read
#endif // TTK_ENABLE_OPENMP
        pTau = partners[tau];
        if(pTau == -1 || s2Boundaries[s2Mapping[pTau]].empty()) {
          break;
        }
      } while(*s2Boundaries[s2Mapping[pTau]].begin() != tau);
 
      critical = true;
    }
    if(pTau == -1) {
      // tau is critical and not paired
 
      // compare-and-swap from "Towards Lockfree Persistent Homology"
      // using locks over 1-saddles instead of atomics (OpenMP compatibility)
      s1Locks[s1Mapping[tau]].lock();
      const auto cap = partners[tau];
      if(partners[tau] == -1) {
        partners[tau] = s2;
      }
      s1Locks[s1Mapping[tau]].unlock();
 
      // cleanup before exiting
      clearOnBoundary();
      s2Locks[s2Mapping[s2]].unlock();
      if(cap == -1) {
        return tau;
      } else {
        return this->eliminateBoundariesSandwich(
          s2, onBoundary, s2Boundaries, s2Mapping, s1Mapping, partners, s1Locks,
          s2Locks, triangulation);
      }
 
    } else {
      // expand boundary
      if(critical && s2Mapping[pTau] != -1) {
        if(s2Mapping[pTau] < s2Mapping[s2]) {
          // pTau is an already-paired 2-saddle
          // merge pTau boundary into s2 boundary
 
          // make sure that pTau boundary is not modified by another
          // thread while we merge the two boundaries...
          s2Locks[s2Mapping[pTau]].lock();
          for(const auto e : s2Boundaries[s2Mapping[pTau]]) {
            addBoundary(e);
          }
          s2Locks[s2Mapping[pTau]].unlock();
          if(this->Compute2SaddlesChildren) {
            this->s2Children_[s2Mapping[s2]].emplace_back(s2Mapping[pTau]);
          }
 
        } else if(s2Mapping[pTau] > s2Mapping[s2]) {
 
          // compare-and-swap from "Towards Lockfree Persistent
          // Homology" using locks over 1-saddles
          s1Locks[s1Mapping[tau]].lock();
          const auto cap = partners[tau];
          if(partners[tau] == pTau) {
            partners[tau] = s2;
          }
          s1Locks[s1Mapping[tau]].unlock();
 
          if(cap == pTau) {
            // cleanup before exiting
            clearOnBoundary();
            s2Locks[s2Mapping[s2]].unlock();
            return this->eliminateBoundariesSandwich(
              pTau, onBoundary, s2Boundaries, s2Mapping, s1Mapping, partners,
              s1Locks, s2Locks, triangulation);
          }
        }
      } else { // pTau is a regular triangle
        // add pTau triangle boundary (3 edges)
        for(SimplexId i = 0; i < 3; ++i) {
          SimplexId e{};
          triangulation.getTriangleEdge(pTau, i, e);
          addBoundary(e);
        }
      }
    }
  }
 
  // cleanup before exiting
  clearOnBoundary();
  s2Locks[s2Mapping[s2]].unlock();
  return -1;
}
 
template <typename triangulationType>
void ttk::DiscreteMorseSandwich::getSaddleSaddlePairs(
  std::vector<PersistencePair> &pairs,
  std::vector<bool> &paired1Saddles,
  std::vector<bool> &paired2Saddles,
  const bool exportBoundaries,
  std::vector<GeneratorType> &boundaries,
  const std::vector<SimplexId> &critical1Saddles,
  const std::vector<SimplexId> &critical2Saddles,
  const std::vector<SimplexId> &crit1SaddlesOrder,
  const triangulationType &triangulation) const {
 
  Timer tm2{};
  const auto nSadExtrPairs = pairs.size();
 
  // 1- and 2-saddles yet to be paired
  std::vector<SimplexId> saddles1{}, saddles2{};
  // filter out already paired 1-saddles (edge id)
  for(const auto s1 : critical1Saddles) {
    if(!paired1Saddles[s1]) {
      saddles1.emplace_back(s1);
    }
  }
  // filter out already paired 2-saddles (triangle id)
  for(const auto s2 : critical2Saddles) {
    if(!paired2Saddles[s2]) {
      saddles2.emplace_back(s2);
    }
  }
 
  if(this->Compute2SaddlesChildren) {
    this->s2Children_.resize(saddles2.size());
  }
 
  // sort every triangulation edges by filtration order
  const auto &edgesFiltrOrder{crit1SaddlesOrder};
 
  auto &onBoundary{this->onBoundary_};
  auto &edgeTrianglePartner{this->edgeTrianglePartner_};
 
  const auto cmpEdges
    = [&edgesFiltrOrder](const SimplexId a, const SimplexId b) {
        return edgesFiltrOrder[a] > edgesFiltrOrder[b];
      };
  using Container = std::set<SimplexId, decltype(cmpEdges)>;
  std::vector<Container> s2Boundaries(saddles2.size(), Container(cmpEdges));
 
  // unpaired critical triangle id -> index in saddle2 vector
  auto &s2Mapping{this->s2Mapping_};
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel for num_threads(threadNumber_)
#endif // TTK_ENABLE_OPENMP
  for(size_t i = 0; i < saddles2.size(); ++i) {
    s2Mapping[saddles2[i]] = i;
  }
 
  // unpaired critical edge id -> index in saddle1 vector
  auto &s1Mapping{this->s1Mapping_};
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel for num_threads(threadNumber_)
#endif // TTK_ENABLE_OPENMP
  for(size_t i = 0; i < saddles1.size(); ++i) {
    s1Mapping[saddles1[i]] = i;
  }
 
  // one lock per 1-saddle
  std::vector<Lock> s1Locks(saddles1.size());
  // one lock per 2-saddle
  std::vector<Lock> s2Locks(saddles2.size());
 
  // compute 2-saddles boundaries in parallel
 
#ifdef TTK_ENABLE_OPENMP4
#pragma omp parallel for num_threads(threadNumber_) schedule(dynamic) \
  firstprivate(onBoundary)
#endif // TTK_ENABLE_OPENMP4
  for(size_t i = 0; i < saddles2.size(); ++i) {
    // 2-saddles sorted in increasing order
    const auto s2 = saddles2[i];
    this->eliminateBoundariesSandwich(s2, onBoundary, s2Boundaries, s2Mapping,
                                      s1Mapping, edgeTrianglePartner, s1Locks,
                                      s2Locks, triangulation);
  }
 
  Timer tmseq{};
 
  // extract saddle-saddle pairs from computed boundaries
  for(size_t i = 0; i < saddles2.size(); ++i) {
    if(!s2Boundaries[i].empty()) {
      const auto s2 = saddles2[i];
      const auto s1 = *s2Boundaries[i].begin();
      // we found a pair
      pairs.emplace_back(s1, s2, 1);
      paired1Saddles[s1] = true;
      paired2Saddles[s2] = true;
    }
  }
 
  if(exportBoundaries) {
    boundaries.resize(s2Boundaries.size());
    for(size_t i = 0; i < boundaries.size(); ++i) {
      const auto &boundSet{s2Boundaries[i]};
      if(boundSet.empty()) {
        continue;
      }
      boundaries[i] = {
        {boundSet.begin(), boundSet.end()},
        saddles2[i],
        std::array<SimplexId, 2>{
          this->dg_.getCellGreaterVertex(Cell{2, saddles2[i]}, triangulation),
          this->dg_.getCellGreaterVertex(
            Cell{1, *boundSet.begin()}, triangulation),
        }};
    }
  }
 
  const auto nSadSadPairs = pairs.size() - nSadExtrPairs;
 
  this->printMsg(
    "Computed " + std::to_string(nSadSadPairs) + " saddle-saddle pairs", 1.0,
    tm2.getElapsedTime(), this->threadNumber_);
 
  this->printMsg("saddle-saddle pairs sequential part", 1.0,
                 tmseq.getElapsedTime(), 1, debug::LineMode::NEW,
                 debug::Priority::VERBOSE);
}
 
template <typename triangulationType>
void ttk::DiscreteMorseSandwich::extractCriticalCells(
  std::array<std::vector<SimplexId>, 4> &criticalCellsByDim,
  std::array<std::vector<SimplexId>, 4> &critCellsOrder,
  const SimplexId *const offsets,
  const triangulationType &triangulation,
  const bool sortEdges) const {
 
  Timer tm{};
 
  this->dg_.getCriticalPoints(criticalCellsByDim, triangulation);
 
  this->printMsg("Extracted critical cells", 1.0, tm.getElapsedTime(),
                 this->threadNumber_, debug::LineMode::NEW,
                 debug::Priority::VERBOSE);
 
  // memory allocations
  auto &critEdges{this->critEdges_};
  if(!sortEdges) {
    critEdges.resize(criticalCellsByDim[1].size());
  }
  std::vector<TriangleSimplex> critTriangles(criticalCellsByDim[2].size());
  std::vector<TetraSimplex> critTetras(criticalCellsByDim[3].size());
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel num_threads(threadNumber_)
#endif // TTK_ENABLE_OPENMP
  {
    if(sortEdges) {
#ifdef TTK_ENABLE_OPENMP
#pragma omp for nowait
#endif // TTK_ENABLE_OPENMP
      for(size_t i = 0; i < critEdges.size(); ++i) {
        critEdges[i].fillEdge(i, offsets, triangulation);
      }
    } else {
#ifdef TTK_ENABLE_OPENMP
#pragma omp for nowait
#endif // TTK_ENABLE_OPENMP
      for(size_t i = 0; i < critEdges.size(); ++i) {
        critEdges[i].fillEdge(criticalCellsByDim[1][i], offsets, triangulation);
      }
    }
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp for nowait
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critTriangles.size(); ++i) {
      critTriangles[i].fillTriangle(
        criticalCellsByDim[2][i], offsets, triangulation);
    }
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp for
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critTetras.size(); ++i) {
      critTetras[i].fillTetra(criticalCellsByDim[3][i], offsets, triangulation);
    }
  }
 
  TTK_PSORT(this->threadNumber_, critEdges.begin(), critEdges.end());
  TTK_PSORT(this->threadNumber_, critTriangles.begin(), critTriangles.end());
  TTK_PSORT(this->threadNumber_, critTetras.begin(), critTetras.end());
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel num_threads(threadNumber_)
#endif // TTK_ENABLE_OPENMP
  {
#ifdef TTK_ENABLE_OPENMP
#pragma omp for nowait
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critEdges.size(); ++i) {
      critCellsOrder[1][critEdges[i].id_] = i;
    }
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp for nowait
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critTriangles.size(); ++i) {
      criticalCellsByDim[2][i] = critTriangles[i].id_;
      critCellsOrder[2][critTriangles[i].id_] = i;
    }
 
#ifdef TTK_ENABLE_OPENMP
#pragma omp for
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critTetras.size(); ++i) {
      criticalCellsByDim[3][i] = critTetras[i].id_;
      critCellsOrder[3][critTetras[i].id_] = i;
    }
  }
 
  if(sortEdges) {
    TTK_PSORT(this->threadNumber_, criticalCellsByDim[1].begin(),
              criticalCellsByDim[1].end(),
              [&critCellsOrder](const SimplexId a, const SimplexId b) {
                return critCellsOrder[1][a] < critCellsOrder[1][b];
              });
  } else {
#ifdef TTK_ENABLE_OPENMP
#pragma omp parallel for num_threads(threadNumber_)
#endif // TTK_ENABLE_OPENMP
    for(size_t i = 0; i < critEdges.size(); ++i) {
      criticalCellsByDim[1][i] = critEdges[i].id_;
    }
  }
 
  this->printMsg("Extracted & sorted critical cells", 1.0, tm.getElapsedTime(),
                 this->threadNumber_, debug::LineMode::NEW,
                 debug::Priority::DETAIL);
}
 
template <typename triangulationType>
int ttk::DiscreteMorseSandwich::computePersistencePairs(
  std::vector<PersistencePair> &pairs,
  const SimplexId *const offsets,
  const triangulationType &triangulation,
  const bool ignoreBoundary,
  const bool compute2SaddlesChildren) {
 
  // allocate memory
  this->alloc(triangulation);
 
  Timer tm{};
  pairs.clear();
  const auto dim = this->dg_.getDimensionality();
  this->Compute2SaddlesChildren = compute2SaddlesChildren;
 
  // get every critical cell sorted them by dimension
  std::array<std::vector<SimplexId>, 4> criticalCellsByDim{};
  // holds the critical cells order
  auto &critCellsOrder{this->critCellsOrder_};
 
  this->extractCriticalCells(
    criticalCellsByDim, critCellsOrder, offsets, triangulation, dim == 3);
 
  // if minima are paired
  auto &pairedMinima{this->pairedCritCells_[0]};
  // if 1-saddles are paired
  auto &paired1Saddles{this->pairedCritCells_[1]};
  // if 2-saddles are paired
  auto &paired2Saddles{this->pairedCritCells_[dim - 1]};
  // if maxima are paired
  auto &pairedMaxima{this->pairedCritCells_[dim]};
 
  // connected components (global min/max pair)
  size_t nConnComp{};
 
  if(this->ComputeMinSad) {
    // minima - saddle pairs
    this->getMinSaddlePairs(pairs, pairedMinima, paired1Saddles,
                            criticalCellsByDim[1], critCellsOrder[1], offsets,
                            triangulation);
 
    // non-paired minima
    for(const auto min : criticalCellsByDim[0]) {
      if(!pairedMinima[min]) {
        pairs.emplace_back(min, -1, 0);
        pairedMinima[min] = true;
        nConnComp++;
      }
    }
  } else {
    // still extract the global pair
    const auto globMin{*std::min_element(
      criticalCellsByDim[0].begin(), criticalCellsByDim[0].end(),
      [offsets](const SimplexId a, const SimplexId b) {
        return offsets[a] < offsets[b];
      })};
    pairs.emplace_back(globMin, -1, 0);
    pairedMinima[globMin] = true;
    nConnComp++;
  }
 
  if(dim > 1 && this->ComputeSadMax) {
    // saddle - maxima pairs
    this->getMaxSaddlePairs(
      pairs, pairedMaxima, paired2Saddles, criticalCellsByDim[dim - 1],
      critCellsOrder[dim - 1], critCellsOrder[dim], triangulation);
  }
 
  if(ignoreBoundary) {
    // post-process saddle-max pairs: remove the one with the global
    // maximum (if it exists) to be (more) compatible with FTM
    const auto it
      = std::find_if(pairs.begin(), pairs.end(), [&](const PersistencePair &p) {
          if(p.type < dim - 1) {
            return false;
          }
          const Cell cmax{dim, p.death};
          const auto vmax{this->getCellGreaterVertex(cmax, triangulation)};
          return offsets[vmax] == triangulation.getNumberOfVertices() - 1;
        });
 
    if(it != pairs.end()) {
      // remove saddle-max pair with global maximum
      paired2Saddles[it->birth] = false;
      pairedMaxima[it->death] = false;
      pairs.erase(it);
    }
  }
 
  // saddle - saddle pairs
  if(dim == 3 && !criticalCellsByDim[1].empty()
     && !criticalCellsByDim[2].empty() && this->ComputeSadSad) {
    std::vector<GeneratorType> tmp{};
    this->getSaddleSaddlePairs(
      pairs, paired1Saddles, paired2Saddles, false, tmp, criticalCellsByDim[1],
      criticalCellsByDim[2], critCellsOrder[1], triangulation);
  }
 
  if(std::is_same<triangulationType, ttk::ExplicitTriangulation>::value) {
    // create infinite pairs from non-paired 1-saddles, 2-saddles and maxima
    size_t nHandles{}, nCavities{}, nNonPairedMax{};
    if((dim == 2 && !ignoreBoundary && this->ComputeMinSad
        && this->ComputeSadMax)
       || (dim == 3 && this->ComputeMinSad && this->ComputeSadSad)) {
      // non-paired 1-saddles
      for(const auto s1 : criticalCellsByDim[1]) {
        if(!paired1Saddles[s1]) {
          paired1Saddles[s1] = true;
          // topological handles
          pairs.emplace_back(s1, -1, 1);
          nHandles++;
        }
      }
    }
    if(dim == 3 && !ignoreBoundary && this->ComputeSadMax
       && this->ComputeSadSad) {
      // non-paired 2-saddles
      for(const auto s2 : criticalCellsByDim[2]) {
        if(!paired2Saddles[s2]) {
          paired2Saddles[s2] = true;
          // cavities
          pairs.emplace_back(s2, -1, 2);
          nCavities++;
        }
      }
    }
    if(dim == 2 && !ignoreBoundary && this->ComputeSadMax) {
      // non-paired maxima
      for(const auto max : criticalCellsByDim[dim]) {
        if(!pairedMaxima[max]) {
          pairs.emplace_back(max, -1, 2);
          nNonPairedMax++;
        }
      }
    }
 
    int nBoundComp
      = (dim == 3 ? nCavities : nHandles) + nConnComp - nNonPairedMax;
    nBoundComp = std::max(nBoundComp, 0);
 
    // print Betti numbers
    const std::vector<std::vector<std::string>> rows{
      {" #Connected components", std::to_string(nConnComp)},
      {" #Topological handles", std::to_string(nHandles)},
      {" #Cavities", std::to_string(nCavities)},
      {" #Boundary components", std::to_string(nBoundComp)},
    };
 
    this->printMsg(rows, debug::Priority::DETAIL);
  }
 
  this->printMsg(
    "Computed " + std::to_string(pairs.size()) + " persistence pairs", 1.0,
    tm.getElapsedTime(), this->threadNumber_);
 
  this->displayStats(pairs, criticalCellsByDim, pairedMinima, paired1Saddles,
                     paired2Saddles, pairedMaxima);
 
  // free memory
  this->clear();
 
  return 0;
}