LevelSchedule.cpp

/**
Copyright (c) 2015, Intel Corporation. All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
    * Redistributions of source code must retain the above copyright
      notice, this list of conditions and the following disclaimer.
    * Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.
    * Neither the name of the Intel Corporation nor the
      names of its contributors may be used to endorse or promote products
      derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL INTEL CORPORATION BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/

#include <cstring>
#include <cassert>
#include <cmath>
#include <algorithm>
#include <omp.h>

#include "LevelSchedule.hpp"
#include "Utils.hpp"
#include "MemoryPool.hpp"

#ifdef LOADIMBA
#include "synk/loadimba.hpp"
#else
#include "synk/barrier.hpp"
#endif

using namespace std;

namespace SpMP
{

  using namespace std;

#ifdef TRACE_TASK_TIMES
  unsigned long long *taskTimes[NUM_MAX_THREADS] = {NULL};
  unsigned long long taskTimeCnt[NUM_MAX_THREADS];
#endif

#ifdef MEASURE_SPIN_TIME
  unsigned long long spin_times[NUM_MAX_THREADS] = {0};
#ifdef MEASURE_TASK_TIME
  unsigned long long task_time_sum[65536] = {0};
  int task_time_cnt[65536] = {0};
#endif

#ifdef TRACE_SPIN_TIME
  unsigned long long *spin_traces[NUM_MAX_THREADS] = {NULL};
  int spin_trace_counts[NUM_MAX_THREADS];
#endif
#endif

#ifdef TRACE_SPIN_TIME
  void dumpTrace(const char *fileName)
  {
    ofstream o(fileName);
    for (int ti = 0; ti < omp_get_max_threads(); ++ti)
    {
      for (int i = 1; i < spin_trace_counts[ti]; i += 2)
      {
        o << "=Worker " << ti << "| type 0| start:" << (spin_traces[ti][i] - spin_traces[ti][0]) << "| end:" << (spin_traces[ti][i + 1] - spin_traces[ti][0]) << "| task 0" << endl;
      }
    }
  }
#endif

  LevelSchedule::LevelSchedule()
      : nparentsForward(NULL), nparentsBackward(NULL),
        parentsForward(NULL), parentsBackward(NULL),
        taskFinished(NULL),
        useBarrier(false), transitiveReduction(true), fuseSpMV(false),
#ifdef __MIC__
        aggregateForVectorization(true),
#else
        aggregateForVectorization(false),
#endif
        useMemoryPool(false)
  {
    init_();
  }

  LevelSchedule::~LevelSchedule()
  {
    if (!useMemoryPool)
    {
      FREE(nparentsForward);
      FREE(nparentsBackward);
      FREE(parentsForward);
      FREE(parentsBackward);

      int *tempTaskFinished = (int *)taskFinished;
      FREE(tempTaskFinished);

      FREE(origToThreadContPerm);
      FREE(threadContToOrigPerm);
      FREE(parentsBuf[0]);
      FREE(parentsBuf[1]);

      if (fusedSchedule)
        delete fusedSchedule;
    }
  }

  void LevelSchedule::init_()
  {
    origToThreadContPerm = NULL;
    threadContToOrigPerm = NULL;
    parentsBuf[0] = parentsBuf[1] = NULL;
    fusedSchedule = NULL;
  }

  template <int BASE = 0>
  static int *constructDiagPtr_(
      int m, const int *rowptr, const int *colidx)
  {
    int *diagptr = MALLOC(int, m);
#pragma omp parallel for
    for (int i = 0; i < m; ++i)
    {
      for (int j = rowptr[i] - BASE; j < rowptr[i + 1] - BASE; ++j)
      {
        if (colidx[j] - BASE >= i)
        {
          diagptr[i] = j + BASE;
          break;
        }
      }
    }
    return diagptr;
  }

  int *constructDiagPtr_(int m, const int *rowptr, const int *colidx, int base)
  {
    if (0 == base)
    {
      return constructDiagPtr_<0>(m, rowptr, colidx);
    }
    else
    {
      assert(1 == base);
      return constructDiagPtr_<1>(m, rowptr, colidx);
    }
  }

  void CSR::constructDiagPtr()
  {
    if (!diagptr)
    {
      diagptr = constructDiagPtr_(m, rowptr, colidx, getBase());
    }
  }

  void LevelSchedule::constructTaskGraph(CSR &A)
  {
    constructTaskGraph(A, PrefixSumCostFunction(A.rowptr));
  }

  void LevelSchedule::constructTaskGraph(CSR &A, const CostFunction &costFunction)
  {
    A.constructDiagPtr();
    assert(A.isSymmetric(false));
    constructTaskGraph(A.m, A.rowptr, A.diagptr, A.extptr, A.colidx, costFunction);
  }

  void LevelSchedule::constructTaskGraph(
      int m, const int *rowptr, const int *colidx, const CostFunction &costFunction)
  {
    int *diagptr = constructDiagPtr_(m, rowptr, colidx, rowptr[0]);

    constructTaskGraph(m, rowptr, diagptr, NULL, colidx, costFunction);

    FREE(diagptr);
  }

  void LevelSchedule::constructTaskGraph(
      int m, const int *rowptr, const int *diagptr, const int *colidx,
      const CostFunction &costFunction)
  {
    constructTaskGraph(m, rowptr, diagptr, NULL, colidx, costFunction);
  }

#define NUM_MAX_THREADS (2048)

  template <int BASE = 0>
  static void findLevels_(
      int *qTail, int *nnzPrefixSum, int *nparents, int **q, int **rowPtrs,
      const int *diagptr, const int *extptr,
      const int *colidx,
      int *reversePerm, vector<int> &levIndices,
      int m,
      unsigned long long &tt2, unsigned long long &tt3, unsigned long long &tt4, unsigned long long &tt5, unsigned long long &tt6)
  {
    colidx -= BASE;
    // 每个线程的的qTail存储的入度为0的节点数量，ptr
    int qTailPrefixSum[NUM_MAX_THREADS] = {0};
    volatile int endSynchronized[1] = {0};

    // int nnz_all = 0;

#ifdef LOADIMBA
    synk::LoadImba *bar = synk::LoadImba::getInstance();
#else
    synk::Barrier *bar = synk::Barrier::getInstance();
#endif

#pragma omp parallel
    {
      int tid = omp_get_thread_num();
      int nthreads = omp_get_num_threads();

      bool prevIterSerial = false;
      unsigned long long tTemp = __rdtsc();

      int end = 0;

      while (true)
      { // until all nodes are visited
        bool currIterSerial = false;
        int begin = end;

        // 将上一轮迭代产生的level中的节点进行拷贝到reversePerm, 并且计算下一轮迭代需要的的数据
        if (prevIterSerial)
        {
          if (0 == tid)
          {
            if (qTail[0] == 0)
            {
              bar->wait(tid); // wake up other threads
              break;
            }
#define THRESH 64
            // currIterSerial = qTail[0] < THRESH * 2;
            currIterSerial = true;
            if (!currIterSerial) // 本次iter是并行的
            {
              *endSynchronized = begin;
              bar->wait(tid); // wake up other threads
              end = begin + qTail[0];

              qTailPrefixSum[1] = qTail[0];
              bar->wait(tid);
            }
            else
            {
              end = begin + qTail[0];
            }
          }
          else
          {
            currIterSerial = false;

            begin = *endSynchronized;
            end = begin + qTail[0];
            qTailPrefixSum[tid + 1] = qTail[0];
            nnzPrefixSum[tid + 1] = nnzPrefixSum[1];
            bar->wait(tid);
          }
        }
        else
        {
          tTemp = __rdtsc();
          bar->wait(tid);
          if (0 == tid)
            tt2 += __rdtsc() - tTemp;

          tTemp = __rdtsc();

          // compute prefix sum of # of discovered nodes and nnzs associated with them
#pragma omp single
          {
            for (int t = 0; t < nthreads; ++t)
            {
              qTailPrefixSum[t + 1] = qTailPrefixSum[t] + qTail[t * 16];
              nnzPrefixSum[t + 1] += nnzPrefixSum[t];
            }
          }

          if (0 == tid)
            tt3 += __rdtsc() - tTemp;

          if (qTailPrefixSum[nthreads] == 0) // 迭代结束了没有需要更新入度的节点
            break;
          // currIterSerial = qTailPrefixSum[nthreads] < THRESH;
          currIterSerial = true;

          tTemp = __rdtsc();

          int begin = end;
          end = begin + qTailPrefixSum[nthreads];

          // 线程拷贝的数据范围
          int iPerThread = (qTailPrefixSum[nthreads] + nthreads - 1) / nthreads;
          int iBegin = min(begin + iPerThread * tid, end);
          int iEnd = min(iBegin + iPerThread, end);

          // 拷贝的数据来源的线程范围
          int tBegin = upper_bound(
                           qTailPrefixSum, qTailPrefixSum + nthreads + 1,
                           iPerThread * tid) -
                       qTailPrefixSum - 1;
          int tEnd = upper_bound(
                         qTailPrefixSum, qTailPrefixSum + nthreads + 1,
                         iPerThread * (tid + 1)) -
                     qTailPrefixSum - 1;

          for (int t = tBegin; t <= min(tEnd, nthreads - 1); ++t)
          {
            int b = max(0, iBegin - begin - qTailPrefixSum[t]);
            int e = min((int)qTail[t * 16], iEnd - begin - qTailPrefixSum[t]);

            memcpy(
                &reversePerm[0] + begin + qTailPrefixSum[t] + b,
                q[t] + b,
                (e - b) * sizeof(int));
          }
        }

        if (0 == tid)
        {
          levIndices.push_back(end);
        }

        if (currIterSerial)
        {
          if (!prevIterSerial)
          {
            bar->wait(tid);
          }

          if (0 == tid)
          {
            tTemp = __rdtsc();

            int *tailPtr = &reversePerm[end];
            int *rowPtr = rowPtrs[tid];
            *rowPtr = 0;

            for (int i = begin; i < end; ++i)
            {
              int pred = reversePerm[i];
              int jBegin = diagptr[pred] + 1;
              int jEnd = extptr[pred];
              for (int j = jBegin; j < jEnd; ++j)
              {
                int succ = colidx[j] - BASE; // 对称矩阵
                if (succ >= m)
                  continue; // for a "fat" matrices with halo
                --nparents[succ];
                assert(nparents[succ] >= 0);
                if (nparents[succ] == 0)
                {
                  *tailPtr = succ;
                  *(rowPtr + 1) = *rowPtr + extptr[succ] - diagptr[succ] - 1;

                  ++tailPtr;
                  ++rowPtr;
                }
              }
            } // for each ready row

            qTail[0] = tailPtr - &reversePerm[end];
            nnzPrefixSum[1] = *rowPtr;

            tt6 += __rdtsc() - tTemp;
          }
          else
          {
            assert(!prevIterSerial);
            bar->wait(tid); // waiting for the master thread
            if (qTail[0] == 0)
              break;
          }
        }
        else
        {
          // nnz_all = 0;
          int nnzPerThread = (nnzPrefixSum[nthreads] + nthreads - 1) / nthreads;
          // 确定数据在哪个线程复负责的范围内
          int tBegin = upper_bound(
                           nnzPrefixSum, nnzPrefixSum + nthreads + 1,
                           nnzPerThread * tid) -
                       nnzPrefixSum - 1;
          int tEnd = upper_bound(
                         nnzPrefixSum, nnzPrefixSum + nthreads + 1,
                         nnzPerThread * (tid + 1)) -
                     nnzPrefixSum - 1;

          int iBegin, iEnd;
          if (0 == tid)
          {
            iBegin = 0;
          }
          else if (tBegin == nthreads)
          {
            iBegin = qTailPrefixSum[nthreads];
          }
          else
          { // 查找第一个不大于这个边界的值
            iBegin = upper_bound(
                         rowPtrs[tBegin], rowPtrs[tBegin] + qTail[tBegin * 16],
                         nnzPerThread * tid - nnzPrefixSum[tBegin]) -
                     rowPtrs[tBegin] - 1 +
                     qTailPrefixSum[tBegin];
          }

          if (tEnd == nthreads)
          {
            iEnd = qTailPrefixSum[nthreads];
          }
          else
          {
            iEnd = upper_bound(
                       rowPtrs[tEnd], rowPtrs[tEnd] + qTail[tEnd * 16],
                       nnzPerThread * (tid + 1) - nnzPrefixSum[tEnd]) -
                   rowPtrs[tEnd] - 1 +
                   qTailPrefixSum[tEnd];
          }

          // Note: add
          assert(iBegin <= qTailPrefixSum[nthreads]);
          assert(iEnd <= qTailPrefixSum[nthreads]);

          iBegin += begin;
          iEnd += begin;

          if (0 == tid)
            tt4 += __rdtsc() - tTemp;

          tTemp = __rdtsc();
          bar->wait(tid);
          if (0 == tid)
            tt5 += __rdtsc() - tTemp;

          tTemp = __rdtsc();
          int *tailPtr = q[tid];
          int *rowPtr = rowPtrs[tid];
          *rowPtr = 0;

          for (int i = iBegin; i < iEnd; ++i)
          {
            int pred = reversePerm[i];
            int jBegin = diagptr[pred] + 1;
            int jEnd = extptr[pred];
            for (int j = jBegin; j < jEnd; ++j)
            {
              int succ = colidx[j] - BASE;
              if (succ >= m)
                continue; // for a "fat" matrices with halo
              // __sync_fetch_and_add(&nnz_all, 1);
              // #pragma omp atomic
              // nnz_all += 1;

              if (__sync_fetch_and_add(nparents + succ, -1) == 1) // return oldValue
              {
                *tailPtr = succ;
                *(rowPtr + 1) =
                    *rowPtr + extptr[succ] - diagptr[succ] - 1;

                ++tailPtr;
                ++rowPtr;
              }
              // else{
              //   exit(2);
              // }
              assert(nparents[succ] >= 0);
            }
          } // for each ready row

          qTail[tid * 16] = tailPtr - q[tid];
          nnzPrefixSum[tid + 1] = *rowPtr;
          if (0 == tid)
            tt6 += __rdtsc() - tTemp;
            // Note: add
// #pragma omp barrier
//           assert(nnz_all == nnzPrefixSum[nthreads]);
        }

        prevIterSerial = currIterSerial;
      } // while true
    } // omp parallel

#ifndef NDEBUG
    for (int i = 0; i < m; ++i)
    {
      assert(0 == nparents[i]);
    }
#endif
    assert(isPerm(reversePerm, m));
  }

  // This routine doesn't depend on that colidx of each row is fully sorted.
  // It only needs to be partially sorted that those smaller than its row index
  // appear before diagptr, those larger than # of rows appear after extptr,
  // and all others appear in between.
  template <int BASE = 0>
  static void findLevels_(
      LevelSchedule *schedule,
      int m, const int *rowptr, const int *diagptr, const int *extptr, const int *colidx,
      const CostFunction &costFunction)
  {
    vector<int> &levIndices = schedule->levIndices;
    vector<int> &taskBoundaries = schedule->taskBoundaries;
    vector<int> &threadBoundaries = schedule->threadBoundaries;

    bool useBarrier = schedule->useBarrier;
    bool aggregateForVectorization = schedule->aggregateForVectorization;

    bool useMemoryPool = schedule->useMemoryPool;

#ifndef NDEBUG
    CSR A(m, m, (int *)rowptr, (int *)colidx, (double *)NULL);
    A.extptr = (int *)extptr;
    assert(A.isSymmetric(false, true));
    A.extptr = NULL;
    A.~CSR();
#endif

    // extptr points to the beginning of non-local columns (when MPI is used).
    // When MPI is not used, exptr will be set to NULL and we use rowptr + 1 as
    // extptr.
    if (NULL == extptr)
    {
      extptr = rowptr + 1;
    }

    // check columns are partially sorted that lower triangular parts appear before
    // diagptr and upper triangular parts appear after diagptr
#ifndef NDEBUG
    for (int i = 0; i < m; ++i)
    {
      for (int j = rowptr[i] - BASE; j < diagptr[i] - BASE; ++j)
      {
        assert(colidx[j] - BASE < i);
      }
      for (int j = diagptr[i] - BASE; j < extptr[i] - BASE; ++j)
      {
        assert(colidx[j] - BASE >= i);
      }
      for (int j = extptr[i] - BASE; j < rowptr[i] - BASE; ++j)
      {
        assert(colidx[j] - BASE >= m);
      }
    }
#endif

// #define PRINT_TIME_BREAKDOWN
#ifdef PRINT_TIME_BREAKDOWN
    unsigned long long tt0 = 0, tt1 = 0;
#endif
    unsigned long long tt2 = 0, tt3 = 0, tt4 = 0, tt5 = 0, tt6 = 0;

#ifdef PRINT_TIME_BREAKDOWN
    unsigned long long t = __rdtsc();
#endif
    int n = m;
    int nthreads = omp_get_max_threads();

    MemoryPool *memoryPool = MemoryPool::getSingleton();

    int nAligned = (n + 15) / 16 * 16;
    int *levContToOrigPerm;
    levContToOrigPerm = schedule->allocate<int>(nAligned + 128);

    size_t bufferBegin = memoryPool->getTail();
    int *buffer = schedule->allocate<int>(5 * max(nthreads, n + nthreads - 1) + NUM_MAX_THREADS * (2 + 16 + 2 + 2)); // max required in case n == 0
    assert(buffer);
    int *tempBuffer = buffer + max(nthreads, n + nthreads - 1) * 4;
    // between buffer and tempBuffer, the followings are allocated
    // q : max(nthreads, n + nthreads - 1)*2
    // rowPtrs : max(nthreads, n + nthreads - 1)*2

    int *nparents = tempBuffer;
    tempBuffer += n;
    int **q = (int **)tempBuffer;
    tempBuffer += NUM_MAX_THREADS * (sizeof(int *) / sizeof(int));
    int *qTail = tempBuffer;
    tempBuffer += NUM_MAX_THREADS * 16;
    int **rowPtrs = (int **)tempBuffer;
    tempBuffer += NUM_MAX_THREADS * (sizeof(int *) / sizeof(int));
    int *nnzPrefixSum = tempBuffer;
    tempBuffer += NUM_MAX_THREADS * 2;

    nnzPrefixSum[0] = 0;
    nnzPrefixSum[nthreads + 1] = 0;

    levIndices.clear();
    levIndices.push_back(0);

#ifdef PRINT_TIME_BREAKDOWN
    tt0 = __rdtsc() - t;
    unsigned long long ttt;
#endif

#pragma omp parallel num_threads(nthreads)
    {
      int tid = omp_get_thread_num();
#ifdef PRINT_TIME_BREAKDOWN
      if (0 == tid)
        ttt = __rdtsc();
      unsigned long long tTemp = __rdtsc();
#endif
      // 每个线程缓存本线程扫描得到的parents == 0 的节点空间， 每个线程扫描的最大节点数量平均为 n / nthread，因此这个部分的空间不会超
      q[tid] = buffer + max(nthreads, n + nthreads - 1) / nthreads * 2 * tid;
      // 前面的部分为q指针的结尾地址，后半部分为线程扫描的最大节点数量平均 n / nthread， 用以存储对称矩阵的对角右侧的nnz便宜，
      // 这个部分的nnz对应的colidx就是其successor节点的rowidx
      rowPtrs[tid] = buffer + max(nthreads, n + nthreads - 1) * 2 + max(nthreads, n + nthreads - 1) / nthreads * 2 * tid;

      int *tailPtr = q[tid];
      int *rowPtr = rowPtrs[tid];
      *rowPtr = 0;
/** 并行计算所有节点的父节点数量，将父节点数量为0的节点加入到队列中 */
#pragma omp for
      for (int i = 0; i < n; ++i)
      {
        nparents[i] = diagptr[i] - rowptr[i];
        if (nparents[i] == 0)
        {
          *tailPtr = i;
          *(rowPtr + 1) = *rowPtr + extptr[i] - diagptr[i] - 1;

          ++tailPtr;
          ++rowPtr;
        }
      }
      // 存储每个线程的队列的尾指针，这里存储的是尾指针在队列相对线程第一个任务的偏移
      qTail[tid * 16] = tailPtr - q[tid]; // 存储每个线程的q中缓存了多少个此轮迭代入度为0的节点
      nnzPrefixSum[tid + 1] = *rowPtr;

#ifdef PRINT_TIME_BREAKDOWN
      if (0 == tid)
        tt1 = __rdtsc() - tTemp;
#endif
    } // omp parallel

    assert(q[0]);
    /** 调用基础版本的findLevels函数，执行BFS的操作 */
    findLevels_<BASE>(
        qTail, nnzPrefixSum, nparents, q, rowPtrs,
        diagptr, extptr, colidx,
        levContToOrigPerm, levIndices, m,
        tt2, tt3, tt4, tt5, tt6);

#ifdef PRINT_TIME_BREAKDOWN
    ttt = __rdtsc() - ttt;

    printf("t1 = %f\n", (__rdtsc() - t) / get_cpu_freq());
    printf("ttt = %f\n", ttt / get_cpu_freq());
    printf("tt0 = %f (init)\n", tt0 / get_cpu_freq());
    printf("tt1 = %f (first level)\n", tt1 / get_cpu_freq());
    printf("tt2 = %f (first barrier)\n", tt2 / get_cpu_freq());
    printf("tt3 = %f (prefix sum)\n", tt3 / get_cpu_freq());
    printf("tt4 = %f (copy)\n", tt4 / get_cpu_freq());
    printf("tt5 = %f (second barrier)\n", tt5 / get_cpu_freq());
    printf("tt6 = %f (traverse)\n", tt6 / get_cpu_freq());
#undef PRINT_TIME_BREAKDOWN
#endif

    if (useMemoryPool)
    {
      memoryPool->setTail(bufferBegin);
      buffer = NULL;
    }
    else
    {
      FREE(buffer);
    }

    // int ippNumThreads = 0;
    // ippGetNumThreads(&ippNumThreads);
    // printf("ippGetNumthreads = %d\n", ippNumThreads);
    // ippSetNumThreads(1);

    int *reversePerm = levContToOrigPerm;

    size_t taskRowsEnd = memoryPool->getHead();
    pair<int, int> *taskRows =
        schedule->allocateFront<pair<int, int>>(nthreads * levIndices.size() - 1);

    /*copy(levIndices.begin(), levIndices.end(), ostream_iterator<int>(cout, " "));
    printf("\n");*/

    size_t bBufferBegin = memoryPool->getTail();
    int *bBuffer = schedule->allocate<int>(n + nthreads);

#ifdef _OPENMP
#pragma omp parallel for
#endif
    for (int l = 0; l < levIndices.size() - 1; ++l)
    {
      int levelBegin = levIndices[l], levelEnd = levIndices[l + 1];
      int *b = &bBuffer[0];

      sort(&reversePerm[levelBegin], &reversePerm[levelEnd]);

      // partition this level.
      int nnz = 0;
      /** 计算每个行的cost (nnz数量) */
      for (int i = levelBegin; i < levelEnd; ++i)
      {
        b[i] = nnz;
        nnz += costFunction.getCostOf(reversePerm[i]);
      }
      int nnzPerThread = (nnz + nthreads - 1) / nthreads;

      int prevEnd = levelBegin;
      int r = levelBegin;
      nnz = 0;
      int t;
      for (t = 0; t < nthreads; ++t)
      {
        // 查找第一个不低于阈值的索引，lower_bound为std中的函数，返回一个迭代器，指向范围中第一个不小于给定值的元素
        int newr = lower_bound(&b[r], &b[levelEnd], (t + 1) * nnzPerThread) - &b[0];
        if (aggregateForVectorization)
        {
          // make task size a multiple of 8 as much as possible
          if (0 == t)
          {
            r = min(r + (newr - r + 7) / 8 * 8, levelEnd);
          }
          else
          {
            r = min(r - 1 + (newr - r + 1 + 7) / 8 * 8, levelEnd);
          }
        }
        else
        {
          r = newr;
        }
        // for ( ; r < levelEnd && nnz < (t + 1)*nnzPerThread; r++) {
        // nnz += L.rowptr[reversePerm[r] + 1] - L.rowptr[reversePerm[r]];
        //}
        // assert(r == r2);

        int begin = prevEnd;
        int end = min(r, levelEnd);
        prevEnd = end;
        // 将每个线程负责的每个level中对应的行，组织为taskRows存储起来
        taskRows[t * (levIndices.size() - 1) + l] = make_pair(begin, end);
        if (aggregateForVectorization && end >= levelEnd)
          break;

        // printf("(%d, %d) (%d, %d) (%d, %d)\n", l, t, begin, end, levelBegin, levelEnd);
        // if (r < levelEnd)
        // nnz += L.rowptr[reversePerm[r] + 1] - L.rowptr[reversePerm[r]];
        ++r; // make sure first threads execute some tasks -> improve locality
      } // for each thread

      if (aggregateForVectorization)
      {
        // shift tasks to the last threads so that earlier threads
        // have tasks whose number of rows is a multiple of 8.

        // we have (t + 1) non-empty tasks in this level
        // copy [0, t] to [nthreads - 1 - t, nthreads - 1]
        for (int i = t; i >= 0; --i)
        {
          taskRows[(nthreads - 1 - t + i) * (levIndices.size() - 1) + l] =
              taskRows[i * (levIndices.size() - 1) + l];
        }
        // fill [0, nthreads - 1 - t) empty
        for (int i = 0; i < nthreads - t - 1; ++i)
        {
          taskRows[i * (levIndices.size() - 1) + l] = make_pair(levelBegin, levelBegin);
        }
      }
    } // for each level

    if (useMemoryPool)
    {
      memoryPool->setTail(bBufferBegin);
      bBuffer = NULL;
    }
    else
    {
      FREE(bBuffer);
    }

    // printf("t2 = %f\n", (__rdtsc() - t)/get_cpu_freq());

    // new permutation so that rows for a thread are contiguous
    vector<int> rowPartialSum(nthreads + 1);
    rowPartialSum[0] = 0;

    threadBoundaries.resize(nthreads + 1);
    threadBoundaries[0] = 0;
    if (useBarrier)
    {
#pragma omp parallel for
      for (int tid = 0; tid < nthreads; ++tid)
      {
        int sum = 0;
        int cnt = 0;
        for (int i = 0; i < levIndices.size() - 1; ++i)
        {
          int diff = taskRows[tid * (levIndices.size() - 1) + i].second - taskRows[tid * (levIndices.size() - 1) + i].first;
          sum += diff;
          ++cnt;
        }
        rowPartialSum[tid + 1] = sum;
        threadBoundaries[tid + 1] = cnt;
      }
    }
    else
    {
#pragma omp parallel for
      for (int tid = 0; tid < nthreads; ++tid)
      {
        int sum = 0;
        int cnt = 0;
        for (int i = 0; i < levIndices.size() - 1; ++i)
        {
          int diff = taskRows[tid * (levIndices.size() - 1) + i].second - taskRows[tid * (levIndices.size() - 1) + i].first;
          sum += diff;
          if (diff)
            ++cnt;
        }
        rowPartialSum[tid + 1] = sum;
        threadBoundaries[tid + 1] = cnt;
      }
    }

    for (int tid = 0; tid < nthreads; ++tid)
    {
      rowPartialSum[tid + 1] += rowPartialSum[tid];
      threadBoundaries[tid + 1] += threadBoundaries[tid];
    }

    FREE(schedule->origToThreadContPerm);
    FREE(schedule->threadContToOrigPerm);
    schedule->origToThreadContPerm = schedule->allocate<int>(m + 64);
    schedule->threadContToOrigPerm = schedule->allocate<int>(m + 64);
    int *origToThreadContPerm = schedule->origToThreadContPerm;
    int *threadContToOrigPerm = schedule->threadContToOrigPerm;

    taskBoundaries.resize(threadBoundaries[nthreads] + 1);

    if (useBarrier)
    {
#pragma omp parallel for
      for (int tid = 0; tid < nthreads; ++tid)
      {
        int rowOffset = rowPartialSum[tid];
        int taskOffset = threadBoundaries[tid];
        for (int i = 0; i < levIndices.size() - 1; ++i)
        {
          taskBoundaries[taskOffset] = rowOffset;
          ++taskOffset;

          for (int j = taskRows[tid * (levIndices.size() - 1) + i].first; j < taskRows[tid * (levIndices.size() - 1) + i].second; ++j)
          {
            origToThreadContPerm[reversePerm[j]] = rowOffset;
            threadContToOrigPerm[rowOffset] = reversePerm[j];
            rowOffset++;
          }
        } // for each task
      } // for each thread
    } // useBarrier
    else
    {
#pragma omp parallel for
      for (int tid = 0; tid < nthreads; ++tid)
      {
        int rowOffset = rowPartialSum[tid];
        int taskOffset = threadBoundaries[tid];
        for (int i = 0; i < levIndices.size() - 1; ++i)
        {
          if (taskRows[tid * (levIndices.size() - 1) + i].second > taskRows[tid * (levIndices.size() - 1) + i].first)
          {
            taskBoundaries[taskOffset] = rowOffset;
            ++taskOffset;

            for (int j = taskRows[tid * (levIndices.size() - 1) + i].first; j < taskRows[tid * (levIndices.size() - 1) + i].second; ++j)
            {
              origToThreadContPerm[reversePerm[j]] = rowOffset;
              assert(reversePerm[j] >= 0);
              threadContToOrigPerm[rowOffset] = reversePerm[j];
              rowOffset++;
            }
          }
        } // for each task
      } // for each thread
    } // !useBarrier
    taskBoundaries[threadBoundaries[nthreads]] = m;

    assert(isPerm(origToThreadContPerm, m));
    assert(isPerm(threadContToOrigPerm, m));

    if (!useMemoryPool)
    {
      FREE(taskRows);
    }
    else
    {
      memoryPool->setHead(taskRowsEnd);
      taskRows = NULL;
    }

    if (!useMemoryPool)
    {
      FREE(levContToOrigPerm);
    }

    schedule->ntasks = threadBoundaries.back();
  }

  template <int BASE = 0>
  static void constructTaskGraph_(
      LevelSchedule *schedule,
      int m,
      const int *rowptr, const int *diagptr, const int *extptr, const int *colidx,
      const CostFunction &costFunction)
  {
    vector<int> &taskBoundaries = schedule->taskBoundaries;
    vector<int> &threadBoundaries = schedule->threadBoundaries;

    int **parentsBuf = schedule->parentsBuf;

    bool transitiveReduction = schedule->transitiveReduction;
    bool fuseSpMV = schedule->fuseSpMV;

    bool useMemoryPool = schedule->useMemoryPool;
    /**
     * 执行基本的findLevels的操作（并行和串行），以及实现每个level内的任务划分到线程。
     */
    findLevels_<BASE>(schedule, m, rowptr, diagptr, extptr, colidx, costFunction);

        int ntasks = schedule->ntasks;
        int *threadContToOrigPerm = schedule->threadContToOrigPerm;

        if (NULL == extptr)
        {
          extptr = rowptr + 1;
        }
        colidx -= BASE;

        int nnz = rowptr[m] - BASE;

    #ifdef TRSOLVER_LOG
        unsigned long long t;
        double spmvTime = nnz * 12 / 60 / 1e9;
    #endif
    #ifdef _OPENMP
        int nthreads = omp_get_max_threads();
    #else
        int nthreads = 1;
    #endif

        MemoryPool *memoryPool = MemoryPool::getSingleton();
        size_t origRowIdToTaskIdEnd = memoryPool->getHead();
        int *origRowIdToTaskId = schedule->allocateFront<int>(m);
    #pragma omp parallel for num_threads(nthreads)
        for (int task = 0; task < ntasks; ++task)
        {
          for (int row = taskBoundaries[task]; row < taskBoundaries[task + 1]; ++row)
          {
            origRowIdToTaskId[threadContToOrigPerm[row]] = task;
          }
        }

        schedule->nparentsForward = schedule->allocate<short>(ntasks);
        schedule->parentsForward = schedule->allocate<int *>(ntasks);
        short *nparentsForward = schedule->nparentsForward;
        int **parentsForward = schedule->parentsForward;

        schedule->nparentsBackward = schedule->allocate<short>(ntasks);
        schedule->parentsBackward = schedule->allocate<int *>(ntasks);
        short *nparentsBackward = schedule->nparentsBackward;
        int **parentsBackward = schedule->parentsBackward;

        schedule->taskFinished = schedule->allocate<int>(ntasks);
        volatile int *taskFinished = schedule->taskFinished;

    #pragma omp parallel for
        for (int i = 0; i < ntasks; ++i)
        {
          nparentsForward[i] = 0;
          parentsForward[i] = NULL;

          nparentsBackward[i] = 0;
          parentsBackward[i] = NULL;

          taskFinished[i] = 0;
        }

    #ifdef LOG_MEMORY_ALLOCATION
        printf("Allocate %d MB in %s\n", (sizeof(short) * ntasks + sizeof(int *) * ntasks + sizeof(int) * ntasks) / 1024 / 1024, __func__);
        fflush(stdout);
    #endif

        size_t childrenBufEnd = 0;
        short *nparentsTemp;
        vector<vector<int>> tempParents;

        // unsigned long long barrierTimes[NUM_MAX_THREADS] = { 0 };

    #ifdef TRSOLVER_LOG
        t = __rdtsc();
    #endif

        size_t adjacencyBegin = memoryPool->getTail();
        CSR taskAdjacency, taskInvAdjacency;
        taskAdjacency.rowptr = schedule->allocate<int>(ntasks + 1);
        taskAdjacency.colidx = schedule->allocate<int>((nnz - m) / 2 + m);
        taskInvAdjacency.rowptr = schedule->allocate<int>(ntasks + 1);
        taskInvAdjacency.colidx = schedule->allocate<int>((nnz - m) / 2 + m + ntasks);

        int *taskInvAdjacencyLengths = schedule->allocate<int>(ntasks);
    #pragma omp parallel for num_threads(nthreads)
        for (int i = 0; i < ntasks; ++i)
        {
          taskInvAdjacencyLengths[i] = 0;
        }

    #ifdef LOG_MEMORY_ALLOCATION
        printf(
            "Allocate %d MB in %s\n",
            sizeof(int) * (ntasks * 3 + (nnz - m) + 2 * m) / 1024 / 1024,
            __func__);
        fflush(stdout);
    #endif

        vector<int> nnzPerRow(ntasks);

        int perThreadOrigRowPtrSum[NUM_MAX_THREADS] = {0};
        int perThreadOrigInvRowPtrSum[NUM_MAX_THREADS] = {0};
        int perThreadRowPtrSum[NUM_MAX_THREADS] = {0};
    #ifdef TRSOLVER_LOG
        int numNewEdges[NUM_MAX_THREADS] = {0};
    #endif
        int *childrenBuf;
        int **children;
        short *numOfChildren;

        int v1Boundaries[NUM_MAX_THREADS] = {0}, v1Boundaries2[NUM_MAX_THREADS] = {0};

    #ifdef _OPENMP
    #pragma omp parallel num_threads(nthreads)
    #endif
        {
          int tid = omp_get_thread_num();
          int v1PerThread = (ntasks + nthreads - 1) / nthreads;
          int v1Begin = min(tid * v1PerThread, ntasks);
          int v1End = min(v1Begin + v1PerThread, ntasks);

          // count nnzs to load balance
          int origRowPtrCnt = 0, origInvRowPtrCnt = 0;
          for (int v1 = v1Begin; v1 < v1End; ++v1)
          {
            int size = 0;
            int invSize = 0;
            for (int i1Perm = taskBoundaries[v1]; i1Perm < taskBoundaries[v1 + 1]; ++i1Perm)
            {
              int i1 = threadContToOrigPerm[i1Perm];
              size += extptr[i1] - diagptr[i1] - 1;
              invSize += diagptr[i1] - rowptr[i1] + 1; // additional 1 for intra-thread
            }
            taskAdjacency.rowptr[v1] = origRowPtrCnt;
            taskInvAdjacency.rowptr[v1] = origInvRowPtrCnt;
            origRowPtrCnt += size;
            origInvRowPtrCnt += invSize;
            nnzPerRow[v1] = size;
          }
          perThreadOrigRowPtrSum[tid + 1] = origRowPtrCnt;
          perThreadOrigInvRowPtrSum[tid + 1] = origInvRowPtrCnt;

          // compute prefix sums
    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
          {
            for (int tid = 0; tid < nthreads; ++tid)
            {
              perThreadOrigRowPtrSum[tid + 1] += perThreadOrigRowPtrSum[tid];
              perThreadOrigInvRowPtrSum[tid + 1] += perThreadOrigInvRowPtrSum[tid];
            }
          }

          for (int v1 = v1Begin; v1 < v1End; ++v1)
          {
            taskInvAdjacency.rowptr[v1] += perThreadOrigInvRowPtrSum[tid];
          }

          // load-balanced partition
          if (tid == nthreads - 1)
          {
            v1Boundaries[tid + 1] = ntasks;
          }
          else
          {
            int nnzPerThread = ((nnz - m) / 2 + nthreads - 1) / nthreads;
            // thread
            int boundaryTid =
                upper_bound(
                    perThreadOrigRowPtrSum, perThreadOrigRowPtrSum + nthreads - 1,
                    nnzPerThread * (tid + 1)) -
                perThreadOrigRowPtrSum - 1;
            // row boundary
            v1Boundaries[tid + 1] =
                upper_bound(
                    taskAdjacency.rowptr + min(boundaryTid * v1PerThread, ntasks),
                    taskAdjacency.rowptr + min((boundaryTid + 1) * v1PerThread, ntasks),
                    nnzPerThread * (tid + 1) - perThreadOrigRowPtrSum[boundaryTid]) -
                taskAdjacency.rowptr - 1;
            v1Boundaries[tid + 1] = max(0, v1Boundaries[tid + 1]);
          }

    #ifdef _OPENMP
    #pragma omp barrier
    #endif

          v1Begin = v1Boundaries[tid];
          v1End = v1Boundaries[tid + 1];
          // thread boundary
          int t1Begin = distance(
                            threadBoundaries.begin(),
                            upper_bound(threadBoundaries.begin(), threadBoundaries.end(), v1Begin)) -
                        1;
          int t1End = distance(
                          threadBoundaries.begin(),
                          upper_bound(threadBoundaries.begin(), threadBoundaries.end(), v1End)) -
                      1;
          t1End = min(t1End, nthreads - 1);

          origRowPtrCnt = 0;
          int maxSize = 0;
          for (int v1 = v1Begin; v1 < v1End; ++v1)
          {
            assert(v1 >= 0 && v1 < ntasks);
            int size = nnzPerRow[v1];
            maxSize = max(size, maxSize);
            origRowPtrCnt += size;
          }
          perThreadOrigRowPtrSum[tid + 1] = origRowPtrCnt;

    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
          {
            for (int tid = 0; tid < nthreads; ++tid)
            {
              perThreadOrigRowPtrSum[tid + 1] += perThreadOrigRowPtrSum[tid];
            }
          }

          // construct task dependency graph
          unsigned long long t11 = 0, t2 = 0, t3 = 0; //, t4 = 0, t5 = 0;
          int rowPtrCnt = 0;
          vector<int> i2s(maxSize);
          int maxL = 0;
          for (int t1 = t1Begin; t1 <= t1End; ++t1)
          {
            for (int v1 = max(threadBoundaries[t1], v1Begin); v1 < min(threadBoundaries[t1 + 1], v1End); ++v1)
            {
              unsigned long long tempTimer = __rdtsc();

              int
                  boundaryBegin = threadBoundaries[t1],
                  boundaryEnd = threadBoundaries[t1 + 1];

              int size = 0;
              for (int i1Perm = taskBoundaries[v1]; i1Perm < taskBoundaries[v1 + 1]; ++i1Perm)
              {
                int i1 = threadContToOrigPerm[i1Perm];

                int begin = diagptr[i1] + 1;  // 因为是对称矩阵，这里是上三角的行部分，对应下三角的列
                int end = extptr[i1];
                for (int j = begin; j < end; ++j)
                {
                  int v = origRowIdToTaskId[colidx[j] - BASE];
                  if (v < boundaryBegin || v >= boundaryEnd)  //non-intra-thread edge
                  {
                    i2s[size] = v;
                    ++size;
                  }
                  // else {
                  //__sync_fetch_and_add(&intraThreadDependencyCount, 1);
                  //}
                }
              }

              t11 += __rdtsc() - tempTimer;
              tempTimer = __rdtsc();

              sort(i2s.begin(), i2s.begin() + size);

              t2 += __rdtsc() - tempTimer;
              tempTimer = __rdtsc();

              int oldV2 = -1;

              taskAdjacency.rowptr[v1] = perThreadOrigRowPtrSum[tid] + rowPtrCnt;
              int *adjList = taskAdjacency.colidx + perThreadOrigRowPtrSum[tid] + rowPtrCnt;
              int l = 0;
              bool intraAdded = v1 == threadBoundaries[t1 + 1] - 1;
              for (int i2 = 0; i2 < size; ++i2)
              {
                int v2 = i2s[i2];
                if (v2 == oldV2)
                  continue;
                oldV2 = v2;

                adjList[l] = v2;
                ++l;

                if (!intraAdded && transitiveReduction && v2 >= v1 + 1)
                {
                  if (v2 > v1 + 1)// 内部依赖没有添加，先添加线程内部的依赖，这样添加的依赖在colidx上是顺序递增的
                  {
                    int tempL = __sync_fetch_and_add(&taskInvAdjacencyLengths[v1 + 1], 1);
                    taskInvAdjacency.colidx[taskInvAdjacency.rowptr[v1 + 1] + tempL] = v1;
                  }
                  intraAdded = true;
                }
                // add reverse dependency
                int tempL = __sync_fetch_and_add(&taskInvAdjacencyLengths[v2], 1);
                assert(taskInvAdjacency.rowptr[v2] + tempL >= 0);
                assert(taskInvAdjacency.rowptr[v2] + tempL < (nnz - m) / 2 + m + ntasks);
                taskInvAdjacency.colidx[taskInvAdjacency.rowptr[v2] + tempL] = v1;  // 添加依赖 v1-> v2到CSR的v2上
              }
              if (!intraAdded && transitiveReduction)
              {// v1不是本线程的最后一个task因此 添加v1 -> v1 + 1 确保可以进行transitive reduction
                int tempL = __sync_fetch_and_add(&taskInvAdjacencyLengths[v1 + 1], 1);
                assert(taskInvAdjacency.rowptr[v1 + 1] + tempL >= 0);
                assert(taskInvAdjacency.rowptr[v1 + 1] + tempL < (nnz - m) / 2 + m + ntasks);
                taskInvAdjacency.colidx[taskInvAdjacency.rowptr[v1 + 1] + tempL] = v1;
              }

              t3 += __rdtsc() - tempTimer;
              rowPtrCnt += l;
              nnzPerRow[v1] = l;
              maxL = max(maxL, l);
            }
          }

          perThreadRowPtrSum[tid + 1] = rowPtrCnt;  // task DAG row ptr

          unsigned long long tempTimer = __rdtsc();

    #ifdef _OPENMP
    #pragma omp barrier
    #endif

          // t4 = __rdtsc() - tempTimer;
          // barrierTimes[tid] = t4;
          tempTimer = __rdtsc();

          // compute prefix sum
    #ifdef _OPENMP
    #pragma omp single
    #endif
          {
            for (int tid = 0; tid < nthreads; ++tid)
            {
              perThreadRowPtrSum[tid + 1] += perThreadRowPtrSum[tid];
            }
            taskAdjacency.rowptr[ntasks] = perThreadOrigRowPtrSum[nthreads];
          }

          for (int task = v1Begin; task < v1End; ++task)
          {
            sort(
                taskInvAdjacency.colidx + taskInvAdjacency.rowptr[task],
                taskInvAdjacency.colidx + taskInvAdjacency.rowptr[task] + taskInvAdjacencyLengths[task]);
          }

          // t5 = __rdtsc() - tempTimer;
    #ifndef _OPENMP
    #define _OPENMP
    #endif
    #ifdef _OPENMP
          // construct FusedGSAndSpMVSchedule
          if (fuseSpMV)
          {
    #pragma omp barrier
    #pragma omp single // FIXME - parallelize this
            {
              schedule->fusedSchedule = new FusedGSAndSpMVSchedule(schedule);

              tempParents.resize(ntasks);
              for (int task = 0; task < ntasks; ++task)
              {
                for (int j = 0; j < nnzPerRow[task]; ++j)
                {
                  int child = taskAdjacency.colidx[taskAdjacency.rowptr[task] + j];
                  if (tempParents[child].empty())
                  {
                    tempParents[child].push_back(task);
                  }
                  else
                  {
                    // if there's already a parent, append only if the new parent is assigned
                    // to a different thread
                    int t = distance(
                                threadBoundaries.begin(),
                                upper_bound(threadBoundaries.begin(), threadBoundaries.end(), task)) -
                            1;
                    int tPrev = distance(
                                    threadBoundaries.begin(),
                                    upper_bound(threadBoundaries.begin(), threadBoundaries.end(), tempParents[child].back())) -
                                1;
                    if (t != tPrev)
                    {
                      tempParents[child].push_back(task);
                    }
                  }
                } // for each child of this task
              } // for each task
            } // omp single

            // add intra-thread dependency
    #pragma omp for
            for (int t = 0; t < nthreads; ++t)
            {
              for (int task = threadBoundaries[t]; task < threadBoundaries[t + 1]; ++task)
              {
                int minParentTask = task;
                for (int i = taskBoundaries[task]; i < taskBoundaries[task + 1]; ++i)
                {
                  int row = threadContToOrigPerm[i];
                  int begin = rowptr[row], end = diagptr[row];
                  for (int j = begin; j < end; ++j)
                  {
                    int parentTask = origRowIdToTaskId[colidx[j] - BASE];
                    if (parentTask >= threadBoundaries[t] && parentTask < threadBoundaries[t + 1])
                    {
                      if (parentTask < minParentTask)
                      {
                        minParentTask = parentTask;
                      }
                    }
    #if 0 // ndef NDEBUG
                else {
                  bool found = false;
                  for (int k = 0; k < fusedSchedule->nparents[task]; ++k) {
                    if (fusedSchedule->parents[task][k] == parentTask) {
                      found = true;
                      break;
                    }
                  }
                  if (!found) {
                    printf("%d->%d is missing, row = %d, parentRow = %d\n", parentTask, task, row, colidx[j]);
                  }
                  assert(found);
                }
    #endif
                  } // for each parent
                } // for each row

                tempParents[task].push_back(minParentTask);
                // fusedSchedule->parents[task][fusedSchedule->nparents[task]] = minParentTask;
                // fusedSchedule->nparents[task]++;
              } // for each task
            } // for each thread

            // remove transitive dependencies
            // v->u is redundant when
            // case a : v ...-> q -> p -> u
            //        (v is executed before q in the same thread)
            // case b : v -> q ...-> p -> u
    #pragma omp for
            for (int u = 0; u < ntasks; ++u)
            {
              sort(tempParents[u].begin(), tempParents[u].end());

              vector<int> parentsToRemove;
              for (auto v = tempParents[u].begin(); v != tempParents[u].end(); ++v)
              {
                int tOfV = distance(
                               threadBoundaries.begin(),
                               upper_bound(threadBoundaries.begin(), threadBoundaries.end(), *v)) -
                           1;
                bool redundant = false;

                for (auto p = tempParents[u].begin(); p != tempParents[u].end(); ++p)
                {
                  // case a
                  for (int j = 0; j < nnzPerRow[*p]; ++j)
                  {
                    int q = taskAdjacency.colidx[taskAdjacency.rowptr[*p] + j];
                    if (q >= threadBoundaries[tOfV] && q <= *v)
                    {
                      redundant = true;
                      break;
                    }
                  }

                  // case b : this happens very rarely
                  /*int tOfP = distance(
                    threadBoundaries.begin(),
                    upper_bound(threadBoundaries.begin(), threadBoundaries.end(), *p)) - 1;

                  for (int j = 0; j < taskInvAdjacencyLengths[*v]; ++j) {
                    int q = taskInvAdjacency.colidx[taskInvAdjacency.rowptr[*v] + j];
                    if (q >= *p && q < threadBoundaries[tOfP]) {
                      redundant = true;
                      break;
                    }
                  }*/
                }

                if (redundant)
                {
                  parentsToRemove.push_back(*v);
                }
              } // for each parent v of u

              sort(parentsToRemove.begin(), parentsToRemove.end());
              vector<int> remainder;
              set_difference(
                  tempParents[u].begin(), tempParents[u].end(),
                  parentsToRemove.begin(), parentsToRemove.end(),
                  back_inserter(remainder));
              tempParents[u].swap(remainder);
            } // for each u

    #pragma omp single
            {
              FusedGSAndSpMVSchedule *fusedSchedule = schedule->fusedSchedule;
              fusedSchedule->parents = MALLOC(int *, ntasks);
              fusedSchedule->parentsBuf = MALLOC(int, perThreadOrigRowPtrSum[nthreads] + 2 * ntasks);

              fusedSchedule->nparents = MALLOC(short, ntasks);
              memset((void *)fusedSchedule->nparents, 0, sizeof(short) * ntasks);

              int cnt = 0;
              for (int task = 0; task < ntasks; ++task)
              {
                fusedSchedule->parents[task] = fusedSchedule->parentsBuf + cnt;
                fusedSchedule->nparents[task] = tempParents[task].size();
                copy(tempParents[task].begin(), tempParents[task].end(), fusedSchedule->parents[task]);
                cnt += taskInvAdjacencyLengths[task];
              }

              /*int cnt = 0;
              for (int task = 0; task < ntasks; ++task) {
                fusedSchedule->parents[task] = fusedSchedule->parentsBuf + cnt;
                cnt += taskInvAdjacencyLengths[task] + 1;
              }
              assert(cnt <= perThreadOrigRowPtrSum[nthreads] + ntasks);

              for (int task = 0; task < ntasks; ++task) {
                for (int j = 0; j < nnzPerRow[task]; ++j) {
                  int child = taskAdjacency.colidx[taskAdjacency.rowptr[task] + j];
                  if (fusedSchedule->nparents[child] > 0) {
                    int t = distance(
                      threadBoundaries.begin(),
                      upper_bound(threadBoundaries.begin(), threadBoundaries.end(), task)) - 1;
                    int tPrev = distance(
                      threadBoundaries.begin(),
                      upper_bound(threadBoundaries.begin(), threadBoundaries.end(), fusedSchedule->parents[child][fusedSchedule->nparents[child] - 1])) - 1;
                    if (t != tPrev) {
                      fusedSchedule->parents[child][fusedSchedule->nparents[child]] = task;
                      fusedSchedule->nparents[child]++;
                    }
                  }
                  else {
                    fusedSchedule->parents[child][fusedSchedule->nparents[child]] = task;
                    fusedSchedule->nparents[child]++;
                  }
                }
              } // for each task

              // add intra-thread dependency
              for (int t = 0; t < nthreads; ++t) {
                for (int task = threadBoundaries[t]; task < threadBoundaries[t + 1]; ++task) {
                  int minParentTask = task;
                  for (int i = taskBoundaries[task]; i < taskBoundaries[task + 1]; ++i) {
                    int row = threadContToOrigPerm[i];
                    int begin = rowptr[row], end = diagptr[row];
                    for (int j = begin; j < end; ++j) {
                      int parentTask = origRowIdToTaskId[colidx[j]];
                      if (parentTask >= threadBoundaries[t] && parentTask < threadBoundaries[t + 1]) {
                        if (parentTask < minParentTask) {
                          minParentTask = parentTask;
                        }
                      }
        #if 0 // ndef NDEBUG
                      else {
                        bool found = false;
                        for (int k = 0; k < fusedSchedule->nparents[task]; ++k) {
                          if (fusedSchedule->parents[task][k] == parentTask) {
                            found = true;
                            break;
                          }
                        }
                        if (!found) {
                          printf("%d->%d is missing, row = %d, parentRow = %d\n", parentTask, task, row, colidx[j]);
                        }
                        assert(found);
                      }
        #endif
                    } // for each parent
                  } // for each row

                  fusedSchedule->parents[task][fusedSchedule->nparents[task]] = minParentTask;
                  fusedSchedule->nparents[task]++;
                } // for each task
              } // for each thread*/

    // #define PRINT_AVG_DEGREE_OF_SPMV_TASKS
    #ifdef PRINT_AVG_DEGREE_OF_SPMV_TASKS
              int nDeps = 0;
              for (int task = 0; task < ntasks; ++task)
              {
                nDeps += fusedSchedule->nparents[task];
              }
              printf("average degree = %f\n", (double)nDeps / ntasks);
    #endif

              /*for (int task = 0; task < ntasks; ++task) {
                printf("(");
                for (int j = 0; j < fusedSchedule->nparents[task]; ++j) {
                  int parent = fusedSchedule->parents[task][j];
                  int t = distance(
                    threadBoundaries.begin(),
                    upper_bound(threadBoundaries.begin(), threadBoundaries.end(), parent)) - 1;
                  printf("%d(%d:%d) ", parent, t, parent - threadBoundaries[t]);
                }
                printf(")->%d\n", task);
              }*/
            } // omp single
          } // construct FusedGSAndSpMVSchedule
    #endif // !_OPENMP

    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
          {

    #ifdef TRSOLVER_LOG
            double dt = (__rdtsc() - t) / get_cpu_freq();
            printf(
                "construct task dependency graph takes %f (%f SpMVs)\n", dt, dt / spmvTime);

            printf(
                "average degree = %f\n",
                (double)perThreadRowPtrSum[nthreads] / (schedule->levIndices.size() - 2) / nthreads);
            fflush(stdout);

            t = __rdtsc();
    #endif

          } // omp single

          // load-balanced partition
          if (tid == nthreads - 1)
          {
            v1Boundaries2[tid + 1] = ntasks;
          }
          else
          {
            int nnzPerThread = (perThreadRowPtrSum[nthreads] + nthreads - 1) / nthreads;
            int boundaryTid =
                upper_bound(
                    perThreadRowPtrSum, perThreadRowPtrSum + nthreads - 1,
                    nnzPerThread * (tid + 1)) -
                perThreadRowPtrSum - 1;

            v1Boundaries2[tid + 1] =
                upper_bound(
                    taskAdjacency.rowptr + v1Boundaries[boundaryTid],
                    taskAdjacency.rowptr + v1Boundaries[boundaryTid + 1],
                    nnzPerThread * (tid + 1) + perThreadOrigRowPtrSum[boundaryTid] - perThreadRowPtrSum[boundaryTid]) -
                taskAdjacency.rowptr - 1;
            v1Boundaries2[tid + 1] = max(0, v1Boundaries2[tid + 1]);
          }

    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
          {
            for (int i = 0; i < nthreads; ++i)
            {
            }
          }

          v1Begin = v1Boundaries2[tid];
          v1End = v1Boundaries2[tid + 1];

          t1Begin = distance(
                        threadBoundaries.begin(),
                        upper_bound(threadBoundaries.begin(), threadBoundaries.end(), v1Begin)) -
                    1;
          t1End = distance(
                      threadBoundaries.begin(),
                      upper_bound(threadBoundaries.begin(), threadBoundaries.end(), v1End)) -
                  1;
          t1End = min(t1End, nthreads - 1);

          rowPtrCnt = 0;
          maxL = 0;
          for (int v1 = v1Begin; v1 < v1End; ++v1)
          {
            rowPtrCnt += nnzPerRow[v1];
            maxL = max(maxL, nnzPerRow[v1]);
          }
          perThreadRowPtrSum[tid + 1] = rowPtrCnt;

    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
          {
            for (int tid = 0; tid < nthreads; ++tid)
            {
              perThreadRowPtrSum[tid + 1] += perThreadRowPtrSum[tid];
            }
          }

          rowPtrCnt = 0;

    #ifdef _OPENMP
          if (transitiveReduction)
    #endif
          {
    #ifdef _OPENMP
    #pragma omp barrier
    #pragma omp single
    #endif
            {
              childrenBufEnd = memoryPool->getHead();

              childrenBuf = schedule->allocateFront<int>(perThreadRowPtrSum[nthreads]);
              children = schedule->allocateFront<int *>(ntasks);
              numOfChildren = schedule->allocateFront<short>(ntasks);
            }

    #ifdef _OPENMP
    #pragma omp for
    #endif
            for (int i = 0; i < ntasks; ++i)
            {
              children[i] = NULL;
              numOfChildren[i] = 0;
            }

            t11 = 0;
            t2 = 0;
            vector<int> tempAdj(maxL + 1);
            // int itr1DistanceSum = 0, itr2DistanceSum = 0, itrDistanceCnt = 0;
            // int itr1ContSum = 0, itr1ContCnt = 0, itr2ContSum = 0, itr2ContCnt = 0;
            // int skipSum = 0, skipCnt = 0;
            for (int t = t1Begin; t <= t1End; ++t)
            {
              for (int j = max(threadBoundaries[t], v1Begin); j < min(threadBoundaries[t + 1], v1End); ++j)
              {
                unsigned long long tempTimer = __rdtsc();

                children[j] = childrenBuf + perThreadRowPtrSum[tid] + rowPtrCnt;
                int *adj = taskAdjacency.colidx + taskAdjacency.rowptr[j];
                int adjSize = nnzPerRow[j];

                int size;
                if (j < threadBoundaries[t + 1] - 1)
                {
                  int idx = lower_bound(adj, adj + adjSize, j + 1) - adj;
                  if (idx < adjSize && adj[idx] == j + 1)
                  {
                    size = adjSize;
                    memcpy(&tempAdj[0], adj, sizeof(adj[0]) * adjSize);
                  }
                  else
                  {
                    size = adjSize + 1;
                    memcpy(&tempAdj[0], adj, sizeof(adj[0]) * idx);
                    tempAdj[idx] = j + 1;  // 补充intra dependency
                    memcpy(&tempAdj[idx + 1], adj + idx, sizeof(adj[0]) * (adjSize - idx));
                  }
                }
                else
                {
                  size = adjSize;
                  memcpy(&tempAdj[0], adj, sizeof(adj[0]) * adjSize);
                }

                t11 += __rdtsc() - tempTimer;
                tempTimer = __rdtsc();

                int idx = 0;
                for (int *succItr = adj; succItr != adj + adjSize; ++succItr)
                {
                  int i = *succItr;  // i

                  int *itr1 = taskInvAdjacency.colidx + taskInvAdjacency.rowptr[i]; // k
                  int *itr1End = itr1 + taskInvAdjacencyLengths[i];
                  int *itr2 = &tempAdj[0];   // j
                  int *itr2End = itr2 + size;
                  // i->j, i-> k, 寻找是否存在j->k

                  bool toRemove = false;

                  if (itr2 != itr2End)
                  {
                    itr1 = lower_bound(itr1, itr1End, *itr2);
                  }
                  if (itr1 != itr1End)
                  {
                    itr2 = lower_bound(itr2, itr2End, *itr1);

                    /*itr1End = upper_bound(itr1, itr1End, *(itr2End - 1));
                    itr2End = upper_bound(itr2, itr2End, *(itr1End - 1));*/
                  }

                  while (itr1 != itr1End && itr2 != itr2End)
                  {
                    int cmp = *itr1 - *itr2;
                    if (0 == cmp)
                    {
                      toRemove = true;
                      break;
                    }
                    else if (cmp < 0)
                      ++itr1;
                    else
                      ++itr2;
                  }

                  if (!toRemove)
                  {
                    children[j][idx] = *succItr;
                    __sync_fetch_and_add(&nparentsForward[*succItr], 1);
                    ++idx;
                  }
                }

                numOfChildren[j] = idx;
                rowPtrCnt += idx;

                t2 += __rdtsc() - tempTimer;
              } // for each j
            } // for each t

            if (20 == tid)
            {
              // printf("skip = %f\n", (double)skipSum/skipCnt);
              /*printf(
                "itr1Distance = %f, itr2Distance = %f, itr1Cont = %f, itr2Cont = %f\n",
                (double)itr1DistanceSum/itrDistanceCnt,
                (double)itr2DistanceSum/itrDistanceCnt,
                (double)itr1ContSum/itr1ContCnt,
                (double)itr2ContSum/itr2ContCnt);*/
            }

    #ifdef TRSOLVER_LOG
            numNewEdges[tid] = rowPtrCnt;
    #endif
            tempAdj.clear();

            tempTimer = __rdtsc();
    #ifdef _OPENMP
    #pragma omp barrier
    #endif
            t3 = __rdtsc() - tempTimer;
            // barrierTimes[tid] = t3;

    #ifdef _OPENMP
    #pragma omp barrier
    #endif
            // convert numOfChildren and children
            // to nparents and parents
            size_t nparentsTempBegin;

    #ifdef _OPENMP
    #pragma omp master
    #endif
            {
              if (!useMemoryPool)
              {
                FREE(taskAdjacency.rowptr);
                FREE(taskAdjacency.colidx);

                FREE(taskInvAdjacency.rowptr);
                FREE(taskInvAdjacency.colidx);
              }

              taskAdjacency.~CSR();
              taskInvAdjacency.~CSR();
              if (useMemoryPool)
              {
                memoryPool->setTail(adjacencyBegin);
              }
              else
              {
                FREE(taskInvAdjacencyLengths);
              }

              int cForward = 0, cBackward = 0;
              for (int i = 0; i < ntasks; ++i)
              {
                cForward += nparentsForward[i];
                cBackward += numOfChildren[i];

                nparentsBackward[i] = numOfChildren[i];
              }

              parentsBuf[0] = schedule->allocate<int>(cForward);
              parentsBuf[1] = schedule->allocate<int>(cBackward);

              cForward = 0, cBackward = 0;
              for (int i = 0; i < ntasks; ++i)
              {
                parentsForward[i] = parentsBuf[0] + cForward;
                cForward += nparentsForward[i];

                parentsBackward[i] = parentsBuf[1] + cBackward;
                cBackward += nparentsBackward[i];
              }

              nparentsTempBegin = memoryPool->getTail();
              nparentsTemp = schedule->allocate<short>(ntasks);
            } // omp single
    #ifdef _OPENMP
    #pragma omp barrier

    #pragma omp for
    #endif
            for (int i = 0; i < ntasks; ++i)
            {
              nparentsTemp[i] = 0;
            }

    #ifdef _OPENMP
    #pragma omp for
    #endif
            for (int i = 0; i < ntasks; ++i)
            {
              for (int j = 0; j < numOfChildren[i]; ++j)
              {
                int child = children[i][j];
                int idx = __sync_fetch_and_add(nparentsTemp + child, 1);
                parentsForward[child][idx] = i;
                parentsBackward[i][j] = child;
              }
            }

    #ifdef _OPENMP
    #pragma omp master
    #endif
            {
              if (useMemoryPool)
              {
                memoryPool->setTail(nparentsTempBegin);
                memoryPool->setHead(childrenBufEnd);
                nparentsTemp = NULL;
                children = NULL;
                numOfChildren = NULL;
                childrenBuf = NULL;
              }
              else
              {
                FREE(nparentsTemp);
                FREE(children);
                FREE(numOfChildren);
                FREE(childrenBuf);
              }
            } // omp single
    #ifdef _OPENMP
    #pragma omp barrier
    #endif
          } // if (transitiveReduction)
    #ifdef _OPENMP
          else
          {
    #pragma omp barrier
    #pragma omp single
            {
              int cForward = 0, cBackward = 0;
              for (int i = 0; i < ntasks; ++i)
              {
                nparentsForward[i] = taskInvAdjacencyLengths[i];
                nparentsBackward[i] = nnzPerRow[i];

                cForward += nparentsForward[i];
                cBackward += nparentsBackward[i];
              }
              FREE(taskInvAdjacencyLengths);

              parentsBuf[0] = schedule->allocate<int>(cForward);
              parentsBuf[1] = schedule->allocate<int>(cBackward);

    #ifdef LOG_MEMORY_ALLOCATION
              printf("Allocate %d MB in %s\n", sizeof(int) * cForward / 1024 / 1024, __func__);
    #endif

              cForward = 0, cBackward = 0;
              for (int i = 0; i < ntasks; ++i)
              {
                parentsForward[i] = parentsBuf[0] + cForward;
                cForward += nparentsForward[i];

                parentsBackward[i] = parentsBuf[1] + cBackward;
                cBackward += nparentsBackward[i];

                for (int j = 0; j < nparentsForward[i]; ++j)
                {
                  parentsForward[i][j] = taskInvAdjacency.colidx[taskInvAdjacency.rowptr[i] + j];
                }

                for (int j = 0; j < nparentsBackward[i]; ++j)
                {
                  parentsBackward[i][j] = taskAdjacency.colidx[taskAdjacency.rowptr[i] + j];
                }
              }

              taskAdjacency.~CSR();
              taskInvAdjacency.~CSR();
            } // omp single
          } // !transitiveReduction
    #endif
        } // omp parallel

    #ifdef TRSOLVER_LOG
        double dt = (__rdtsc() - t) / get_cpu_freq();
        printf("transitive reduction takes %f (%f SpMVs) \n", dt, dt / spmvTime);

        int numNewEdgesSum = 0;
        for (int i = 0; i < nthreads; ++i)
        {
          numNewEdgesSum += numNewEdges[i];
        }
        printf(
            "average degree after reduction = %f\n",
            (double)numNewEdgesSum / (schedule->levIndices.size() - 2) / nthreads);
    #endif

        if (useMemoryPool)
        {
          memoryPool->setHead(origRowIdToTaskIdEnd);
        }
        else
        {
          FREE(origRowIdToTaskId);
        }

        /*for (int u = 0; u < ntasks; ++u) {
          int d = taskAdjacency[u].size();
          numOfChildren[u] = d;
          children[u] = (int *)_mm_malloc(sizeof(int)*d, 64);

          auto itr(taskAdjacency[u].begin());
          for (int i = 0; itr != taskAdjacency[u].end(); ++itr, ++i) {
            children[u][i] = *itr;
            nparents[*itr]++;
          }
        }*/

        // count mergeable tasks
        /*int mergeables = 0;
        for (int tid = 0; tid < nthreads; ++tid) {
          for (int task = threadBoundariesForward[tid]; task < threadBoundariesForward[tid + 1] - 1; ++task) {
            if (numOfChildrenForward[task] == 0 && nparentsForward[task + 1] == 0) {
              mergeables++;
            }
          }
        }
        printf("mergables = %f\n", (double)mergeables/threadBoundariesForward[nthreads]);*/

    #ifdef TRSOLVER_LOG
        printf("parallelism = %f\n", (double)m / (schedule->levIndices.size() - 1));
    #endif
  }

  void LevelSchedule::constructTaskGraph(
      int m,
      const int *rowptr, const int *diagptr, const int *extptr, const int *colidx,
      const CostFunction &costFunction)
  {
    if (0 == rowptr[0])
    {
      return constructTaskGraph_<0>(this, m, rowptr, diagptr, extptr, colidx, costFunction);
    }
    else
    {
      assert(1 == rowptr[0]);
      return constructTaskGraph_<1>(this, m, rowptr, diagptr, extptr, colidx, costFunction);
    }
  }

} // namespace SpMP