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ortools-clone/ortools/lp_data/sparse.cc
Laurent Perron 16687e6575 speedup glop; better respect the time limit
2025-02-05 18:11:16 +01:00

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// Copyright 2010-2025 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#include "ortools/lp_data/sparse.h"
#include <algorithm>
#include <cstdlib>
#include <initializer_list>
#include <string>
#include <utility>
#include <vector>
#include "absl/log/check.h"
#include "absl/strings/str_format.h"
#include "absl/types/span.h"
#include "ortools/lp_data/lp_types.h"
#include "ortools/lp_data/permutation.h"
#include "ortools/lp_data/sparse_column.h"
#include "ortools/util/return_macros.h"
namespace operations_research {
namespace glop {
namespace {
using ::util::Reverse;
template <typename Matrix>
EntryIndex ComputeNumEntries(const Matrix& matrix) {
EntryIndex num_entries(0);
const ColIndex num_cols(matrix.num_cols());
for (ColIndex col(0); col < num_cols; ++col) {
num_entries += matrix.column(col).num_entries();
}
return num_entries;
}
// Computes the 1-norm of the matrix.
// The 1-norm |A| is defined as max_j sum_i |a_ij| or
// max_col sum_row |a(row,col)|.
template <typename Matrix>
Fractional ComputeOneNormTemplate(const Matrix& matrix) {
Fractional norm(0.0);
const ColIndex num_cols(matrix.num_cols());
for (ColIndex col(0); col < num_cols; ++col) {
Fractional column_norm(0);
for (const SparseColumn::Entry e : matrix.column(col)) {
// Compute sum_i |a_ij|.
column_norm += fabs(e.coefficient());
}
// Compute max_j sum_i |a_ij|
norm = std::max(norm, column_norm);
}
return norm;
}
// Computes the oo-norm (infinity-norm) of the matrix.
// The oo-norm |A| is defined as max_i sum_j |a_ij| or
// max_row sum_col |a(row,col)|.
template <typename Matrix>
Fractional ComputeInfinityNormTemplate(const Matrix& matrix) {
DenseColumn row_sum(matrix.num_rows(), 0.0);
const ColIndex num_cols(matrix.num_cols());
for (ColIndex col(0); col < num_cols; ++col) {
for (const SparseColumn::Entry e : matrix.column(col)) {
// Compute sum_j |a_ij|.
row_sum[e.row()] += fabs(e.coefficient());
}
}
// Compute max_i sum_j |a_ij|
Fractional norm = 0.0;
const RowIndex num_rows(matrix.num_rows());
for (RowIndex row(0); row < num_rows; ++row) {
norm = std::max(norm, row_sum[row]);
}
return norm;
}
} // namespace
// --------------------------------------------------------
// SparseMatrix
// --------------------------------------------------------
SparseMatrix::SparseMatrix() : columns_(), num_rows_(0) {}
#if (!defined(_MSC_VER) || (_MSC_VER >= 1800))
SparseMatrix::SparseMatrix(
std::initializer_list<std::initializer_list<Fractional>> init_list) {
ColIndex num_cols(0);
num_rows_ = RowIndex(init_list.size());
RowIndex row(0);
for (std::initializer_list<Fractional> init_row : init_list) {
num_cols = std::max(num_cols, ColIndex(init_row.size()));
columns_.resize(num_cols, SparseColumn());
ColIndex col(0);
for (Fractional value : init_row) {
if (value != 0.0) {
columns_[col].SetCoefficient(row, value);
}
++col;
}
++row;
}
}
#endif
void SparseMatrix::Clear() {
columns_.clear();
num_rows_ = RowIndex(0);
}
bool SparseMatrix::IsEmpty() const {
return columns_.empty() || num_rows_ == 0;
}
void SparseMatrix::CleanUp() {
const ColIndex num_cols(columns_.size());
for (ColIndex col(0); col < num_cols; ++col) {
columns_[col].CleanUp();
}
}
bool SparseMatrix::CheckNoDuplicates() const {
DenseBooleanColumn boolean_column;
const ColIndex num_cols(columns_.size());
for (ColIndex col(0); col < num_cols; ++col) {
if (!columns_[col].CheckNoDuplicates(&boolean_column)) return false;
}
return true;
}
bool SparseMatrix::IsCleanedUp() const {
const ColIndex num_cols(columns_.size());
for (ColIndex col(0); col < num_cols; ++col) {
if (!columns_[col].IsCleanedUp()) return false;
}
return true;
}
void SparseMatrix::SetNumRows(RowIndex num_rows) { num_rows_ = num_rows; }
ColIndex SparseMatrix::AppendEmptyColumn() {
const ColIndex result = columns_.size();
columns_.push_back(SparseColumn());
return result;
}
void SparseMatrix::AppendUnitVector(RowIndex row, Fractional value) {
DCHECK_LT(row, num_rows_);
SparseColumn new_col;
new_col.SetCoefficient(row, value);
columns_.push_back(std::move(new_col));
}
void SparseMatrix::Swap(SparseMatrix* matrix) {
// We do not need to swap the different mutable scratchpads we use.
columns_.swap(matrix->columns_);
std::swap(num_rows_, matrix->num_rows_);
}
void SparseMatrix::PopulateFromZero(RowIndex num_rows, ColIndex num_cols) {
columns_.resize(num_cols, SparseColumn());
for (ColIndex col(0); col < num_cols; ++col) {
columns_[col].Clear();
}
num_rows_ = num_rows;
}
void SparseMatrix::PopulateFromIdentity(ColIndex num_cols) {
PopulateFromZero(ColToRowIndex(num_cols), num_cols);
for (ColIndex col(0); col < num_cols; ++col) {
const RowIndex row = ColToRowIndex(col);
columns_[col].SetCoefficient(row, Fractional(1.0));
}
}
template <typename Matrix>
void SparseMatrix::PopulateFromTranspose(const Matrix& input) {
Reset(RowToColIndex(input.num_rows()), ColToRowIndex(input.num_cols()));
// We do a first pass on the input matrix to resize the new columns properly.
StrictITIVector<RowIndex, EntryIndex> row_degree(input.num_rows(),
EntryIndex(0));
for (ColIndex col(0); col < input.num_cols(); ++col) {
for (const SparseColumn::Entry e : input.column(col)) {
++row_degree[e.row()];
}
}
for (RowIndex row(0); row < input.num_rows(); ++row) {
columns_[RowToColIndex(row)].Reserve(row_degree[row]);
}
for (ColIndex col(0); col < input.num_cols(); ++col) {
const RowIndex transposed_row = ColToRowIndex(col);
for (const SparseColumn::Entry e : input.column(col)) {
const ColIndex transposed_col = RowToColIndex(e.row());
columns_[transposed_col].SetCoefficient(transposed_row, e.coefficient());
}
}
DCHECK(IsCleanedUp());
}
void SparseMatrix::PopulateFromSparseMatrix(const SparseMatrix& matrix) {
Reset(ColIndex(0), matrix.num_rows_);
columns_ = matrix.columns_;
}
template <typename Matrix>
void SparseMatrix::PopulateFromPermutedMatrix(
const Matrix& a, const RowPermutation& row_perm,
const ColumnPermutation& inverse_col_perm) {
const ColIndex num_cols = a.num_cols();
Reset(num_cols, a.num_rows());
for (ColIndex col(0); col < num_cols; ++col) {
for (const auto e : a.column(inverse_col_perm[col])) {
columns_[col].SetCoefficient(row_perm[e.row()], e.coefficient());
}
}
DCHECK(CheckNoDuplicates());
}
void SparseMatrix::PopulateFromLinearCombination(Fractional alpha,
const SparseMatrix& a,
Fractional beta,
const SparseMatrix& b) {
DCHECK_EQ(a.num_cols(), b.num_cols());
DCHECK_EQ(a.num_rows(), b.num_rows());
const ColIndex num_cols = a.num_cols();
Reset(num_cols, a.num_rows());
const RowIndex num_rows = a.num_rows();
RandomAccessSparseColumn dense_column(num_rows);
for (ColIndex col(0); col < num_cols; ++col) {
for (const SparseColumn::Entry e : a.columns_[col]) {
dense_column.AddToCoefficient(e.row(), alpha * e.coefficient());
}
for (const SparseColumn::Entry e : b.columns_[col]) {
dense_column.AddToCoefficient(e.row(), beta * e.coefficient());
}
dense_column.PopulateSparseColumn(&columns_[col]);
columns_[col].CleanUp();
dense_column.Clear();
}
}
void SparseMatrix::PopulateFromProduct(const SparseMatrix& a,
const SparseMatrix& b) {
const ColIndex num_cols = b.num_cols();
const RowIndex num_rows = a.num_rows();
Reset(num_cols, num_rows);
RandomAccessSparseColumn tmp_column(num_rows);
for (ColIndex col_b(0); col_b < num_cols; ++col_b) {
for (const SparseColumn::Entry eb : b.columns_[col_b]) {
if (eb.coefficient() == 0.0) {
continue;
}
const ColIndex col_a = RowToColIndex(eb.row());
for (const SparseColumn::Entry ea : a.columns_[col_a]) {
const Fractional value = ea.coefficient() * eb.coefficient();
tmp_column.AddToCoefficient(ea.row(), value);
}
}
// Populate column col_b.
tmp_column.PopulateSparseColumn(&columns_[col_b]);
columns_[col_b].CleanUp();
tmp_column.Clear();
}
}
void SparseMatrix::DeleteColumns(const DenseBooleanRow& columns_to_delete) {
if (columns_to_delete.empty()) return;
ColIndex new_index(0);
const ColIndex num_cols = columns_.size();
for (ColIndex col(0); col < num_cols; ++col) {
if (col >= columns_to_delete.size() || !columns_to_delete[col]) {
columns_[col].Swap(&(columns_[new_index]));
++new_index;
}
}
columns_.resize(new_index);
}
void SparseMatrix::DeleteRows(RowIndex new_num_rows,
const RowPermutation& permutation) {
DCHECK_EQ(num_rows_, permutation.size());
for (RowIndex row(0); row < num_rows_; ++row) {
DCHECK_LT(permutation[row], new_num_rows);
}
const ColIndex end = num_cols();
for (ColIndex col(0); col < end; ++col) {
columns_[col].ApplyPartialRowPermutation(permutation);
}
SetNumRows(new_num_rows);
}
bool SparseMatrix::AppendRowsFromSparseMatrix(const SparseMatrix& matrix) {
const ColIndex end = num_cols();
if (end != matrix.num_cols()) {
return false;
}
const RowIndex offset = num_rows();
for (ColIndex col(0); col < end; ++col) {
const SparseColumn& source_column = matrix.columns_[col];
columns_[col].AppendEntriesWithOffset(source_column, offset);
}
SetNumRows(offset + matrix.num_rows());
return true;
}
void SparseMatrix::ApplyRowPermutation(const RowPermutation& row_perm) {
const ColIndex num_cols(columns_.size());
for (ColIndex col(0); col < num_cols; ++col) {
columns_[col].ApplyRowPermutation(row_perm);
}
}
Fractional SparseMatrix::LookUpValue(RowIndex row, ColIndex col) const {
return columns_[col].LookUpCoefficient(row);
}
bool SparseMatrix::Equals(const SparseMatrix& a, Fractional tolerance) const {
if (num_cols() != a.num_cols() || num_rows() != a.num_rows()) {
return false;
}
RandomAccessSparseColumn dense_column(num_rows());
RandomAccessSparseColumn dense_column_a(num_rows());
const ColIndex num_cols = a.num_cols();
for (ColIndex col(0); col < num_cols; ++col) {
// Store all entries of current matrix in a dense column.
for (const SparseColumn::Entry e : columns_[col]) {
dense_column.AddToCoefficient(e.row(), e.coefficient());
}
// Check all entries of a are those stored in the dense column.
for (const SparseColumn::Entry e : a.columns_[col]) {
if (fabs(e.coefficient() - dense_column.GetCoefficient(e.row())) >
tolerance) {
return false;
}
}
// Store all entries of matrix a in a dense column.
for (const SparseColumn::Entry e : a.columns_[col]) {
dense_column_a.AddToCoefficient(e.row(), e.coefficient());
}
// Check all entries are those stored in the dense column a.
for (const SparseColumn::Entry e : columns_[col]) {
if (fabs(e.coefficient() - dense_column_a.GetCoefficient(e.row())) >
tolerance) {
return false;
}
}
dense_column.Clear();
dense_column_a.Clear();
}
return true;
}
void SparseMatrix::ComputeMinAndMaxMagnitudes(Fractional* min_magnitude,
Fractional* max_magnitude) const {
RETURN_IF_NULL(min_magnitude);
RETURN_IF_NULL(max_magnitude);
*min_magnitude = kInfinity;
*max_magnitude = 0.0;
for (ColIndex col(0); col < num_cols(); ++col) {
for (const SparseColumn::Entry e : columns_[col]) {
const Fractional magnitude = fabs(e.coefficient());
if (magnitude != 0.0) {
*min_magnitude = std::min(*min_magnitude, magnitude);
*max_magnitude = std::max(*max_magnitude, magnitude);
}
}
}
if (*max_magnitude == 0.0) {
*min_magnitude = 0.0;
}
}
EntryIndex SparseMatrix::num_entries() const {
return ComputeNumEntries(*this);
}
Fractional SparseMatrix::ComputeOneNorm() const {
return ComputeOneNormTemplate(*this);
}
Fractional SparseMatrix::ComputeInfinityNorm() const {
return ComputeInfinityNormTemplate(*this);
}
std::string SparseMatrix::Dump() const {
std::string result;
const ColIndex num_cols(columns_.size());
for (RowIndex row(0); row < num_rows_; ++row) {
result.append("{ ");
for (ColIndex col(0); col < num_cols; ++col) {
absl::StrAppendFormat(&result, "%g ", ToDouble(LookUpValue(row, col)));
}
result.append("}\n");
}
return result;
}
void SparseMatrix::Reset(ColIndex num_cols, RowIndex num_rows) {
Clear();
columns_.resize(num_cols, SparseColumn());
num_rows_ = num_rows;
}
EntryIndex MatrixView::num_entries() const { return ComputeNumEntries(*this); }
Fractional MatrixView::ComputeOneNorm() const {
return ComputeOneNormTemplate(*this);
}
Fractional MatrixView::ComputeInfinityNorm() const {
return ComputeInfinityNormTemplate(*this);
}
// Instantiate needed templates.
template void SparseMatrix::PopulateFromTranspose<SparseMatrix>(
const SparseMatrix& input);
template void SparseMatrix::PopulateFromPermutedMatrix<SparseMatrix>(
const SparseMatrix& a, const RowPermutation& row_perm,
const ColumnPermutation& inverse_col_perm);
template void SparseMatrix::PopulateFromPermutedMatrix<CompactSparseMatrixView>(
const CompactSparseMatrixView& a, const RowPermutation& row_perm,
const ColumnPermutation& inverse_col_perm);
void CompactSparseMatrix::PopulateFromMatrixView(const MatrixView& input) {
num_cols_ = input.num_cols();
num_rows_ = input.num_rows();
const EntryIndex num_entries = input.num_entries();
starts_.assign(num_cols_ + 1, EntryIndex(0));
coefficients_.assign(num_entries, 0.0);
rows_.assign(num_entries, RowIndex(0));
EntryIndex index(0);
for (ColIndex col(0); col < input.num_cols(); ++col) {
starts_[col] = index;
for (const SparseColumn::Entry e : input.column(col)) {
coefficients_[index] = e.coefficient();
rows_[index] = e.row();
++index;
}
}
starts_[input.num_cols()] = index;
}
void CompactSparseMatrix::PopulateFromSparseMatrixAndAddSlacks(
const SparseMatrix& input) {
const int input_num_cols = input.num_cols().value();
num_cols_ = input_num_cols + RowToColIndex(input.num_rows());
num_rows_ = input.num_rows();
const EntryIndex num_entries =
input.num_entries() + EntryIndex(num_rows_.value());
starts_.assign(num_cols_ + 1, EntryIndex(0));
coefficients_.resize(num_entries, 0.0);
rows_.resize(num_entries, RowIndex(0));
EntryIndex index(0);
for (ColIndex col(0); col < input_num_cols; ++col) {
starts_[col] = index;
for (const SparseColumn::Entry e : input.column(col)) {
coefficients_[index] = e.coefficient();
rows_[index] = e.row();
++index;
}
}
for (RowIndex row(0); row < num_rows_; ++row) {
starts_[input_num_cols + RowToColIndex(row)] = index;
coefficients_[index] = 1.0;
rows_[index] = row;
++index;
}
DCHECK_EQ(index, num_entries);
starts_[num_cols_] = index;
}
void CompactSparseMatrix::PopulateFromTranspose(
const CompactSparseMatrix& input) {
num_cols_ = RowToColIndex(input.num_rows());
num_rows_ = ColToRowIndex(input.num_cols());
// Fill the starts_ vector by computing the number of entries of each rows and
// then doing a cumulative sum. After this step starts_[col + 1] will be the
// actual start of the column col when we are done.
const ColIndex start_size = num_cols_ + 2;
starts_.assign(start_size, EntryIndex(0));
for (const RowIndex row : input.rows_) {
++starts_[RowToColIndex(row) + 2];
}
for (ColIndex col(2); col < start_size; ++col) {
starts_[col] += starts_[col - 1];
}
coefficients_.resize(starts_.back(), 0.0);
rows_.resize(starts_.back(), kInvalidRow);
starts_.pop_back();
// Use starts_ to fill the matrix. Note that starts_ is modified so that at
// the end it has its final values.
const auto entry_rows = rows_.view();
const auto input_entry_rows = input.rows_.view();
const auto entry_coefficients = coefficients_.view();
const auto input_entry_coefficients = input.coefficients_.view();
const auto num_cols = input.num_cols();
const auto starts = starts_.view();
for (ColIndex col(0); col < num_cols; ++col) {
const RowIndex transposed_row = ColToRowIndex(col);
for (const EntryIndex i : input.Column(col)) {
const ColIndex transposed_col = RowToColIndex(input_entry_rows[i]);
const EntryIndex index = starts[transposed_col + 1]++;
entry_coefficients[index] = input_entry_coefficients[i];
entry_rows[index] = transposed_row;
}
}
DCHECK_EQ(starts_.front(), 0);
DCHECK_EQ(starts_.back(), rows_.size());
}
void TriangularMatrix::PopulateFromTranspose(const TriangularMatrix& input) {
CompactSparseMatrix::PopulateFromTranspose(input);
// This takes care of the triangular special case.
diagonal_coefficients_ = input.diagonal_coefficients_;
all_diagonal_coefficients_are_one_ = input.all_diagonal_coefficients_are_one_;
// The elimination structure of the transpose is not the same.
pruned_ends_.resize(num_cols_, EntryIndex(0));
for (ColIndex col(0); col < num_cols_; ++col) {
pruned_ends_[col] = starts_[col + 1];
}
// Compute first_non_identity_column_. Note that this is not necessarily the
// same as input.first_non_identity_column_ for an upper triangular matrix.
first_non_identity_column_ = 0;
const ColIndex end = diagonal_coefficients_.size();
while (first_non_identity_column_ < end &&
ColumnNumEntries(first_non_identity_column_) == 0 &&
diagonal_coefficients_[first_non_identity_column_] == 1.0) {
++first_non_identity_column_;
}
}
void CompactSparseMatrix::Reset(RowIndex num_rows) {
num_rows_ = num_rows;
num_cols_ = 0;
rows_.clear();
coefficients_.clear();
starts_.clear();
starts_.push_back(EntryIndex(0));
}
void TriangularMatrix::Reset(RowIndex num_rows, ColIndex col_capacity) {
CompactSparseMatrix::Reset(num_rows);
first_non_identity_column_ = 0;
all_diagonal_coefficients_are_one_ = true;
pruned_ends_.resize(col_capacity);
diagonal_coefficients_.resize(col_capacity);
starts_.resize(col_capacity + 1);
// Non-zero entries in the first column always have an offset of 0.
starts_[ColIndex(0)] = 0;
}
void CompactSparseMatrix::AddEntryToCurrentColumn(RowIndex row,
Fractional coeff) {
rows_.push_back(row);
coefficients_.push_back(coeff);
}
void CompactSparseMatrix::CloseCurrentColumn() {
starts_.push_back(rows_.size());
++num_cols_;
}
ColIndex CompactSparseMatrix::AddDenseColumn(const DenseColumn& dense_column) {
return AddDenseColumnPrefix(dense_column.const_view(), RowIndex(0));
}
ColIndex CompactSparseMatrix::AddDenseColumnPrefix(
DenseColumn::ConstView dense_column, RowIndex start) {
const RowIndex num_rows(dense_column.size());
for (RowIndex row(start); row < num_rows; ++row) {
if (dense_column[row] != 0.0) {
rows_.push_back(row);
coefficients_.push_back(dense_column[row]);
}
}
starts_.push_back(rows_.size());
++num_cols_;
return num_cols_ - 1;
}
ColIndex CompactSparseMatrix::AddDenseColumnWithNonZeros(
const DenseColumn& dense_column, absl::Span<const RowIndex> non_zeros) {
if (non_zeros.empty()) return AddDenseColumn(dense_column);
for (const RowIndex row : non_zeros) {
const Fractional value = dense_column[row];
if (value != 0.0) {
rows_.push_back(row);
coefficients_.push_back(value);
}
}
starts_.push_back(rows_.size());
++num_cols_;
return num_cols_ - 1;
}
ColIndex CompactSparseMatrix::AddAndClearColumnWithNonZeros(
DenseColumn* column, std::vector<RowIndex>* non_zeros) {
for (const RowIndex row : *non_zeros) {
const Fractional value = (*column)[row];
if (value != 0.0) {
rows_.push_back(row);
coefficients_.push_back(value);
(*column)[row] = 0.0;
}
}
non_zeros->clear();
starts_.push_back(rows_.size());
++num_cols_;
return num_cols_ - 1;
}
void CompactSparseMatrix::Swap(CompactSparseMatrix* other) {
std::swap(num_rows_, other->num_rows_);
std::swap(num_cols_, other->num_cols_);
coefficients_.swap(other->coefficients_);
rows_.swap(other->rows_);
starts_.swap(other->starts_);
}
void TriangularMatrix::Swap(TriangularMatrix* other) {
CompactSparseMatrix::Swap(other);
diagonal_coefficients_.swap(other->diagonal_coefficients_);
std::swap(first_non_identity_column_, other->first_non_identity_column_);
std::swap(all_diagonal_coefficients_are_one_,
other->all_diagonal_coefficients_are_one_);
}
EntryIndex CompactSparseMatrixView::num_entries() const {
return ComputeNumEntries(*this);
}
Fractional CompactSparseMatrixView::ComputeOneNorm() const {
return ComputeOneNormTemplate(*this);
}
Fractional CompactSparseMatrixView::ComputeInfinityNorm() const {
return ComputeInfinityNormTemplate(*this);
}
// Internal function used to finish adding one column to a triangular matrix.
// This sets the diagonal coefficient to the given value, and prepares the
// matrix for the next column addition.
void TriangularMatrix::CloseCurrentColumn(Fractional diagonal_value) {
DCHECK_NE(diagonal_value, 0.0);
// The vectors diagonal_coefficients, pruned_ends, and starts_ should have all
// been preallocated by a call to SetTotalNumberOfColumns().
DCHECK_LT(num_cols_, diagonal_coefficients_.size());
diagonal_coefficients_[num_cols_] = diagonal_value;
// TODO(user): This is currently not used by all matrices. It will be good
// to fill it only when needed.
DCHECK_LT(num_cols_, pruned_ends_.size());
const EntryIndex num_entries = coefficients_.size();
pruned_ends_[num_cols_] = num_entries;
++num_cols_;
DCHECK_LT(num_cols_, starts_.size());
starts_[num_cols_] = num_entries;
if (first_non_identity_column_ == num_cols_ - 1 && diagonal_value == 1.0 &&
num_entries == 0) {
first_non_identity_column_ = num_cols_;
}
all_diagonal_coefficients_are_one_ =
all_diagonal_coefficients_are_one_ && (diagonal_value == 1.0);
}
void TriangularMatrix::AddDiagonalOnlyColumn(Fractional diagonal_value) {
CloseCurrentColumn(diagonal_value);
}
void TriangularMatrix::AddTriangularColumn(const ColumnView& column,
RowIndex diagonal_row) {
Fractional diagonal_value = 0.0;
for (const SparseColumn::Entry e : column) {
if (e.row() == diagonal_row) {
diagonal_value = e.coefficient();
} else {
DCHECK_NE(0.0, e.coefficient());
rows_.push_back(e.row());
coefficients_.push_back(e.coefficient());
}
}
CloseCurrentColumn(diagonal_value);
}
void TriangularMatrix::AddAndNormalizeTriangularColumn(
const SparseColumn& column, RowIndex diagonal_row,
Fractional diagonal_coefficient) {
// TODO(user): use division by a constant using multiplication.
for (const SparseColumn::Entry e : column) {
if (e.row() != diagonal_row) {
if (e.coefficient() != 0.0) {
rows_.push_back(e.row());
coefficients_.push_back(e.coefficient() / diagonal_coefficient);
}
} else {
DCHECK_EQ(e.coefficient(), diagonal_coefficient);
}
}
CloseCurrentColumn(1.0);
}
void TriangularMatrix::AddTriangularColumnWithGivenDiagonalEntry(
const SparseColumn& column, RowIndex diagonal_row,
Fractional diagonal_value) {
for (SparseColumn::Entry e : column) {
DCHECK_NE(e.row(), diagonal_row);
rows_.push_back(e.row());
coefficients_.push_back(e.coefficient());
}
CloseCurrentColumn(diagonal_value);
}
void TriangularMatrix::PopulateFromTriangularSparseMatrix(
const SparseMatrix& input) {
Reset(input.num_rows(), input.num_cols());
for (ColIndex col(0); col < input.num_cols(); ++col) {
AddTriangularColumn(ColumnView(input.column(col)), ColToRowIndex(col));
}
DCHECK(IsLowerTriangular() || IsUpperTriangular());
}
bool TriangularMatrix::IsLowerTriangular() const {
for (ColIndex col(0); col < num_cols_; ++col) {
if (diagonal_coefficients_[col] == 0.0) return false;
for (EntryIndex i : Column(col)) {
if (rows_[i] <= ColToRowIndex(col)) return false;
}
}
return true;
}
bool TriangularMatrix::IsUpperTriangular() const {
for (ColIndex col(0); col < num_cols_; ++col) {
if (diagonal_coefficients_[col] == 0.0) return false;
for (EntryIndex i : Column(col)) {
if (rows_[i] >= ColToRowIndex(col)) return false;
}
}
return true;
}
void TriangularMatrix::ApplyRowPermutationToNonDiagonalEntries(
const RowPermutation& row_perm) {
EntryIndex num_entries = rows_.size();
for (EntryIndex i(0); i < num_entries; ++i) {
rows_[i] = row_perm[rows_[i]];
}
}
void TriangularMatrix::CopyColumnToSparseColumn(ColIndex col,
SparseColumn* output) const {
output->Clear();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (const EntryIndex i : Column(col)) {
output->SetCoefficient(entry_rows[i], entry_coefficients[i]);
}
output->SetCoefficient(ColToRowIndex(col), diagonal_coefficients_[col]);
output->CleanUp();
}
void TriangularMatrix::CopyToSparseMatrix(SparseMatrix* output) const {
output->PopulateFromZero(num_rows_, num_cols_);
for (ColIndex col(0); col < num_cols_; ++col) {
CopyColumnToSparseColumn(col, output->mutable_column(col));
}
}
void TriangularMatrix::LowerSolve(DenseColumn* rhs) const {
LowerSolveStartingAt(ColIndex(0), rhs);
}
void TriangularMatrix::LowerSolveStartingAt(ColIndex start,
DenseColumn* rhs) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
LowerSolveStartingAtInternal<true>(start, rhs->view());
} else {
LowerSolveStartingAtInternal<false>(start, rhs->view());
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::LowerSolveStartingAtInternal(
ColIndex start, DenseColumn::View rhs) const {
const ColIndex begin = std::max(start, first_non_identity_column_);
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
const auto diagonal_coefficients = diagonal_coefficients_.view();
const ColIndex end = diagonal_coefficients.size();
for (ColIndex col(begin); col < end; ++col) {
const Fractional value = rhs[ColToRowIndex(col)];
if (value == 0.0) continue;
const Fractional coeff =
diagonal_of_ones ? value : value / diagonal_coefficients[col];
if (!diagonal_of_ones) {
rhs[ColToRowIndex(col)] = coeff;
}
for (const EntryIndex i : Column(col)) {
rhs[entry_rows[i]] -= coeff * entry_coefficients[i];
}
}
}
void TriangularMatrix::UpperSolve(DenseColumn* rhs) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
UpperSolveInternal<true>(rhs->view());
} else {
UpperSolveInternal<false>(rhs->view());
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::UpperSolveInternal(DenseColumn::View rhs) const {
const ColIndex end = first_non_identity_column_;
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
const auto diagonal_coefficients = diagonal_coefficients_.view();
const auto starts = starts_.view();
for (ColIndex col(diagonal_coefficients.size() - 1); col >= end; --col) {
const Fractional value = rhs[ColToRowIndex(col)];
if (value == 0.0) continue;
const Fractional coeff =
diagonal_of_ones ? value : value / diagonal_coefficients[col];
if (!diagonal_of_ones) {
rhs[ColToRowIndex(col)] = coeff;
}
// It is faster to iterate this way (instead of i : Column(col)) because of
// cache locality. Note that the floating-point computations are exactly the
// same in both cases.
const EntryIndex i_end = starts[col];
for (EntryIndex i(starts[col + 1] - 1); i >= i_end; --i) {
rhs[entry_rows[i]] -= coeff * entry_coefficients[i];
}
}
}
void TriangularMatrix::TransposeUpperSolve(DenseColumn* rhs) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
TransposeUpperSolveInternal<true>(rhs->view());
} else {
TransposeUpperSolveInternal<false>(rhs->view());
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::TransposeUpperSolveInternal(
DenseColumn::View rhs) const {
const ColIndex end = num_cols_;
const auto starts = starts_.view();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
const auto diagonal_coefficients = diagonal_coefficients_.view();
EntryIndex i = starts_[first_non_identity_column_];
for (ColIndex col(first_non_identity_column_); col < end; ++col) {
Fractional sum = rhs[ColToRowIndex(col)];
// Note that this is a bit faster than the simpler
// for (const EntryIndex i : Column(col)) {
// EntryIndex i is explicitly not modified in outer iterations, since
// the last entry in column col is stored contiguously just before the
// first entry in column col+1.
const EntryIndex i_end = starts[col + 1];
const EntryIndex shifted_end = i_end - 3;
for (; i < shifted_end; i += 4) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]] +
entry_coefficients[i + 1] * rhs[entry_rows[i + 1]] +
entry_coefficients[i + 2] * rhs[entry_rows[i + 2]] +
entry_coefficients[i + 3] * rhs[entry_rows[i + 3]];
}
if (i < i_end) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]];
if (i + 1 < i_end) {
sum -= entry_coefficients[i + 1] * rhs[entry_rows[i + 1]];
if (i + 2 < i_end) {
sum -= entry_coefficients[i + 2] * rhs[entry_rows[i + 2]];
}
}
i = i_end;
}
rhs[ColToRowIndex(col)] =
diagonal_of_ones ? sum : sum / diagonal_coefficients[col];
}
}
void TriangularMatrix::TransposeLowerSolve(DenseColumn* rhs) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
TransposeLowerSolveInternal<true>(rhs->view());
} else {
TransposeLowerSolveInternal<false>(rhs->view());
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::TransposeLowerSolveInternal(
DenseColumn::View rhs) const {
const ColIndex end = first_non_identity_column_;
// We optimize a bit the solve by skipping the last 0.0 positions.
ColIndex col = num_cols_ - 1;
while (col >= end && rhs[ColToRowIndex(col)] == 0.0) {
--col;
}
const auto starts = starts_.view();
const auto diagonal_coeffs = diagonal_coefficients_.view();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
EntryIndex i = starts[col + 1] - 1;
for (; col >= end; --col) {
Fractional sum = rhs[ColToRowIndex(col)];
// Note that this is a bit faster than the simpler
// for (const EntryIndex i : Column(col)) {
// mainly because we iterate in a good direction for the cache.
// EntryIndex i is explicitly not modified in outer iterations, since
// the last entry in column col is stored contiguously just before the
// first entry in column col+1.
const EntryIndex i_end = starts[col];
const EntryIndex shifted_end = i_end + 3;
for (; i >= shifted_end; i -= 4) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]] +
entry_coefficients[i - 1] * rhs[entry_rows[i - 1]] +
entry_coefficients[i - 2] * rhs[entry_rows[i - 2]] +
entry_coefficients[i - 3] * rhs[entry_rows[i - 3]];
}
if (i >= i_end) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]];
if (i >= i_end + 1) {
sum -= entry_coefficients[i - 1] * rhs[entry_rows[i - 1]];
if (i >= i_end + 2) {
sum -= entry_coefficients[i - 2] * rhs[entry_rows[i - 2]];
}
}
i = i_end - 1;
}
rhs[ColToRowIndex(col)] =
diagonal_of_ones ? sum : sum / diagonal_coeffs[col];
}
}
void TriangularMatrix::HyperSparseSolve(DenseColumn* rhs,
RowIndexVector* non_zero_rows) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
HyperSparseSolveInternal<true>(rhs->view(), non_zero_rows);
} else {
HyperSparseSolveInternal<false>(rhs->view(), non_zero_rows);
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::HyperSparseSolveInternal(
DenseColumn::View rhs, RowIndexVector* non_zero_rows) const {
int new_size = 0;
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (const RowIndex row : *non_zero_rows) {
if (rhs[row] == 0.0) continue;
const ColIndex row_as_col = RowToColIndex(row);
const Fractional coeff =
diagonal_of_ones ? rhs[row]
: rhs[row] / diagonal_coefficients_[row_as_col];
rhs[row] = coeff;
for (const EntryIndex i : Column(row_as_col)) {
rhs[entry_rows[i]] -= coeff * entry_coefficients[i];
}
(*non_zero_rows)[new_size] = row;
++new_size;
}
non_zero_rows->resize(new_size);
}
void TriangularMatrix::HyperSparseSolveWithReversedNonZeros(
DenseColumn* rhs, RowIndexVector* non_zero_rows) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
HyperSparseSolveWithReversedNonZerosInternal<true>(rhs->view(),
non_zero_rows);
} else {
HyperSparseSolveWithReversedNonZerosInternal<false>(rhs->view(),
non_zero_rows);
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::HyperSparseSolveWithReversedNonZerosInternal(
DenseColumn::View rhs, RowIndexVector* non_zero_rows) const {
int new_start = non_zero_rows->size();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (const RowIndex row : Reverse(*non_zero_rows)) {
if (rhs[row] == 0.0) continue;
const ColIndex row_as_col = RowToColIndex(row);
const Fractional coeff =
diagonal_of_ones ? rhs[row]
: rhs[row] / diagonal_coefficients_[row_as_col];
rhs[row] = coeff;
for (const EntryIndex i : Column(row_as_col)) {
rhs[entry_rows[i]] -= coeff * entry_coefficients[i];
}
--new_start;
(*non_zero_rows)[new_start] = row;
}
non_zero_rows->erase(non_zero_rows->begin(),
non_zero_rows->begin() + new_start);
}
void TriangularMatrix::TransposeHyperSparseSolve(
DenseColumn* rhs, RowIndexVector* non_zero_rows) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
TransposeHyperSparseSolveInternal<true>(rhs->view(), non_zero_rows);
} else {
TransposeHyperSparseSolveInternal<false>(rhs->view(), non_zero_rows);
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::TransposeHyperSparseSolveInternal(
DenseColumn::View rhs, RowIndexVector* non_zero_rows) const {
int new_size = 0;
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (const RowIndex row : *non_zero_rows) {
Fractional sum = rhs[row];
const ColIndex row_as_col = RowToColIndex(row);
// Note that we do the loop in exactly the same way as
// in TransposeUpperSolveInternal().
EntryIndex i = starts_[row_as_col];
const EntryIndex i_end = starts_[row_as_col + 1];
const EntryIndex shifted_end = i_end - 3;
for (; i < shifted_end; i += 4) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]] +
entry_coefficients[i + 1] * rhs[entry_rows[i + 1]] +
entry_coefficients[i + 2] * rhs[entry_rows[i + 2]] +
entry_coefficients[i + 3] * rhs[entry_rows[i + 3]];
}
if (i < i_end) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]];
if (i + 1 < i_end) {
sum -= entry_coefficients[i + 1] * rhs[entry_rows[i + 1]];
if (i + 2 < i_end) {
sum -= entry_coefficients[i + 2] * rhs[entry_rows[i + 2]];
}
}
}
rhs[row] =
diagonal_of_ones ? sum : sum / diagonal_coefficients_[row_as_col];
if (sum != 0.0) {
(*non_zero_rows)[new_size] = row;
++new_size;
}
}
non_zero_rows->resize(new_size);
}
void TriangularMatrix::TransposeHyperSparseSolveWithReversedNonZeros(
DenseColumn* rhs, RowIndexVector* non_zero_rows) const {
RETURN_IF_NULL(rhs);
if (all_diagonal_coefficients_are_one_) {
TransposeHyperSparseSolveWithReversedNonZerosInternal<true>(rhs->view(),
non_zero_rows);
} else {
TransposeHyperSparseSolveWithReversedNonZerosInternal<false>(rhs->view(),
non_zero_rows);
}
}
template <bool diagonal_of_ones>
void TriangularMatrix::TransposeHyperSparseSolveWithReversedNonZerosInternal(
DenseColumn::View rhs, RowIndexVector* non_zero_rows) const {
int new_start = non_zero_rows->size();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (const RowIndex row : Reverse(*non_zero_rows)) {
Fractional sum = rhs[row];
const ColIndex row_as_col = RowToColIndex(row);
// We do the loop this way so that the floating point operations are exactly
// the same as the ones performed by TransposeLowerSolveInternal().
EntryIndex i = starts_[row_as_col + 1] - 1;
const EntryIndex i_end = starts_[row_as_col];
const EntryIndex shifted_end = i_end + 3;
for (; i >= shifted_end; i -= 4) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]] +
entry_coefficients[i - 1] * rhs[entry_rows[i - 1]] +
entry_coefficients[i - 2] * rhs[entry_rows[i - 2]] +
entry_coefficients[i - 3] * rhs[entry_rows[i - 3]];
}
if (i >= i_end) {
sum -= entry_coefficients[i] * rhs[entry_rows[i]];
if (i >= i_end + 1) {
sum -= entry_coefficients[i - 1] * rhs[entry_rows[i - 1]];
if (i >= i_end + 2) {
sum -= entry_coefficients[i - 2] * rhs[entry_rows[i - 2]];
}
}
}
rhs[row] =
diagonal_of_ones ? sum : sum / diagonal_coefficients_[row_as_col];
if (sum != 0.0) {
--new_start;
(*non_zero_rows)[new_start] = row;
}
}
non_zero_rows->erase(non_zero_rows->begin(),
non_zero_rows->begin() + new_start);
}
void TriangularMatrix::PermutedLowerSolve(
const SparseColumn& rhs, const RowPermutation& row_perm,
const RowMapping& partial_inverse_row_perm, SparseColumn* lower,
SparseColumn* upper) const {
DCHECK(all_diagonal_coefficients_are_one_);
RETURN_IF_NULL(lower);
RETURN_IF_NULL(upper);
initially_all_zero_scratchpad_.resize(num_rows_, 0.0);
for (const SparseColumn::Entry e : rhs) {
initially_all_zero_scratchpad_[e.row()] = e.coefficient();
}
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
const RowIndex end_row(partial_inverse_row_perm.size());
for (RowIndex row(ColToRowIndex(first_non_identity_column_)); row < end_row;
++row) {
const RowIndex permuted_row = partial_inverse_row_perm[row];
const Fractional pivot = initially_all_zero_scratchpad_[permuted_row];
if (pivot == 0.0) continue;
for (EntryIndex i : Column(RowToColIndex(row))) {
initially_all_zero_scratchpad_[entry_rows[i]] -=
entry_coefficients[i] * pivot;
}
}
lower->Clear();
const RowIndex num_rows = num_rows_;
for (RowIndex row(0); row < num_rows; ++row) {
if (initially_all_zero_scratchpad_[row] != 0.0) {
if (row_perm[row] < 0) {
lower->SetCoefficient(row, initially_all_zero_scratchpad_[row]);
} else {
upper->SetCoefficient(row, initially_all_zero_scratchpad_[row]);
}
initially_all_zero_scratchpad_[row] = 0.0;
}
}
DCHECK(lower->CheckNoDuplicates());
}
void TriangularMatrix::PermutedLowerSparseSolve(const ColumnView& rhs,
const RowPermutation& row_perm,
SparseColumn* lower_column,
SparseColumn* upper_column) {
DCHECK(all_diagonal_coefficients_are_one_);
RETURN_IF_NULL(lower_column);
RETURN_IF_NULL(upper_column);
// Compute the set of rows that will be non zero in the result (lower_column,
// upper_column).
PermutedComputeRowsToConsider(rhs, row_perm, &lower_column_rows_,
&upper_column_rows_);
// Copy rhs into initially_all_zero_scratchpad_.
initially_all_zero_scratchpad_.resize(num_rows_, 0.0);
for (const auto e : rhs) {
initially_all_zero_scratchpad_[e.row()] = e.coefficient();
}
// We clear lower_column first in case upper_column and lower_column point to
// the same underlying SparseColumn.
num_fp_operations_ = 0;
lower_column->Clear();
// rows_to_consider_ contains the row to process in reverse order. Note in
// particular that each "permuted_row" will never be touched again and so its
// value is final. We copy the result in (lower_column, upper_column) and
// clear initially_all_zero_scratchpad_ at the same time.
upper_column->Reserve(upper_column->num_entries() +
EntryIndex(upper_column_rows_.size()));
for (const RowIndex permuted_row : Reverse(upper_column_rows_)) {
const Fractional pivot = initially_all_zero_scratchpad_[permuted_row];
if (pivot == 0.0) continue;
// Note that permuted_row will not appear in the loop below so we
// already know the value of the solution at this position.
initially_all_zero_scratchpad_[permuted_row] = 0.0;
const ColIndex row_as_col = RowToColIndex(row_perm[permuted_row]);
DCHECK_GE(row_as_col, 0);
upper_column->SetCoefficient(permuted_row, pivot);
DCHECK_EQ(diagonal_coefficients_[row_as_col], 1.0);
num_fp_operations_ += 1 + ColumnNumEntries(row_as_col).value();
for (const auto e : column(row_as_col)) {
initially_all_zero_scratchpad_[e.row()] -= e.coefficient() * pivot;
}
}
// TODO(user): The size of lower is exact, so we could be slighly faster here.
lower_column->Reserve(EntryIndex(lower_column_rows_.size()));
for (const RowIndex permuted_row : lower_column_rows_) {
const Fractional pivot = initially_all_zero_scratchpad_[permuted_row];
initially_all_zero_scratchpad_[permuted_row] = 0.0;
lower_column->SetCoefficient(permuted_row, pivot);
}
DCHECK(lower_column->CheckNoDuplicates());
DCHECK(upper_column->CheckNoDuplicates());
}
// The goal is to find which rows of the working column we will need to look
// at in PermutedLowerSparseSolve() when solving P^{-1}.L.P.x = rhs, 'P' being a
// row permutation, 'L' a lower triangular matrix and 'this' being 'P^{-1}.L'.
// Note that the columns of L that are identity columns (this is the case for
// the ones corresponding to a kNonPivotal in P) can be skipped since they will
// leave the working column unchanged.
//
// Let G denote the graph G = (V,E) of the column-to-row adjacency of A:
// - 'V' is the set of nodes, one node i corresponds to a both a row
// and a column (the matrix is square).
// - 'E' is the set of arcs. There is an arc from node i to node j iff the
// coefficient of i-th column, j-th row of A = P^{-1}.L.P is non zero.
//
// Let S denote the set of nodes i such that rhs_i != 0.
// Let R denote the set of all accessible nodes from S in G.
// x_k is possibly non-zero iff k is in R, i.e. if k is not in R then x_k = 0
// for sure, and there is no need to look a the row k during the solve.
//
// So, to solve P^{-1}.L.P.x = rhs, only rows corresponding to P.R have to be
// considered (ignoring the one that map to identity column of L). A topological
// sort of P.R is used to decide in which order one should iterate on them. This
// will be given by upper_column_rows_ and it will be populated in reverse
// order.
void TriangularMatrix::PermutedComputeRowsToConsider(
const ColumnView& rhs, const RowPermutation& row_perm,
RowIndexVector* lower_column_rows, RowIndexVector* upper_column_rows) {
stored_.Resize(num_rows_);
marked_.resize(num_rows_, false);
lower_column_rows->clear();
upper_column_rows->clear();
nodes_to_explore_.clear();
for (SparseColumn::Entry e : rhs) {
const ColIndex col = RowToColIndex(row_perm[e.row()]);
if (col < 0) {
stored_.Set(e.row());
lower_column_rows->push_back(e.row());
} else {
nodes_to_explore_.push_back(e.row());
}
}
// Topological sort based on Depth-First-Search.
// A few notes:
// - By construction, if the matrix can be permuted into a lower triangular
// form, there is no cycle. This code does nothing to test for cycles, but
// there is a DCHECK() to detect them during debugging.
// - This version uses sentinels (kInvalidRow) on nodes_to_explore_ to know
// when a node has been explored (i.e. when the recursive dfs goes back in
// the call stack). This is faster than an alternate implementation that
// uses another Boolean array to detect when we go back in the
// depth-first search.
const auto entry_rows = rows_.view();
while (!nodes_to_explore_.empty()) {
const RowIndex row = nodes_to_explore_.back();
// If the depth-first search from the current node is finished (i.e. there
// is a sentinel on the stack), we store the node (which is just before on
// the stack). This will store the nodes in reverse topological order.
if (row < 0) {
nodes_to_explore_.pop_back();
const RowIndex explored_row = nodes_to_explore_.back();
nodes_to_explore_.pop_back();
DCHECK(!stored_[explored_row]);
stored_.Set(explored_row);
upper_column_rows->push_back(explored_row);
// Unmark and prune the nodes that are already unmarked. See the header
// comment on marked_ for the algorithm description.
//
// Complexity note: The only difference with the "normal" DFS doing no
// pruning is this extra loop here and the marked_[entry_row] = true in
// the loop later in this function. On an already pruned graph, this is
// probably between 1 and 2 times slower than the "normal" DFS.
const ColIndex col = RowToColIndex(row_perm[explored_row]);
EntryIndex i = starts_[col];
EntryIndex end = pruned_ends_[col];
while (i < end) {
const RowIndex entry_row = entry_rows[i];
if (!marked_[entry_row]) {
--end;
// Note that we could keep the pruned row in a separate vector and
// not touch the triangular matrix. But the current solution seems
// better cache-wise and memory-wise.
std::swap(rows_[i], rows_[end]);
std::swap(coefficients_[i], coefficients_[end]);
} else {
marked_[entry_row] = false;
++i;
}
}
pruned_ends_[col] = end;
continue;
}
// If the node is already stored, skip.
if (stored_[row]) {
nodes_to_explore_.pop_back();
continue;
}
// Expand only if we are not on a kNonPivotal row.
// Otherwise we can store the node right away.
const ColIndex col = RowToColIndex(row_perm[row]);
if (col < 0) {
stored_.Set(row);
lower_column_rows->push_back(row);
nodes_to_explore_.pop_back();
continue;
}
// Go one level forward in the depth-first search, and store the 'adjacent'
// node on nodes_to_explore_ for further processing.
nodes_to_explore_.push_back(kInvalidRow);
const EntryIndex end = pruned_ends_[col];
for (EntryIndex i = starts_[col]; i < end; ++i) {
const RowIndex entry_row = entry_rows[i];
if (!stored_[entry_row]) {
nodes_to_explore_.push_back(entry_row);
}
marked_[entry_row] = true;
}
// The graph contains cycles? this is not supposed to happen.
DCHECK_LE(nodes_to_explore_.size(), 2 * num_rows_.value() + rows_.size());
}
// Clear stored_.
for (const RowIndex row : *lower_column_rows) {
stored_.ClearBucket(row);
}
for (const RowIndex row : *upper_column_rows) {
stored_.ClearBucket(row);
}
}
void TriangularMatrix::ComputeRowsToConsiderWithDfs(
RowIndexVector* non_zero_rows) const {
if (non_zero_rows->empty()) return;
// We don't start the DFS if the initial number of non-zeros is under the
// sparsity_threshold. During the DFS, we abort it if the number of floating
// points operations get larger than the num_ops_threshold.
//
// In both cases, we make sure to clear non_zero_rows so that the solving part
// will use the non-hypersparse version of the code.
//
// TODO(user): Investigate the best thresholds.
const int sparsity_threshold =
static_cast<int>(0.025 * static_cast<double>(num_rows_.value()));
const int num_ops_threshold =
static_cast<int>(0.05 * static_cast<double>(num_rows_.value()));
int num_ops = non_zero_rows->size();
if (num_ops > sparsity_threshold) {
non_zero_rows->clear();
return;
}
// Initialize using the non-zero positions of the input.
stored_.Resize(num_rows_);
nodes_to_explore_.clear();
nodes_to_explore_.swap(*non_zero_rows);
// Topological sort based on Depth-First-Search.
// Same remarks as the version implemented in PermutedComputeRowsToConsider().
const auto entry_rows = rows_.view();
while (!nodes_to_explore_.empty()) {
const RowIndex row = nodes_to_explore_.back();
// If the depth-first search from the current node is finished, we store the
// node. This will store the node in reverse topological order.
if (row < 0) {
nodes_to_explore_.pop_back();
const RowIndex explored_row = -row - 1;
stored_.Set(explored_row);
non_zero_rows->push_back(explored_row);
continue;
}
// If the node is already stored, skip.
if (stored_[row]) {
nodes_to_explore_.pop_back();
continue;
}
// Go one level forward in the depth-first search, and store the 'adjacent'
// node on nodes_to_explore_ for further processing.
//
// We reverse the sign of nodes_to_explore_.back() to detect when the
// DFS will be back on this node.
nodes_to_explore_.back() = -row - 1;
for (const EntryIndex i : Column(RowToColIndex(row))) {
++num_ops;
const RowIndex entry_row = entry_rows[i];
if (!stored_[entry_row]) {
nodes_to_explore_.push_back(entry_row);
}
}
// Abort if the number of operations is not negligible compared to the
// number of rows. Note that this test also prevents the code from cycling
// in case the matrix is actually not triangular.
if (num_ops > num_ops_threshold) break;
}
// Clear stored_.
for (const RowIndex row : *non_zero_rows) {
stored_.ClearBucket(row);
}
// If we aborted, clear the result.
if (num_ops > num_ops_threshold) non_zero_rows->clear();
}
void TriangularMatrix::ComputeRowsToConsiderInSortedOrder(
RowIndexVector* non_zero_rows) const {
if (non_zero_rows->empty()) return;
// TODO(user): Investigate the best thresholds.
const int sparsity_threshold =
static_cast<int>(0.025 * static_cast<double>(num_rows_.value()));
const int num_ops_threshold =
static_cast<int>(0.05 * static_cast<double>(num_rows_.value()));
int num_ops = non_zero_rows->size();
if (num_ops > sparsity_threshold) {
non_zero_rows->clear();
return;
}
stored_.Resize(num_rows_);
Bitset64<RowIndex>::View stored = stored_.view();
for (const RowIndex row : *non_zero_rows) {
stored.Set(row);
}
const auto matrix_view = view();
const auto entry_rows = rows_.view();
for (int i = 0; i < non_zero_rows->size(); ++i) {
const RowIndex row = (*non_zero_rows)[i];
for (const EntryIndex index : matrix_view.Column(RowToColIndex(row))) {
++num_ops;
const RowIndex entry_row = entry_rows[index];
if (!stored[entry_row]) {
non_zero_rows->push_back(entry_row);
stored.Set(entry_row);
}
}
if (num_ops > num_ops_threshold) break;
}
if (num_ops > num_ops_threshold) {
stored_.ClearAll();
non_zero_rows->clear();
} else {
std::sort(non_zero_rows->begin(), non_zero_rows->end());
for (const RowIndex row : *non_zero_rows) {
stored_.ClearBucket(row);
}
}
}
// A known upper bound for the infinity norm of T^{-1} is the
// infinity norm of y where T'*y = x with:
// - x the all 1s vector.
// - Each entry in T' is the absolute value of the same entry in T.
Fractional TriangularMatrix::ComputeInverseInfinityNormUpperBound() const {
if (first_non_identity_column_ == num_cols_) {
// Identity matrix
return 1.0;
}
const bool is_upper = IsUpperTriangular();
DenseColumn row_norm_estimate(num_rows_, 1.0);
const int num_cols = num_cols_.value();
const auto entry_rows = rows_.view();
const auto entry_coefficients = coefficients_.view();
for (int i = 0; i < num_cols; ++i) {
const ColIndex col(is_upper ? num_cols - 1 - i : i);
DCHECK_NE(diagonal_coefficients_[col], 0.0);
const Fractional coeff = row_norm_estimate[ColToRowIndex(col)] /
std::abs(diagonal_coefficients_[col]);
row_norm_estimate[ColToRowIndex(col)] = coeff;
for (const EntryIndex i : Column(col)) {
row_norm_estimate[entry_rows[i]] +=
coeff * std::abs(entry_coefficients[i]);
}
}
return *std::max_element(row_norm_estimate.begin(), row_norm_estimate.end());
}
Fractional TriangularMatrix::ComputeInverseInfinityNorm() const {
const bool is_upper = IsUpperTriangular();
DenseColumn row_sum(num_rows_, 0.0);
DenseColumn right_hand_side;
for (ColIndex col(0); col < num_cols_; ++col) {
right_hand_side.assign(num_rows_, 0);
right_hand_side[ColToRowIndex(col)] = 1.0;
// Get the col-th column of the matrix inverse.
if (is_upper) {
UpperSolve(&right_hand_side);
} else {
LowerSolve(&right_hand_side);
}
// Compute sum_j |inverse_ij|.
for (RowIndex row(0); row < num_rows_; ++row) {
row_sum[row] += std::abs(right_hand_side[row]);
}
}
// Compute max_i sum_j |inverse_ij|.
Fractional norm = 0.0;
for (RowIndex row(0); row < num_rows_; ++row) {
norm = std::max(norm, row_sum[row]);
}
return norm;
}
} // namespace glop
} // namespace operations_research
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