android-12.0.0_r2/s

/* Copyright 2017 The TensorFlow Authors. All Rights Reserved.

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 "tensorflow/core/grappler/optimizers/dependency_optimizer.h"

#include <unordered_set>

#include "absl/container/flat_hash_map.h"
#include "tensorflow/core/framework/node_def.pb.h"
#include "tensorflow/core/framework/node_def_util.h"
#include "tensorflow/core/framework/op.h"
#include "tensorflow/core/grappler/costs/graph_properties.h"
#include "tensorflow/core/grappler/grappler_item.h"
#include "tensorflow/core/grappler/op_types.h"
#include "tensorflow/core/grappler/optimizers/constant_folding.h"
#include "tensorflow/core/grappler/utils.h"
#include "tensorflow/core/grappler/utils/topological_sort.h"
#include "tensorflow/core/lib/core/errors.h"
#include "tensorflow/core/lib/core/stringpiece.h"
#include "tensorflow/core/lib/gtl/inlined_vector.h"
#include "tensorflow/core/lib/strings/str_util.h"
#include "tensorflow/core/lib/strings/strcat.h"
#include "tensorflow/core/util/device_name_utils.h"

namespace tensorflow {
namespace grappler {

namespace {

bool RemoveControlInput(NodeDef* node, const string& control_input_to_remove,
                        NodeMap* node_map) {
  for (int pos = node->input_size() - 1; pos >= 0; --pos) {
    const string& input = node->input(pos);
    if (input[0] != '^') break;
    if (input == control_input_to_remove) {
      node->mutable_input()->SwapElements(pos, node->input_size() - 1);
      node->mutable_input()->RemoveLast();
      node_map->RemoveOutput(NodeName(input), node->name());
      return true;
    }
  }
  return false;
}

}  // namespace

bool DependencyOptimizer::SafeToRemoveIdentity(const NodeDef& node) const {
  if (!IsIdentity(node) && !IsIdentityN(node)) {
    return true;
  }

  if (nodes_to_preserve_.find(node.name()) != nodes_to_preserve_.end()) {
    return false;
  }
  if (!fetch_nodes_known_) {
    // The output values of this node may be needed.
    return false;
  }
  const NodeDef* input = node_map_->GetNode(NodeName(node.input(0)));
  CHECK(input != nullptr) << "node = " << node.name()
                          << " input = " << node.input(0);
  // Don't remove Identity nodes corresponding to Variable reads or following
  // Recv.
  if (IsVariable(*input) || IsRecv(*input)) {
    return false;
  }
  for (const auto& consumer : node_map_->GetOutputs(node.name())) {
    if (node.input_size() > 1 && (IsRetval(*consumer) || IsMerge(*consumer))) {
      return false;
    }
    if (IsSwitch(*input)) {
      for (const string& consumer_input : consumer->input()) {
        if (consumer_input == AsControlDependency(node.name())) {
          return false;
        }
      }
    }
  }
  return true;
}

bool DependencyOptimizer::SafeToConvertToNoOp(const NodeDef& node) const {
  if (HasRegularOutputs(node, *node_map_)) {
    // The output values of this node may be needed.
    VLOG(3) << "Not safe to convert '" << node.name()
            << " to NoOp. Node has outputs.";
    return false;
  }
  if (!fetch_nodes_known_) {
    VLOG(3) << "Not safe to convert '" << node.name()
            << " to NoOp. Fetches unknown.";
    return false;
  }
  if (nodes_to_preserve_.find(node.name()) != nodes_to_preserve_.end()) {
    VLOG(3) << "Not safe to convert to NoOp: " << node.name()
            << " is in preserve set.";
    return false;
  }
  if (IsMerge(node) || IsSwitch(node) || ModifiesFrameInfo(node)) {
    VLOG(3) << "Not safe to convert '" << node.name()
            << " to NoOp. Node modifies frame info.";
    return false;
  }
  // Ops reading variables are marked as stateful, but are safe to remove if
  // redundant.
  static const absl::flat_hash_set<string>* gather_ops =
      new absl::flat_hash_set<string>{"Gather", "GatherV2", "GatherNd",
                                      "ResourceGather", "ResourceGatherNd"};
  const bool is_variable_read =
      IsReadVariableOp(node) || IsReadVariablesOp(node) ||
      gather_ops->find(node.op()) != gather_ops->end();
  if (!is_variable_read && !IsFreeOfSideEffect(node)) {
    VLOG(3) << "Not safe to convert '" << node.name()
            << " to NoOp. Node has side effect.";
    return false;
  }
  if (node.op().rfind("Submodel", 0) == 0) {
    return false;
  }
  const OpDef* op_def = nullptr;
  Status status = OpRegistry::Global()->LookUpOpDef(node.op(), &op_def);
  if (!status.ok() || op_def->output_arg_size() == 0) {
    return false;
  }
  const std::unordered_set<string> do_not_rewrite_ops{
      "Assert",     "CheckNumerics",         "_Retval",
      "_Arg",       "_ParallelConcatUpdate", "TPUExecute",
      "TPUCompile", "ControlTrigger"};
  if (do_not_rewrite_ops.find(node.op()) != do_not_rewrite_ops.end()) {
    return false;
  }
  if (!SafeToRemoveIdentity(node)) {
    return false;
  }
  return true;
}

int DependencyOptimizer::NumEdgesIfBypassed(
    const NodeDef& node, const std::vector<NodeDef*>& output_nodes) const {
  const bool is_multi_input_identity_n =
      IsIdentityN(node) && !IsIdentityNSingleInput(node);
  const int num_outputs = output_nodes.size();
  const int num_inputs = node.input_size();

  if (is_multi_input_identity_n) {
    // multi-input identity_n with input/output control dependencies will likely
    // increase number of edges after optimization.
    int num_edges_if_bypassed(0);
    for (const string& input_node_name : node.input()) {
      if (IsControlInput(input_node_name)) {
        num_edges_if_bypassed += num_outputs;
      } else {
        ++num_edges_if_bypassed;
      }
    }

    for (auto consumer : output_nodes) {
      for (int j = 0; j < consumer->input_size(); ++j) {
        const TensorId consumer_input = ParseTensorName(consumer->input(j));
        if (consumer_input.node() == node.name()) {
          if (IsControlInput(consumer_input)) {
            num_edges_if_bypassed += num_inputs;
          } else {
            ++num_edges_if_bypassed;
          }
        }
      }
    }
    return num_edges_if_bypassed;
  } else {
    return num_inputs * num_outputs;
  }
}

bool DependencyOptimizer::BypassingNodeIsBeneficial(
    const NodeDef& node, const std::vector<NodeDef*>& input_nodes,
    const std::vector<NodeDef*>& output_nodes) const {
  const bool is_identity = IsIdentity(node) || IsIdentityNSingleInput(node);
  const bool is_multi_input_identity_n =
      IsIdentityN(node) && !IsIdentityNSingleInput(node);
  const int num_outputs = output_nodes.size();
  const int num_inputs = node.input_size();

  if (NumEdgesIfBypassed(node, output_nodes) > num_inputs + num_outputs) {
    return false;
  }

  // Make sure that we don't increase the number of edges that cross
  // device boundaries.
  if ((num_inputs == 1 && num_outputs > 1 &&
       input_nodes[0]->device() != node.device()) ||
      (num_inputs > 1 && num_outputs == 1 &&
       output_nodes[0]->device() != node.device())) {
    return false;
  }

  // TODO(rmlarsen): Not all device crossings are equally expensive.
  // Assign a cost to each based on device affinity and compute a
  // cost before and after.
  const string& node_dev = node.device();
  int num_cross_in = 0;
  for (NodeDef* input_node : input_nodes) {
    num_cross_in += static_cast<int>(input_node->device() != node_dev);
  }
  int num_cross_out = 0;
  for (NodeDef* output_node : output_nodes) {
    num_cross_out += static_cast<int>(output_node->device() != node_dev);
  }

  // Make sure we do not increase the number of device crossings.
  const int num_cross_before = num_cross_in + num_cross_out;
  int num_cross_after = 0;
  for (NodeDef* input_node : input_nodes) {
    for (NodeDef* output_node : output_nodes) {
      num_cross_after +=
          static_cast<int>(input_node->device() != output_node->device());
    }
  }
  if (num_cross_after > num_cross_before) {
    return false;
  }

  if ((is_identity || is_multi_input_identity_n) && num_cross_in > 0 &&
      num_cross_out > 0 && num_cross_after > 0) {
    // This identity node follows a device crossing, so it might be
    // following a _Recv node after partitioning. Do not remove such nodes,
    // unless they only have consumers on the same device as themselves.
    return false;
  }

  return true;
}

void DependencyOptimizer::OptimizeNode(int node_idx,
                                       SetVector<int>* nodes_to_simplify,
                                       std::set<int>* nodes_to_delete) {
  NodeDef* node = optimized_graph_->mutable_node(node_idx);
  const bool is_noop = IsNoOp(*node);
  const bool is_identity = IsIdentity(*node) || IsIdentityNSingleInput(*node);
  const bool is_multi_input_identity =
      IsIdentityN(*node) && !IsIdentityNSingleInput(*node);
  const string node_name = node->name();
  // Constant nodes with no input control dependency are always executed early,
  // so we can prune all their output control dependencies.
  if (IsConstant(*node) && node->input_size() == 0) {
    const auto output_nodes = node_map_->GetOutputs(node_name);
    for (NodeDef* fanout : output_nodes) {
      bool optimize_fanout = false;
      bool data_connection = false;
      for (int i = fanout->input_size() - 1; i >= 0; --i) {
        const TensorId input_tensor = ParseTensorName(fanout->input(i));
        if (input_tensor.node() == node_name) {
          if (input_tensor.index() < 0) {
            fanout->mutable_input()->SwapElements(i, fanout->input_size() - 1);
            fanout->mutable_input()->RemoveLast();
            optimize_fanout = true;
          } else {
            data_connection = true;
          }
        }
      }
      if (optimize_fanout) {
        nodes_to_simplify->PushBack(node_to_idx_[fanout]);
        if (!data_connection) {
          node_map_->RemoveOutput(node_name, fanout->name());
        }
      }
    }
    if (node_map_->GetOutputs(node_name).empty() && fetch_nodes_known_ &&
        nodes_to_preserve_.find(node_name) == nodes_to_preserve_.end()) {
      // Mark the node for deletion.
      nodes_to_delete->insert(node_to_idx_[node]);
    }
    return;
  }

  // Change ops that only have control dependencies as outputs to NoOps.
  if (!is_noop && SafeToConvertToNoOp(*node)) {
    VLOG(2) << "***** Replacing  " << node_name << " (" << node->op()
            << ") with NoOp.";
    // The outputs of this node are not consumed. Replace its inputs with
    // control dependencies and replace the op itself with the NoOp op.
    std::unordered_set<string> ctrl_inputs;
    int pos = 0;
    while (pos < node->input_size()) {
      const string old_input = node->input(pos);
      if (IsControlInput(old_input)) {
        if (!ctrl_inputs.insert(old_input).second) {
          // We found a duplicate control input. Remove it.
          node->mutable_input()->SwapElements(pos, node->input_size() - 1);
          node->mutable_input()->RemoveLast();
        } else {
          ++pos;
        }
        continue;
      }
      // Replace a normal input with a control input.
      const string ctrl_input = ConstantFolding::AddControlDependency(
          old_input, optimized_graph_, node_map_.get());
      ctrl_inputs.insert(ctrl_input);
      node->set_input(pos, ctrl_input);
      node_map_->UpdateInput(node_name, old_input, ctrl_input);
      const NodeDef* old_input_node = node_map_->GetNode(old_input);
      nodes_to_simplify->PushBack(node_to_idx_[old_input_node]);
      ++pos;
    }
    node->set_op("NoOp");
    EraseRegularNodeAttributes(node);
    DedupControlInputs(node);
    nodes_to_simplify->PushBack(node_to_idx_[node]);
    return;
  }

  // Remove NoOp nodes if the product of their fan-in and fan-out is less than
  // or equal to the sum of the fan-in and fan-out. The non-trivial rewrites
  // take the following form:
  //
  // Case a)
  //    x --^> +------+                x --^> +---+
  //    y --^> | NoOp | --^> a   ==>   y --^> | a |
  //    ...    |      |                  ...  |   |
  //    z --^> +------+                z --^> +---+
  //
  // Case b)
  //           +------+ --^> a         +---+ --^> a
  //    x --^> | NoOp | --^> b  ==>    | x | --^> b
  //           |      | ...            |   | ...
  //           +------+ --^> c         +---+ --^> c
  // Case c)
  //           +------+                x ---^> a
  //    x --^> | NoOp | --^> a  ==>      \/
  //    y --^> |      | --^> b           /\
  //           +------+                y ---^> b
  //
  // We only apply this optimization if we don't increase the number of control
  // edges across device boundaries, e.g. in cases a) and b) if NoOp and
  // a and x, respectively, are on the same device. Control edges across device
  // boundaries require inter-device communication (Send/Recv pairs to be
  // inserted in the graph), which is very costly.
  //
  // We also remove identity nodes, subject to the same constraints on number of
  // resulting control edges and device boundary crossings:
  //
  // Case a)
  //          +----------+ ---> a       +---+ ---> a
  //    x --> | Identity | --^> b  ==>  | x | --^> b
  //          |          | ...          |   | ...
  //          +----------+ --^> c       +---+ --^> c
  //
  // Case b)
  //    x ---> +----------+ ---> a      x ---> +---+
  //    y --^> | Identity |        ==>  y --^> | a |
  //    ...    |          |               ...  |   |
  //    z --^> +----------+             z --^> +---+
  //
  // Case c)
  //           +----------+             x ---> +---+
  //    x ---> | Identity | ---> a ==>   \--^> | a |
  //    y --^> |          | --^> b       /\    +---+
  //           +----------+             y --^> b

  if (is_noop || ((is_identity || is_multi_input_identity) &&
                  SafeToRemoveIdentity(*node))) {
    const int num_inputs = node->input_size();
    std::vector<NodeDef*> input_nodes;
    for (int i = 0; i < num_inputs; ++i) {
      NodeDef* input_node = node_map_->GetNode(node->input(i));
      if (input_node == nullptr) {
        LOG(ERROR) << "Invalid input " << node->input(i);
        return;
      }
      input_nodes.push_back(input_node);
    }
    const auto& output_node_set = node_map_->GetOutputs(node_name);
    const std::vector<NodeDef*> output_nodes(output_node_set.begin(),
                                             output_node_set.end());

    if (!BypassingNodeIsBeneficial(*node, input_nodes, output_nodes)) {
      return;
    }

    VLOG(2) << "***** Rerouting input around\n" << node->DebugString();
    // Now remove the node and re-wire its inputs to its outputs.
    for (auto consumer : output_nodes) {
      bool updated_consumer = false;
      VLOG(2) << "consumer before:\n" << consumer->DebugString();
      // Remove dependency on node from consumer.
      for (int i = 0; i < num_inputs; ++i) {
        const NodeDef* input = input_nodes[i];
        // Forward dependency from input to consumer if it doesn't already
        // depend on it.
        if ((is_identity && i == 0) ||
            (is_multi_input_identity && !IsControlInput(node->input(i)))) {
          // Replace regular input from Identity node.
          string new_input;
          const string& input_to_forward = node->input(i);
          CHECK(!IsControlInput(input_to_forward));
          for (int j = 0; j < consumer->input_size(); ++j) {
            const TensorId old_input = ParseTensorName(consumer->input(j));
            if (old_input.node() == node_name) {
              if (old_input.index() == i) {
                // Regular input
                new_input = input_to_forward;
                node_map_->UpdateInput(consumer->name(), old_input.ToString(),
                                       new_input);
                consumer->set_input(j, new_input);
              } else if (old_input.index() == -1) {
                // Control dependency
                new_input = AsControlDependency(NodeName(input_to_forward));
                node_map_->UpdateInput(consumer->name(), old_input.ToString(),
                                       new_input);
                consumer->set_input(j, new_input);
              }
            }
          }
          updated_consumer = true;
        } else {
          // Forward dependency from input to consumer if it doesn't already
          // depend on it.
          if (node_map_->GetOutputs(input->name()).count(consumer) == 0) {
            consumer->add_input(AsControlDependency(input->name()));
            node_map_->AddOutput(input->name(), consumer->name());
            nodes_to_simplify->PushBack(node_to_idx_[input]);
            updated_consumer = true;
          }
        }
      }
      updated_consumer |= RemoveControlInput(
          consumer, AsControlDependency(node_name), node_map_.get());
      if (updated_consumer) {
        nodes_to_simplify->PushBack(node_to_idx_[consumer]);
      }
      VLOG(2) << "consumer after:\n" << consumer->DebugString();
    }
    node_map_->RemoveOutputs(node_name);
    if (fetch_nodes_known_ &&
        nodes_to_preserve_.find(node_name) == nodes_to_preserve_.end()) {
      // Mark the node for deletion.
      nodes_to_delete->insert(node_idx);

      // Disconnect the node from its inputs to enable further optimizations.
      node_map_->RemoveInputs(node_name);
      node->clear_input();
    }
  }
}

void DependencyOptimizer::CleanControlInputs() {
  for (int i = 0; i < optimized_graph_->node_size(); ++i) {
    DedupControlInputs(optimized_graph_->mutable_node(i));
  }
}

Status DependencyOptimizer::OptimizeDependencies() {
  SetVector<int> nodes_to_simplify;
  std::set<int> nodes_to_delete;
  for (int i = 0; i < optimized_graph_->node_size(); ++i) {
    const NodeDef& node = optimized_graph_->node(i);
    if (IsNoOp(node) || IsIdentity(node) || IsIdentityN(node) ||
        IsConstant(node) || SafeToConvertToNoOp(node)) {
      nodes_to_simplify.PushBack(i);
    }
  }
  while (!nodes_to_simplify.Empty()) {
    int node_to_simplify = nodes_to_simplify.PopBack();
    // Discard nodes that were marked for deletion already.
    while (nodes_to_delete.find(node_to_simplify) != nodes_to_delete.end()) {
      node_to_simplify = nodes_to_simplify.PopBack();
    }
    OptimizeNode(node_to_simplify, &nodes_to_simplify, &nodes_to_delete);
  }

  if (fetch_nodes_known_) {
    VLOG(1) << "Deleted " << nodes_to_delete.size() << " out of "
            << optimized_graph_->node_size() << " nodes.";
    EraseNodesFromGraph(nodes_to_delete, optimized_graph_);
    node_map_.reset(new NodeMap(optimized_graph_));
    BuildNodeToIdx();
  }
  return Status::OK();
}

namespace {

enum DistanceFromSource : uint8 { ZERO = 0, ONE = 1, TWO_OR_GREATER = 2 };

void LongestPathsLowerBounds(
    int source, const std::pair<int, int>& target_range,
    const std::vector<std::vector<int>>& outputs,
    std::vector<DistanceFromSource>* longest_distance) {
  std::deque<int> queue;
  queue.emplace_front(source);
  while (!queue.empty()) {
    int node = queue.front();
    queue.pop_front();
    for (int fanout : outputs[node]) {
      // 1) Only nodes in the target range can be on paths from source to one of
      //    its control outputs.
      // 2) Since we only need a lower bound on the longest distance, we can
      //    skip nodes for which we have already proven have a path of
      //    length > 1 from the source.
      if (fanout >= target_range.first && fanout <= target_range.second &&
          (*longest_distance)[fanout] != TWO_OR_GREATER) {
        (*longest_distance)[fanout] =
            (*longest_distance)[fanout] == ZERO ? ONE : TWO_OR_GREATER;
        queue.emplace_front(fanout);
      }
    }
  }
}

}  // namespace

Status DependencyOptimizer::TransitiveReduction() {
  // PRECONDITION: optimized_graph_ must be sorted topologically.
  const int num_nodes = optimized_graph_->node_size();
  // Set up a compressed version of the graph to save a constant factor in the
  // expensive algorithm below. Also cache the set of control outputs and the
  // highest index of a target of any control output from each node.
  int num_controls = 0;
  std::vector<std::vector<int>> outputs(num_nodes);
  std::vector<gtl::InlinedVector<std::pair<int, int>, 2>> control_outputs(
      num_nodes);
  // target_range[i] contains the range of node indices for which to compute
  // longest paths starting from node i.
  std::vector<std::pair<int, int>> target_range(num_nodes, {num_nodes, -1});
  for (int node_idx = 0; node_idx < num_nodes; ++node_idx) {
    const NodeDef& node = optimized_graph_->node(node_idx);
    if (ModifiesFrameInfo(node) || !HasOpDef(node)) {
      // Ignore function nodes and nodes that modify frame info.
      continue;
    }
    for (int input_slot = 0; input_slot < node.input_size(); ++input_slot) {
      const string& input = node.input(input_slot);
      const NodeDef* input_node = node_map_->GetNode(input);
      if (ModifiesFrameInfo(*input_node) || IsMerge(*input_node)) {
        // Ignore edges from nodes that modify frame info and from Merge nodes,
        // because we cannot know which of it's input paths executes.
        continue;
      }
      const int input_node_idx = node_to_idx_[input_node];
      outputs[input_node_idx].push_back(node_idx);
      target_range[input_node_idx].first =
          std::min(target_range[input_node_idx].first, node_idx);
      if (IsControlInput(input)) {
        ++num_controls;
        control_outputs[input_node_idx].emplace_back(node_idx, input_slot);
        target_range[input_node_idx].second =
            std::max(target_range[input_node_idx].second, node_idx);
      }
    }
  }

  // Run the longest path in DAG algorithm for each source node that has control
  // outputs. If, for any target node of a control output, there exists a path
  // of length > 1, we can drop that control dependency.
  int num_controls_removed = 0;
  std::vector<DistanceFromSource> longest_distance(num_nodes);
  // Map from target_index -> set of (input_slot, source_index), representing
  // the control edges to remove. We sort them in reverse order by input slot,
  // such that when we swap them out so we don't clobber the
  // node(target).input() repeated field.
  typedef std::pair<int, int> InputSlotAndSource;
  absl::flat_hash_map<
      int, std::set<InputSlotAndSource, std::greater<InputSlotAndSource>>>
      control_edges_to_remove;
  for (int source = 0; source < num_nodes; ++source) {
    if (target_range[source].first >= target_range[source].second ||
        target_range[source].second <= source) {
      continue;
    }
    // Compute the set of nodes in the transitive fanout of source with
    // topological sort index in [target_range.first : target_range.second]]
    // to which there exists a path of length 2 or more from source.
    std::fill(longest_distance.begin() + target_range[source].first,
              longest_distance.begin() + target_range[source].second + 1, ZERO);
    LongestPathsLowerBounds(source, target_range[source], outputs,
                            &longest_distance);

    // If the longest path from source to target of a control dependency is
    // longer than 1, there exists an alternate path, and we can eliminate the
    // redundant direct control dependency.
    for (const auto& control_output : control_outputs[source]) {
      const int target = control_output.first;
      if (longest_distance[target] == TWO_OR_GREATER) {
        const int input_slot = control_output.second;
        control_edges_to_remove[target].emplace(input_slot, source);
      }
    }
  }
  for (const auto& it : control_edges_to_remove) {
    const int target = it.first;
    NodeDef* target_node = optimized_graph_->mutable_node(target);
    for (const InputSlotAndSource& slot_and_source : it.second) {
      const int input_slot = slot_and_source.first;
      const int source = slot_and_source.second;
      const NodeDef& source_node = optimized_graph_->node(source);
      CHECK_LT(input_slot, target_node->input_size());
      target_node->mutable_input()->SwapElements(input_slot,
                                                 target_node->input_size() - 1);
      node_map_->RemoveOutput(source_node.name(), target_node->name());
      target_node->mutable_input()->RemoveLast();
      ++num_controls_removed;
    }
  }
  VLOG(1) << "Removed " << num_controls_removed << " out of " << num_controls
          << " control dependencies";
  return Status::OK();
}

void DependencyOptimizer::BuildNodeToIdx() {
  // Set up &node -> index map.
  node_to_idx_.clear();
  for (int i = 0; i < optimized_graph_->node_size(); ++i) {
    const NodeDef& node = optimized_graph_->node(i);
    node_to_idx_[&node] = i;
  }
}

// Suppose there are cross-device control inputs to node C from multiple nodes
// that are located on another device, e.g., we have control edges:
// A->C, B->C
// where A and B are on device X and C is on device Y.
// We can reduce cross-device communication by introducing an intermediate
// NoOp node C' on device X and rewriting the control edges to:
// A->C', B->C', C' -> C
void DependencyOptimizer::GroupCrossDeviceControlEdges(bool host_granularity) {
  VLOG(1)
      << "DependencyOptimizer::GroupCrossDeviceControlEdges host_granularity="
      << host_granularity;
  const int num_nodes = optimized_graph_->node_size();
  for (int i = 0; i < num_nodes; ++i) {
    NodeDef* node = optimized_graph_->mutable_node(i);
    if (node->device().empty()) continue;
    string rest, node_device = node->device();
    if (host_granularity) {
      DeviceNameUtils::SplitDeviceName(node->device(), &node_device, &rest);
    }

    // Creates new noop nodes for devices on which multiple control inputs are
    // located.

    // Map keyed by device name to the newly introduced Noop node for that
    // device. A nullptr value means that we have only seen a single node on
    // that device.
    std::map<string, NodeDef*> noops;
    int num_noops = 0;
    for (int j = 0; j < node->input_size(); ++j) {
      if (IsControlInput(node->input(j))) {
        const NodeDef* input = node_map_->GetNode(node->input(j));
        if (input == nullptr || input->device().empty()) continue;
        string input_device = input->device();
        if (host_granularity) {
          DeviceNameUtils::SplitDeviceName(input->device(), &input_device,
                                           &rest);
        }
        if (input_device != node_device) {
          VLOG(2) << "Cross-device " << node->name() << " " << input->device()
                  << " -> " << node->device();
          auto emplace_result = noops.emplace(input_device, nullptr);
          if (!emplace_result.second &&
              emplace_result.first->second == nullptr) {
            VLOG(2) << "Duplicate input device from " << node->name();
            // This is the second cross-device control input from the same
            // device. Creates an intermediate noop node on that device.
            string group_name;
            NodeDef* noop;
            // Creates a fresh node name; there may be conflicting names from
            // a previous iteration of the optimizer.
            do {
              group_name = AddPrefixToNodeName(
                  node->name(),
                  strings::StrCat("GroupCrossDeviceControlEdges_", num_noops));
              noop = node_map_->GetNode(group_name);
              ++num_noops;
            } while (noop != nullptr);
            noop = optimized_graph_->add_node();
            noop->set_name(group_name);
            noop->set_device(input->device());
            noop->set_op("NoOp");
            node_map_->AddNode(noop->name(), noop);
            emplace_result.first->second = noop;
            VLOG(1) << "GroupCrossDeviceControlEdges: Added "
                    << SummarizeNodeDef(*noop);
          }
        }
      }
    }

    // Reroute existing control edges to go via the newly introduced NoOp nodes.
    int pos = 0;
    while (pos < node->input_size()) {
      const string& input_name = node->input(pos);
      if (IsControlInput(input_name)) {
        NodeDef* input = node_map_->GetNode(input_name);
        if (input == nullptr) {
          ++pos;
        } else {
          string input_device = input->device();
          if (host_granularity) {
            DeviceNameUtils::SplitDeviceName(input->device(), &input_device,
                                             &rest);
          }
          auto it = noops.find(input_device);
          if (it == noops.end() || it->second == nullptr) {
            ++pos;
          } else {
            VLOG(2) << "Rewriting input from " << input_name;
            node->mutable_input()->SwapElements(pos, node->input_size() - 1);
            node->mutable_input()->RemoveLast();
            it->second->add_input(AsControlDependency(*input));
            node_map_->UpdateOutput(input_name, node->name(),
                                    it->second->name());
          }
        }
      } else {
        ++pos;
      }
    }
    for (const auto& entry : noops) {
      if (entry.second) {
        node->add_input(AsControlDependency(*entry.second));
        node_map_->AddOutput(entry.second->name(), node->name());
      }
    }
  }
}

Status DependencyOptimizer::Optimize(Cluster* cluster, const GrapplerItem& item,
                                     GraphDef* optimized_graph) {
  optimized_graph_ = optimized_graph;
  *optimized_graph_ = item.graph;
  nodes_to_preserve_ = item.NodesToPreserve();
  fetch_nodes_known_ = !item.fetch.empty();
  CleanControlInputs();

  const int num_iterations = 2;
  for (int iteration = 0; iteration < num_iterations; ++iteration) {
    GRAPPLER_RETURN_IF_DEADLINE_EXCEEDED();
    Status topo_sort_status;
    // Perform topological sort to prepare the graph for transitive reduction.
    topo_sort_status = TopologicalSort(optimized_graph_);
    // Set up index-based graph datastructures to speed up analysis steps below.
    node_map_.reset(new NodeMap(optimized_graph_));
    BuildNodeToIdx();

    if (topo_sort_status.ok()) {
      // Remove redundant control dependencies.
      TF_RETURN_IF_ERROR(TransitiveReduction());
    } else {
      LOG(ERROR) << "Iteration = " << iteration
                 << ", topological sort failed with message: "
                 << topo_sort_status.error_message();
    }
    // Turn nodes with only control outputs into NoOps, prune NoOp and Identity
    // nodes.
    TF_RETURN_IF_ERROR(OptimizeDependencies());

    // Dedup control inputs.
    CleanControlInputs();

    // Merge multiple control edges from the same device.
    GroupCrossDeviceControlEdges(/*host_granularity=*/false);

    // Merge control edges from the same host to reduce RPC traffic.
    GroupCrossDeviceControlEdges(/*host_granularity=*/true);
  }

  return Status::OK();
}

void DependencyOptimizer::Feedback(Cluster* /*cluster*/,
                                   const GrapplerItem& /*item*/,
                                   const GraphDef& /*optimized_graph*/,
                                   double /*result*/) {
  // Nothing to do for DependencyOptimizer.
}

}  // end namespace grappler
}  // end namespace tensorflow