[framework] Improve DiagramBuilder::Build performance (#19746)

When searching the Diagram feedthrough graph, only ever ask each
subsystem for its own feedthrough graph at most once.

systems/framework/diagram.cc

@@ -73,10 +73,11 @@ Diagram<T>::get_output_port_locator(OutputPortIndex port_index) const {
 
 template <typename T>
 std::multimap<int, int> Diagram<T>::GetDirectFeedthroughs() const {
+  std::unordered_map<const System<T>*, std::multimap<int, int>> memoize;
   std::multimap<int, int> pairs;
   for (InputPortIndex u(0); u < this->num_input_ports(); ++u) {
     for (OutputPortIndex v(0); v < this->num_output_ports(); ++v) {
-      if (DiagramHasDirectFeedthrough(u, v)) {
+      if (DiagramHasDirectFeedthrough(u, v, &memoize)) {
         pairs.emplace(u, v);
       }
     }
@@ -1090,12 +1091,15 @@ const SystemBase& Diagram<T>::GetRootSystemBase() const {
 }
 
 template <typename T>
-bool Diagram<T>::DiagramHasDirectFeedthrough(int input_port, int output_port)
+bool Diagram<T>::DiagramHasDirectFeedthrough(
+    int input_port, int output_port,
+    std::unordered_map<const System<T>*, std::multimap<int, int>>* memoize)
     const {
   DRAKE_ASSERT(input_port >= 0);
   DRAKE_ASSERT(input_port < this->num_input_ports());
   DRAKE_ASSERT(output_port >= 0);
   DRAKE_ASSERT(output_port < this->num_output_ports());
+  DRAKE_ASSERT(memoize != nullptr);
 
   const auto input_ids = GetInputPortLocators(InputPortIndex(input_port));
   std::set<InputPortLocator> target_input_ids(input_ids.begin(),
@@ -1108,10 +1112,19 @@ bool Diagram<T>::DiagramHasDirectFeedthrough(int input_port, int output_port)
   active_set.insert(output_port_ids_[output_port]);
   while (!active_set.empty()) {
     const OutputPortLocator current_output_id = *active_set.begin();
-    size_t removed_count = active_set.erase(current_output_id);
+    const size_t removed_count = active_set.erase(current_output_id);
     DRAKE_ASSERT(removed_count == 1);
-    const System<T>* sys = current_output_id.first;
-    for (const auto& [sys_input, sys_output] : sys->GetDirectFeedthroughs()) {
+    const System<T>* const sys = current_output_id.first;
+    auto memoized_feedthroughs = memoize->find(sys);
+    if (memoized_feedthroughs == memoize->end()) {
+      bool inserted{};
+      std::tie(memoized_feedthroughs, inserted) =
+          memoize->emplace(sys, sys->GetDirectFeedthroughs());
+      DRAKE_ASSERT(inserted);
+    }
+    const std::multimap<int, int>& sys_feedthroughs =
+        memoized_feedthroughs->second;
+    for (const auto& [sys_input, sys_output] : sys_feedthroughs) {
       if (sys_output == current_output_id.second) {
         const InputPortLocator curr_input_id(sys, sys_input);
         if (target_input_ids.count(curr_input_id)) {

systems/framework/diagram.h

@@ -5,6 +5,7 @@
 #include <memory>
 #include <optional>
 #include <string>
+#include <unordered_map>
 #include <utility>
 #include <vector>
 
@@ -400,8 +401,13 @@ class Diagram : public System<T>, internal::SystemParentServiceInterface {
   const SystemBase& GetRootSystemBase() const final;
 
   // Returns true if there might be direct feedthrough from the given
-  // @p input_port of the Diagram to the given @p output_port of the Diagram.
-  bool DiagramHasDirectFeedthrough(int input_port, int output_port) const;
+  // `input_port` of the Diagram to the given `output_port` of the Diagram.
+  // The `memoize` dictionary caches the subsystem's reported feedthrough,
+  // to accelerate repeated calls to this function.
+  bool DiagramHasDirectFeedthrough(
+      int input_port, int output_port,
+      std::unordered_map<const System<T>*, std::multimap<int, int>>* memoize)
+      const;
 
   // Allocates a collection of homogeneous events (e.g., publish events) for
   // this Diagram.