Fix `SearchAnchor` triggers unnecessary suggestions builder calls (#143443)

fixes [`SearchAnchor` triggers extra search operations](https://github.com/flutter/flutter/issues/139880)

### Code sample

<details>
<summary>expand to view the code sample</summary> 

```dart
import 'package:flutter/material.dart';

Future<List<String>> createFuture() async {
  return List.generate(1000, (index) => "Hello World!");
}

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  @override
  Widget build(BuildContext context) {
    return const MaterialApp(
      title: 'Flutter Demo',
      home: MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  const MyHomePage({super.key, required this.title});

  final String title;

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  final SearchController controller = SearchController();

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        backgroundColor: Theme.of(context).colorScheme.inversePrimary,
        title: Text(widget.title),
      ),
      body: Center(
        child: Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            SearchAnchor(
              searchController: controller,
              suggestionsBuilder: (suggestionsContext, controller) {
                final resultFuture = createFuture();
                return [
                  FutureBuilder(
                    future: resultFuture,
                    builder: ((context, snapshot) {
                      if (snapshot.connectionState != ConnectionState.done) {
                        return const LinearProgressIndicator();
                      }
                      final result = snapshot.data;
                      if (result == null) {
                        return const LinearProgressIndicator();
                      }
                      return ListView.builder(
                        shrinkWrap: true,
                        physics: const NeverScrollableScrollPhysics(),
                        itemCount: result.length,
                        itemBuilder: (BuildContext context, int index) {
                          final root = result[index];
                          return ListTile(
                            leading: const Icon(Icons.article),
                            title: Text(root),
                            subtitle: Text(
                              root,
                              overflow: TextOverflow.ellipsis,
                              style: TextStyle(
                                color: Theme.of(suggestionsContext)
                                    .colorScheme
                                    .onSurfaceVariant,
                              ),
                            ),
                            onTap: () {},
                          );
                        },
                      );
                    }),
                  ),
                ];
              },
              builder: (BuildContext context, SearchController controller) {
                return IconButton(
                  onPressed: () {
                    controller.openView();
                  },
                  icon: const Icon(Icons.search),
                );
              },
            ),
          ],
        ),
      ),
    );
  }
}

```

</details>

### Before

https://github.com/flutter/flutter/assets/48603081/69f6dfdc-9f92-4d2e-8a3e-984fce25f9e4

### After

https://github.com/flutter/flutter/assets/48603081/be105e2c-51d8-4cb0-a75b-f5f41d948e5e

packages/flutter/lib/src/material/search_anchor.dart

@@ -740,6 +740,8 @@ class _ViewContentState extends State<_ViewContent> {
   late Rect _viewRect;
   late final SearchController _controller;
   Iterable<Widget> result = <Widget>[];
+  String? searchValue;
+  Timer? _timer;
 
   @override
   void initState() {
@@ -770,12 +772,23 @@ class _ViewContentState extends State<_ViewContent> {
         _viewRect = Offset.zero & _screenSize!;
       }
     }
-    unawaited(updateSuggestions());
+
+    if (searchValue != _controller.text) {
+      _timer?.cancel();
+      _timer = Timer(Duration.zero, () async {
+        searchValue = _controller.text;
+        result = await widget.suggestionsBuilder(context, _controller);
+        _timer?.cancel();
+        _timer = null;
+      });
+    }
   }
 
   @override
   void dispose() {
     _controller.removeListener(updateSuggestions);
+    _timer?.cancel();
+    _timer = null;
     super.dispose();
   }
 
@@ -793,6 +806,8 @@ class _ViewContentState extends State<_ViewContent> {
   }
 
   Future<void> updateSuggestions() async {
+    if (searchValue != _controller.text) {
+      searchValue = _controller.text;
       final Iterable<Widget> suggestions = await widget.suggestionsBuilder(context, _controller);
       if (mounted) {
         setState(() {
@@ -800,6 +815,7 @@ class _ViewContentState extends State<_ViewContent> {
         });
       }
     }
+  }
 
   @override
   Widget build(BuildContext context) {

packages/flutter/test/material/search_anchor_test.dart

@@ -3131,6 +3131,104 @@ void main() {
     await tester.pump();
     expect(find.widgetWithIcon(IconButton, Icons.close), findsNothing);
   });
+
+  // This is a regression test for https://github.com/flutter/flutter/issues/139880.
+  testWidgets('suggestionsBuilder with Future is not called twice on layout resize', (WidgetTester tester) async {
+    int suggestionsLoadingCount = 0;
+
+    Future<List<String>> createListData() async {
+      return List<String>.generate(1000, (int index) {
+        return  'Hello World - $index';
+      });
+    }
+
+    await tester.pumpWidget(MaterialApp(
+      home: Scaffold(
+        body: Center(
+          child: SearchAnchor(
+            builder: (BuildContext context, SearchController controller) {
+              return const Icon(Icons.search);
+            },
+            suggestionsBuilder: (BuildContext context, SearchController controller) {
+              return <Widget>[
+                FutureBuilder<List<String>>(
+                  future: createListData(),
+                  builder: (BuildContext context, AsyncSnapshot<List<String>> snapshot) {
+                    if (snapshot.connectionState != ConnectionState.done) {
+                      return const LinearProgressIndicator();
+                    }
+                    final List<String>? result = snapshot.data;
+                    if (result == null) {
+                      return const LinearProgressIndicator();
+                    }
+                    suggestionsLoadingCount++;
+                    return SingleChildScrollView(
+                      child: Column(
+                        children: result.map((String text) {
+                          return ListTile(title: Text(text));
+                        }).toList(),
+                      ),
+                    );
+                  },
+                ),
+              ];
+            },
+          ),
+        ),
+      ),
+    ));
+    await tester.pump();
+    await tester.tap(find.byIcon(Icons.search)); // Open search view.
+    await tester.pumpAndSettle();
+
+    // Simulate the keyboard opening resizing the view.
+    tester.view.viewInsets = const FakeViewPadding(bottom: 500.0);
+    addTearDown(tester.view.reset);
+    await tester.pumpAndSettle();
+
+    expect(suggestionsLoadingCount, 1);
+  });
+
+  // This is a regression test for https://github.com/flutter/flutter/issues/139880.
+  testWidgets('suggestionsBuilder is not called when the search value does not change', (WidgetTester tester) async {
+    int suggestionsBuilderCalledCount = 0;
+
+    await tester.pumpWidget(MaterialApp(
+      home: Scaffold(
+        body: Center(
+          child: SearchAnchor(
+            builder: (BuildContext context, SearchController controller) {
+              return const Icon(Icons.search);
+            },
+            suggestionsBuilder: (BuildContext context, SearchController controller) {
+              suggestionsBuilderCalledCount++;
+              return <Widget>[];
+            },
+          ),
+        ),
+      ),
+    ));
+    await tester.pump();
+    await tester.tap(find.byIcon(Icons.search)); // Open search view.
+    await tester.pumpAndSettle();
+
+    // Simulate the keyboard opening resizing the view.
+    tester.view.viewInsets = const FakeViewPadding(bottom: 500.0);
+    addTearDown(tester.view.reset);
+    // Show the keyboard.
+    await tester.showKeyboard(find.byType(TextField));
+    await tester.pumpAndSettle();
+
+    expect(suggestionsBuilderCalledCount, 2);
+
+    // Remove the viewInset, as if the keyboard were hidden.
+    tester.view.resetViewInsets();
+    // Hide the keyboard.
+    await tester.testTextInput.receiveAction(TextInputAction.done);
+    await tester.pumpAndSettle();
+
+    expect(suggestionsBuilderCalledCount, 2);
+  });
 }
 
 Future<void> checkSearchBarDefaults(WidgetTester tester, ColorScheme colorScheme, Material material) async {