adding gprof page

lectures/optimization/gprof/README.md

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+```markdown
+# C++ Profiling with `gprof`
+
+## What is `gprof`?
+`gprof` is a GNU profiler that helps analyze where a program spends most of its execution time. It provides function call counts and execution time details.
+
+## Installing `gprof`
+```sh
+sudo apt-get install binutils
+```
+
+## Compiling a C++ Program for Profiling
+To use `gprof`, compile your program with the `-pg` flag:
+```sh
+g++ -pg -o my_program my_program.cpp
+```
+
+## Example C++ Program
+Create a file `my_program.cpp` with the following code:
+```cpp
+#include <iostream>
+#include <chrono>
+#include <thread>
+
+void slowFunction() {
+    std::this_thread::sleep_for(std::chrono::seconds(1));
+}
+
+void fastFunction() {
+    for (volatile int i = 0; i < 1000000; ++i);
+}
+
+int main() {
+    for (int i = 0; i < 5; ++i) slowFunction();
+    for (int i = 100; i < 200; ++i) fastFunction();
+    return 0;
+}
+```
+
+## Running and Profiling the Program
+1. Compile the program:
+   ```sh
+   g++ -pg -o my_program my_program.cpp
+   ```
+2. Execute the program to generate `gmon.out`:
+   ```sh
+   ./my_program
+   ```
+3. Analyze the profiling data:
+   ```sh
+   gprof my_program gmon.out > profile.txt
+   cat profile.txt
+   ```
+
+## Understanding the Output
+- **Flat Profile**: Shows execution time spent in each function.
+- **Call Graph**: Displays function call relationships and their execution time.
+
+## Best Practices for Using `gprof`
+- Use `-O2` optimizations but **avoid `-O3`**, which may inline functions and reduce profiling accuracy.
+- Profile with realistic input data to get meaningful results.
+- Optimize the slowest functions first based on the profiling report.
+
+## Conclusion
+`gprof` is a powerful tool for detecting performance bottlenecks in C++ programs. By identifying expensive functions, developers can make targeted optimizations.
+```