In Algorithm Design Lab, measuring the execution time of algorithms is an important practical method for comparing their efficiency. Two algorithms may produce the same output for a given problem, but their execution times can differ significantly because of differences in their design, number of operations, memory access, and input size. Execution-time measurement provides an empirical way to evaluate algorithm performance and helps in understanding the relationship between theoretical time complexity and actual running time. Suppose two algorithms, Algorithm A and Algorithm B, are designed to solve the same problem. To measure their execution time, both algorithms should be implemented using the same programming language, compiler, computer system, and input data. This ensures that the comparison is fair. The primary objective is to determine how much time each algorithm requires to complete its task for different input sizes. The basic procedure begins by selecting appropriate test inputs. For example, if the algorithms are designed for sorting, arrays containing different numbers of elements can be used. Inputs may contain 100, 1,000, 10,000, 50,000, or 100,000 elements. The same input should be supplied to both algorithms so that their execution times can be compared under identical conditions. It is also useful to test different types of inputs, such as random, sorted, reverse-sorted, and duplicate-containing data, because some algorithms behave differently depending on the characteristics of the input. The execution time can be measured using a system clock available in the programming language. In C++, the `` library provides a convenient and accurate method for measuring elapsed time. The clock is started immediately before the algorithm begins execution and stopped immediately after the algorithm finishes. The difference between the two time points represents the execution time of the algorithm. A simple C++ structure for measuring the running time is: ```cpp #include #include using namespace std; void algorithmA(int n) { // Algorithm A implementation } void algorithmB(int n) { // Algorithm B implementation } int main() { int n = 10000; auto startA = chrono::high_resolution_clock::now(); algorithmA(n); auto endA = chrono::high_resolution_clock::now(); auto timeA = chrono::duration_cast< chrono::microseconds>(endA - startA); auto startB = chrono::high_resolution_clock::now(); algorithmB(n); auto endB = chrono::high_resolution_clock::now(); auto timeB = chrono::duration_cast< chrono::microseconds>(endB - startB); cout << "Algorithm A: " << timeA.count() << " microseconds" << endl; cout << "Algorithm B: " << timeB.count() << " microseconds" << endl; return 0; } ``` Here, `high_resolution_clock::now()` records the current time. The clock is recorded once before the algorithm starts and once after it finishes. The subtraction of the two time points gives the elapsed duration. The duration can be expressed in nanoseconds, microseconds, milliseconds, or seconds depending on the requirement and execution time of the algorithm. For very fast algorithms, a single execution may not provide a