Big-Theta (Θ) notation is a fundamental concept in algorithm analysis, used to describe the tight bound of an algorithm's time or space complexity. It signifies that an algorithm's performance is bounded both from above and below by a specific function, meaning its growth rate is precisely matched by that function.

Big-Theta notation, denoted as <i>f(n) = Θ(g(n))</i>, means that for sufficiently large values of <i>n</i>, the function <i>f(n)</i> is bounded both above and below by a constant multiple of <i>g(n)</i>. Mathematically, this means there exist positive constants <i>c1</i>, <i>c2</i>, and <i>n0</i> such that <i>0 ≤ c1 * g(n) ≤ f(n) ≤ c2 * g(n)</i> for all <i>n ≥ n0</i>.

An algorithm is Θ(g(n)) if and only if it is both O(g(n)) and Ω(g(n)). This means the algorithm's growth rate is precisely characterized by g(n), neither faster nor slower in the long run.

Big-Theta notation is crucial for comparing algorithms directly. If Algorithm A is Θ(n^2) and Algorithm B is Θ(n log n), we can definitively say that Algorithm B is more efficient for large inputs because n log n grows slower than n^2. This precise understanding helps in selecting the most optimal algorithm for a given problem.

Consider a simple linear search algorithm. In the worst case, it might examine every element, giving it a Big-O of O(n). In the best case, it finds the element immediately, giving it a Big-Omega of Ω(1). However, on average, it examines about n/2 elements. Since n/2 is proportional to n, the tight bound is Θ(n).

Understanding common complexities is key for competitive exams:

• Θ(1): Constant time (e.g., accessing an array element by index)
• Θ(log n): Logarithmic time (e.g., binary search)
• Θ(n): Linear time (e.g., linear search, iterating through a list)
• Θ(n log n): Log-linear time (e.g., efficient sorting algorithms like Merge Sort, Quick Sort)
• Θ(n^2): Quadratic time (e.g., bubble sort, selection sort)
• Θ(2^n): Exponential time (e.g., brute-force solutions for some problems)