Time Complexity of an algorithm

Time Complexity ek tarika hai jisse hum yeh samajhte hain ki koi algorithm bada input (large input size N) milne par kitna time lega.

  • Yeh time ko seconds me measure nahi karta.
  • Yeh sirf fundamental operations (basic steps) count karta hai jo algorithm input size N ke hisab se perform karti hai.
  • Fundamental Operations me aate hain:
    • Assignment (value dena)
    • Comparison (tulna karna)
    • Arithmetic operations (plus, minus, multiply)

Kyun Zaroori Hai?

Time complexity batati hai ki algorithm kitni efficient hai. Jab input size N bahut badhta hai, toh hum dekhte hain ki algorithm ka execution time kis speed se badh raha hai.

Time Complexity Ko Kaise Denote Karte Hain? (Asymptotic Notations)

Time complexity ko 3 popular notations se express kiya jata hai, jinme Big O sabse zyada use hota hai.

1. Big O Notation (O): Upper Bound

  • Isko Worst-Case Time Complexity bhi bolte hain.
  • Yeh batata hai ki algorithm worst situation me kitna time legi.
  • Yeh guarantee karta hai ki algorithm ka time is limit se aage nahi badhega.
  • Example: Agar complexity O(N²) hai, matlab time N² se tez nahi badhega.

2. Omega Notation (Ω): Lower Bound

  • Isko Best-Case Time Complexity bolte hain.
  • Yeh batata hai ki algorithm kam se kam kitna time leg sakti hai.

3. Theta Notation (Θ): Tight Bound

  • Yeh batata hai ki actual time best aur worst ke beech me hota hai.
  • Accurate representation deta hai.

Common Time Complexities (Aam Time Complexities)

Algorithm design me kuch common time complexities hoti hain jinka use zyada hota hai.

Notation Name Growth with N Example
O(1) Constant Time Same rehta hai (input se farak nahi) Array index se element nikalna
O(log N) Logarithmic Time Bahut dheere badhta hai Binary Search
O(N) Linear Time Seedha badhta hai (input ke barabar) Array print karna
O(N log N) Linearithmic Time Tezi se badhta hai Merge Sort, Heap Sort
O(N²) Quadratic Time Bahut tezi se badhta hai Bubble Sort, Selection Sort
O(2ⁿ) Exponential Time Extreme growth (avoid karo) TSP (Brute Force)

Example: O(N) vs O(N²)

Maan lo N = 1000:

  • O(N): Algorithm ~1000 operations karegi (fast).
  • O(N²): Algorithm ~10,00,000 (1 million) operations karegi (slow).

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