3.1.1
Intuition Ek sentence mein ideaBig-O ek upper bound describe karta hai ki ek function kitni tezi se grow karta hai, constant factors aur chhote inputs ko ignore karke . Jab hum kehte hain "f ( n ) O ( g ( n )) f utni tez se zyada grow nahi karta jitna g n
Intuition Constants kyun hata dete hainMaano Algorithm A 5 n 100 n 5 100 nahi badalta woh hai shape : dono linearly grow karte hain. Big-O shape capture karta hai, machine nahi.
Usi tarah hum chhote n O ( n 2 ) n = 3 O ( n ) scaling ki parwah karte hain, yaani n → ∞
Toh Big-O jawab deta hai: "Jab input size blast hoti hai, cost roughly kitni badh jaati hai?"
Definition Big-O (formal)Maano f , g : N → R ≥ 0 f ( n ) = O ( g ( n )) == c > 0 == == n 0 ≥ 0 == 0 ≤ f ( n ) ≤ c ⋅ g ( n ) for all n ≥ n 0 .
Ise zor se padho: "Koi multiplier c n 0 n 0 f c g
Teen cheezein jo tumhe yaad rakhni CHAHIYE:
c constant multiplier (hidden coefficients ko absorb karta hai).n 0 threshold jiske baad bound hold karta hai (chhote-n Inequality one-sided hai (≤ upper bound hai.
Poora game hai: ek c n 0
Worked example Prove karo3 n + 7 = O ( n ) Goal: c , n 0 3 n + 7 ≤ c n n ≥ n 0
Step 1 — chhote term ko bade term se bound karo.
Yeh step kyun? n ≥ 7 7 ≤ n koi threshold chahiye; 7 3 n + 7 ≤ 3 n + n = 4 n ( n ≥ 7 ) . Yeh step kyun? Right side ab 4 ⋅ n = c g ( n ) g ( n ) = n c = 4 n 0 = 7 ■
(Bahut saare valid choices exist karte hain — jaise c = 10 , n 0 = 1 ek chahiye.)
Worked example Prove karo2 n 2 + 3 n + 1 = O ( n 2 ) Goal: 2 n 2 + 3 n + 1 ≤ c n 2 n ≥ n 0
Step 1 — har chhote term ko n 2 Yeh step kyun? n ≥ 1 3 n ≤ 3 n 2 1 ≤ n 2 2 n 2 + 3 n + 1 ≤ 2 n 2 + 3 n 2 + n 2 = 6 n 2 ( n ≥ 1 ) . c = 6 n 0 = 1 ■
Lesson: ek degree-d O ( n d )
Worked example Prove karof ( n ) = O ( g ( n )) f = n g = n 2 n ≥ 1 n ≤ n 2 c = 1 , n 0 = 1 n = O ( n 2 ) Yeh kyun matter karta hai: Big-O ek upper bound hai, toh ek function kisi aisi cheez ka O strictly faster grow kare. n = O ( n 2 ) loose hai (tight nahi).
Worked example DISPROVE karo:n 2 = O ( n ) Yeh step kyun? Maano yeh true hota: n 2 ≤ c n n ≥ n 0 n > 0 n ≤ c c fixed constant hai aur n n > c ( c , n 0 ) ■ todein : n → ∞
Recall Jawab padhne se pehle forecast karoQ: Kya 1000 n = O ( n )
Verify: HAAN. c = 1000 , n 0 = 0 1000 n ≤ 1000 ⋅ n point hai.
Recall Forecast karoQ: Kya 2 n = O ( n 100 )
Verify: NAHI. Koi bhi exponential eventually kisi bhi polynomial ko overtake kar leta hai: bade n 2 n / n 100 → ∞ c
exact running time hai."
Kyun sahi lagta hai: hum casually kehte hain "yeh O ( n ) n Fix: Big-O ek upper bound hai, equality nahi. f = O ( g ) f slower grow kare. Exact/tight bounds ke liye Θ f = O ( g ) = f ∈ O ( g ) set membership .
Common mistake "Mujhe constants aur lower-order terms rakhne chahiye."Kyun sahi lagta hai: real benchmarks mein constant matter karta hai!
Fix: Asymptotic world mein hum deliberately unhe drop karte hain: 5 n 2 + 99 n + 7 = O ( n 2 ) c
n 0 Kyun sahi lagta hai: lagta hai ki sabse chhota threshold dhundho.
Fix: Koi bhi valid n 0
f = O ( g ) g = O ( f ) Kyun sahi lagta hai: = Fix: Galat — yeh directional hai. n = O ( n 2 ) n 2 = O ( n ) ≤ =
Recall Ek 12-saal ke bacche ko explain karoImagine karo tum time kar rahe ho ki ek class mein candy baantne mein kitna time lagta hai. Agar n n "yeh bacchon ki ginti jaisa grow karta hai." Tumhe parwah nahi ki ek teacher twice as fast hai (woh ek constant hai), aur tumhe chhote setup time ki parwah nahi. Tumhe sirf ek cheez ki parwah hai: agar class double ho jaaye, toh kaam roughly double ho jaata hai. Big-O woh lazy-but-honest tarika hai yeh kehne ka ki kaam scale kaise karta hai jab cheezein bahut badi ho jaati hain.
"O for Over the top, after a while." O Over -estimate (upper bound), jo after a while hold karta hai (past n 0 × c
Big-O formal definition f ( n ) = O ( g ( n )) ∃ c > 0 , n 0 ≥ 0 0 ≤ f ( n ) ≤ c g ( n ) n ≥ n 0
Constant c Hidden coefficients / constant factors, taaki machine- aur implementation-dependent multipliers ignore ho jaayein.
n 0 Ek threshold; bound ko sirf sufficiently large n
Big-O upper bound hai ya lower bound? Sirf upper bound (inequality one-sided ≤ Ω Θ
Prove karo 3 n + 7 = O ( n ) n ≥ 7 3 n + 7 ≤ 3 n + n = 4 n c = 4 , n 0 = 7
n 2 = O ( n ) Require karta n ≤ c n c n → ∞
Big-O ka Sum rule O ( f ) + O ( g ) = O ( max ( f , g ))
Kya 2 n = O ( n 100 ) Nahi; exponentials eventually kisi bhi polynomial ko dominate karte hain.
Kya f = O ( g ) g = O ( f ) Nahi, Big-O directional hai (≤
True ya false: ek degree-d O ( n d ) True; har chhoti power ko n d
Behaviour as n to infinity
Polynomial is O of top power
