Worked examples — Naive pattern matching — O(nm)

3.8.1 · D3 · Coding › String Algorithms › Naive pattern matching — O(nm)

Shuru karne se pehle, ek reminder un vocabulary ka jo hum har jagah reuse karenge (sab parent mein defined hain):

• ==$T$== = text, length ==$n$==. Likha jata hai $T[0],T[1],\dots ,T[n-1]$ (0-indexed).
• ==$P$== = pattern, length ==$m$==, jahan $m\le n$.
• ==shift $s$== = ek candidate starting position; valid hota hai jab $T[s\dots s+m-1]=P$.
• comparison = ek single "kya $T[s+j]$ equal hai $P[j]$ se?" test. Hum kaam measure karne ke liye inhe lete hain.

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Scenario matrix

Naive matching ke har input ko inhi case classes mein se kisi ek mein rakha ja sakta hai. Har row ek alag cheez hai jo galat ho sakti hai ya specially behave karti hai, aur har ek ka ek worked example neeche diya gaya hai.

#	Case class	Isme kya khaas hai	Example
A	Single clean match	pattern exactly ek baar appear karta hai	Ex 1
B	No match at all	answer empty set hai	Ex 2
C	Overlapping matches	matches jinke ranges characters share karte hain	Ex 3
D	Worst case $\Theta (nm)$	har shift fail hone se pehle poora $m$ pay karta hai	Ex 4
E	Best/lucky case $\sim O(n)$	almost har shift char 0 par hi mar jata hai	Ex 5
F	Degenerate: $m=1$	pattern ek single character hai	Ex 6
G	Degenerate: $m=n$	pattern poore text ki length ka hai	Ex 7
H	Degenerate: empty pattern $m=0$	formula ki limiting boundary	Ex 8
I	Real-world word problem	DNA / search feature	Ex 9
J	Exam twist: comparisons exactly gino	"kahan" nahi balki "kitna kaam"	Ex 10

Hum har cell cover karenge.

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Ex 1 — Case A: ek single clean match

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Ex 2 — Case B: koi match nahi

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Ex 3 — Case C: overlapping matches

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Ex 4 — Case D: sach mein worst case $\Theta (nm)$

Dono extremes ko side by side dekho:

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Ex 5 — Case E: lucky / near-$O(n)$ case

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Ex 6 — Case F: single-character pattern ($m=1$)

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Ex 7 — Case G: pattern utna hi lamba jitna text ($m=n$)

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Ex 8 — Case H: empty pattern ($m=0$)

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Ex 9 — Case I: real-world word problem (DNA search)

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Ex 10 — Case J: exam twist — exactly gino, sirf locate mat karo

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Recall Tumhare liye kaunsa cell sabse mushkil tha?

Answers reveal karo inhe lock karne ke liye. Ex 3 vs Ex 9 — kya matches overlap karte hain? ::: Ex 3 overlap karta hai ($s=0,1,2$ letters share karte hain); Ex 9 nahi karta ($s=0,7$ exactly $m$ apart hain). Ex 4 vs Ex 5 — dono ka koi match nahi tha; cost alag kyun? ::: Ex 4 ka pattern prefix AAA text se match karta hai toh break kabhi jaldi fire nahi hota ($\Theta (nm)$); Ex 5 ka pehla letter A rare hai toh har shift char 0 par mar jaata hai ($O(n)$). Length n ke text ke liye kitne empty-pattern matches? ::: $n+1$ (har position $0\dots n$ par ek), $n-m+1$ se $m=0$ ke saath.

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Connections

• Naive pattern matching — O(nm) — wo parent jiske algorithm ko humne har case par exercise kiya.
• KMP Algorithm — Ex 4 ke worst case ko matched info reuse karke beat karta hai.
• Boyer-Moore Algorithm — Ex 5-style cases ko aur bhi zyaada beat karta hai aage skip karke.
• Rabin-Karp Algorithm — char-by-char ki jagah hashing se poori windows compare karta hai.
• Big-O Notation — "$\Theta (nm)$ worst, $O(n)$ lucky" ki language.
• Sliding Window Technique — har example ke peeche "pattern slide karo" ka motion.
• Substring Search Problem — wo umbrella problem jise saare das examples instantiate karte hain.

Case Map