Tabulation (bottom-up DP) — iterative: Question bank

3.7.8 · D5 · Coding › Algorithm Paradigms › Tabulation (bottom-up DP) — iterative

Shuru karne se pehle, teen words jinpe hum baar baar lean karenge — har ek parent note mein earn kiya gaya tha:

True or false — justify

TF1. "Tabulation aur memoization ki time complexity hamesha same hoti hai."

True — dono har distinct subproblem ko exactly ek baar compute karte hain; dono sirf is baat mein differ karte hain ki boxes tak kaise pahuncha jaaye (loop vs recursion), na ki kitne boxes exist karte hain. Dekho Memoization (top-down DP).

TF2. "Tabulation hamesha practice mein memoization se faster hoti hai."

False — asymptotic cost same hai, lekin tabulation call-stack overhead avoid karti hai aur constant factor se jeet sakti hai. Memoization faster ho sakti hai jab woh unreachable subproblems skip karti hai jo tabulation phir bhi fill karti hai.

TF3. "Har recursive DP ko tabulation ke roop mein rewrite kiya ja sakta hai."

Principle mein True — agar tum saare subproblem parameters list kar sako aur unhe is tarah order kar sako ki dependencies pehle aayein, toh tum unhe loop kar sakte ho. Catch yeh hai ki woh valid order dhundha jaaye.

TF4. "Loop shuru hone se pehle base cases fill karne zaroori hain."

True — base cases woh boxes hain jinmein koi incoming arrow nahi hota; recurrence wahan undefined hai. Unke bina loop pehle real box pe garbage read karta hai ya crash karta hai.

TF5. "Agar algorithm ke baad ek table entry

0

rehti hai, toh yeh hamesha ek bug signal karta hai."

False —

0

ek legitimate answer ho sakta hai (jaise dp[i][c]=0 0-1 Knapsack mein jab koi item fit nahi hota). Sirf ek unexpected

0

jahan recurrence fire honi chahiye thi, woh bug hai.

TF6. "Tabulation mein tumhe kabhi recursion stack ki zaroorat nahi hoti, isliye memory use hamesha memoization se kam hoti hai."

False — tum call stack drop karte ho, lekin table itself same size ki hoti hai. Net memory usually similar hoti hai; win stack se hai, table se nahi.

TF7. "Optimal substructure wala problem lekin bina overlapping subproblems ke — yeh ek accha tabulation candidate hai."

False — overlap ke bina reuse karne ke liye kuch nahi hota; plain divide-and-conquer ya greedy simpler hai. Tabulation tab pay off karta hai jab wahi box kaafi baar zaroori hota hai.

TF8. "Tabulation mein final answer hamesha table ke last cell mein hota hai."

False — usually hota hai (dp[n], dp[n][W]), lekin kabhi kabhi yeh kai cells pe max/sum hota hai (jaise longest-run problems jahan answer

max_{i} d p [i]

hota hai).

TF9. "Climbing Stairs aur Fibonacci ka recurrence same hai, isliye yeh same problem hain."

False — same recurrence shape

d p [i] = d p [i - 1] + d p [i - 2]

, lekin alag base cases (stairs ke liye dp[0]=1 vs Fibonacci ke liye dp[0]=0) aur alag meanings, isliye answers differ karte hain. Skeletons reuse hoti hain; states nahi.

Spot the error

SE1. "for i in range(n, 1, -1): dp[i]=dp[i-1]+dp[i-2] — Fibonacci ke liye main yeh loop karta hoon."

Wrong order — yeh pehle large indices fill karta hai, isliye dp[i-1] aur dp[i-2] read karte waqt abhi bhi empty hain. Dependencies pehle fill honi chahiye inhe read karne se pehle, isliye small → large iterate karo.

SE2. "1D knapsack: for c in range(W+1): dp[c]=max(dp[c], v+dp[c-w])."

Inner capacity loop low → high jaata hai, isliye dp[c-w] is pass mein already overwrite ho chuka tha — iska matlab item

i

ko reuse kiya ja raha hai (unbounded knapsack behaviour). 0/1 ke liye capacity high → low loop karo taaki dp[c-w] abhi bhi previous row hold kare. Dekho Space Optimization in DP.

SE3. "Main dp[0]=0 set karta hoon Climbing Stairs ke liye kyunki zero stairs climb karne ke zero ways hain."

Meaning pe off-by-one: dp[0] un ways ko count karta hai ki step 0 pe khade rehna (kuch na karo), jo exactly ek way hai. dp[0]=1 set karo, warna har count half ho jaata hai.

SE4. "Knapsack ke liye maine table dp = [[0]*W for _ in range(n)] size ki."

Do size bugs: capacity axis ko W+1 columns chahiye (indices

0 \dots W

), aur items axis ko n+1 rows chahiye taaki row

0

"no items" base case hold kar sake. Yahan off-by-one silently out of range read karta hai ya base row drop karta hai.

SE5. "Knapsack recurrence mein maine take-case ke liye v_i + dp[i][c-w_i] likha."

Previous row dp[i-1][c-w_i] padhna chahiye — item

i

lene ke baad tum ise dobara consider nahi kar sakte, isliye "pehle

i - 1

items" pe wapas jaate ho. Same row padhne se item

i

reuse ho jaata.

SE6. "Meri recurrence dp[i-2] padhti hai lekin maine sirf dp[0] initialise kiya, aur n=1 ke liye theek kaam karta hai."

n=1 ke liye loop range(2, n+1) empty hai, isliye bug chhupta hai. Yeh n≥2 pe bite karta hai jab dp[1] kabhi seed nahi kiya gaya aur pehli real iteration ek unset box padhti hai. Sirf n=1 nahi, sabse chhota non-trivial n test karo.

SE7. "Maine table sahi fill ki lekin Fibonacci ke liye dp[n-1] return kiya jahan dp[i]

i

-th number hai."

Answer cell pe off-by-one. dp[i] =

i

-th Fibonacci ke saath, answer dp[n] hai, dp[n-1] nahi. Answer cell locate karne ke liye hamesha apna Step-1 state sentence dobara padho.

Why questions

WH1. "Climbing Stairs do cases ka sum kyun hai lekin 0/1 Knapsack max kyun hai?"

Stairs paths ke disjoint sets ko count karta hai (last move 1 tha ya 2 — koi overlap nahi), isliye add karte hain. Knapsack do mutually exclusive options mein se best choose karta hai (take vs skip), isliye maximise karte hain. Counting → add; optimising → min/max.

WH2. "Fill order recursive call order ka reverse kyun hona chahiye?"

Recursion bade se shuru hoke sub-answers ke liye neeche poochta hai; tabulation ko pehle woh sub-answers produce karne chahiye. Isliye woh direction jisme recursion descend karti hai, exactly wahi direction hai jisme tabulation ascend karni chahiye.

WH3. "Optimal substructure recurrence ko justify kyun karta hai?"

Optimal Substructure ka matlab hai ki ek optimal whole, optimal parts se assemble hoti hai. Yahi license hai dp[big] ko dp[smaller] ke terms mein likhne ka — warna sub-answers combine karne se true optimum nahi milta. Dekho Recurrence Relations.

WH4. "Tabulation 'kabhi recompute nahi karta' jabki plain recursion karta hai — kyun?"

Plain recursion har us branch pe same subproblem dobara derive karta hai jisne ise maanga. Tabulation har box ek baar likhta hai aur baad ke har box woh stored value read karta hai — yahi reuse hai jo Dynamic Programming dilata hai.

WH5. "2D version mein knapsack capacity ko kisi bhi order mein iterate kar sakte hain lekin 1D mein nahi — kyun?"

2D mein, har read dp[i-1][...] hai — ek alag, already-complete row — isliye current row ka order irrelevant hai. 1D mein sirf ek hi row hai, isliye read-order achanak matter karta hai: high → low previous-row values protect karta hai.

WH6. "Agar memoization same complexity deti hai toh tabulation se kyun bothered hon?"

Koi recursion depth limit nahi (huge inputs stack overflow nahi karenge), koi call overhead nahi, aur explicit table rolling-array tricks obvious banata hai. Yeh order ke baare mein thodi upfront thinking ke badle robustness deta hai.

WH7. "Tabulation woh kaam kyun waste karta hai jo memoization avoid kar sakti hai?"

Tabulation range mein har box fill karta hai, chahe koi top-level query unhe kabhi na maange. Memoization sirf actually requested boxes ko touch karti hai. Sparse reachable states pe, top-down strictly kam kaam kar sakta hai.

Edge cases

EC1. "fib(0) kya return karta hai, aur loop run bhi karta hai kya?"

n < 2 guard ke through 0 return karta hai; loop range(2, 1) empty hai isliye kabhi execute nahi hota. Guard specifically n ∈ {0, 1} handle karne ke liye exist karta hai kisi bhi recurrence se pehle.

EC2. "Capacity W = 0 wala Knapsack — har answer kya hai?"

Saari dp[i][0] = 0: koi capacity nahi matlab kuch fit nahi hota, isliye take-branch condition w_i ≤ 0 positive weights ke liye false hai aur hum hamesha skip karte hain. Poora pehla column base value

0

pe rehta hai.

EC3. "Knapsack jahan ek item ka weight saari capacities ke liye W se zyada ho — kya hota hai uske saath?"

Yeh kabhi w_i ≤ c satisfy nahi karta, isliye uski row har jagah uske upar wali row ke barabar hai — item effectively invisible ho jaata hai. Sahi behaviour: ek unusable item kuch contribute nahi karta, koi special-casing zaroorat nahi.

EC4. "Climbing Stairs n = 0 ke saath — answer

0

hai ya

1

1 hai (dp[0]=1): already bottom pe khade rehne ka ek "empty" way. Yahan

0

return karna classic base-case sign error hai.

EC5. "Tabulation mein bahut bade n ke liye Fibonacci — pehle kya fail hota hai, aur hum ise kaise prevent karein?"

Size n+1 ki poori dp array memory blow kar deti hai time se bahut pehle; stack overflow nahi hota kyunki koi recursion nahi hai. Fix: do rolling variables (prev, curr) mein collapse karo kyunki sirf last do boxes kabhi bhi read hoti hain.

EC6. "Ek DP jiska base case ek poori row ya column hai, single cell nahi — kya tabulation phir bhi theek hai?"

Haan — tum main loop se pehle poori base row/column seed karte ho (jaise knapsack mein saari c ke liye dp[0][c]=0). "Base case" ka matlab sirf woh har box hai jinmein koi incoming arrow nahi, chahe kitne bhi hon.

EC7. "Agar do alag fill orders dono 'dependencies first' satisfy karte hain — kya answer change ho jaata hai?"

Nahi — dependency arrows ka koi bhi topological order identical table values deta hai, kyunki har box ke inputs fixed hain regardless ki baaki independent boxes kab fill hue. Correctness order ke valid hone pe depend karti hai, unique hone pe nahi.

EC8. "Recurrence mein negative ya fractional indices (jaise i=1 pe dp[i-2]) — inhe read karne se kaise bachein?"

Ya toh base cases ke saath guard karo jo sabse chhote states cover kare, ya loop aise index se shuru karo jahan saari reads in-range hon (range(2, ...)). Python mein ek unguarded dp[-1] silently last element read karta hai — ek subtle, dangerous bug.

Connections

Tabulation (bottom-up DP) — iterative — woh parent note jiske ye traps drill hain
Memoization (top-down DP) — TF1, TF2, WH2 ke liye contrast
Dynamic Programming — woh paradigm; Optimal Substructure, Overlapping Subproblems — recurrence kyun legal hai
0-1 Knapsack, Coin Change, Longest Common Subsequence — take/skip aur order traps ke sources
Space Optimization in DP — high→low capacity trap (SE2, WH5)
Recurrence Relations — WH3 ke peeche ka math