Tabulation Bottom Up DP in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is Tabulation in Dynamic Programming?

Tabulation is a bottom-up approach in dynamic programming where we solve smaller subproblems first and store their results in a table (usually an array) to build up solutions to bigger problems.

beginner

How does Tabulation differ from Memoization?

Tabulation solves problems iteratively from the smallest subproblem up, filling a table. Memoization solves problems recursively and stores results to avoid repeated work.

beginner

In Tabulation, why do we start filling the table from the base cases?

Base cases are the simplest subproblems with known answers. Starting from them ensures that when solving bigger subproblems, the smaller required answers are already computed and available.

intermediate

What is the time complexity advantage of using Tabulation?

Tabulation avoids repeated calculations by storing results in a table, so each subproblem is solved once. This usually reduces exponential time to polynomial time.

beginner

Give an example problem where Tabulation Bottom Up DP is useful.

The Fibonacci sequence calculation is a classic example. Using tabulation, we fill an array from the first two Fibonacci numbers up to the desired number, avoiding repeated recursive calls.

What does Tabulation in DP primarily use to store intermediate results?

A A queue

B An array or table

C A stack

D A linked list

Tabulation uses an array or table to store results of subproblems for bottom-up computation.

Which approach solves subproblems recursively and stores results to avoid recomputation?

A Memoization

B Brute force

C Greedy

D Tabulation

Memoization is the top-down recursive approach that caches results.

In Tabulation, what is the first step to solve a problem?

A Call the recursive function

B Solve the biggest subproblem first

C Initialize the table with base case values

D Sort the input data

Tabulation starts by initializing base cases in the table.

What is a key benefit of Tabulation over naive recursion?

A Does not require base cases

B Uses less memory

C Runs slower

D Avoids repeated calculations

Tabulation avoids repeated calculations by storing subproblem results.

Which problem is commonly solved using Tabulation Bottom Up DP?

A Finding Fibonacci numbers

B Sorting an array

C Binary search

D Depth-first search

Fibonacci sequence calculation is a classic example for tabulation.

Explain how Tabulation Bottom Up DP works using the Fibonacci sequence as an example.

Think about how you can build Fibonacci numbers from the first two numbers upwards.

Describe the main differences between Tabulation and Memoization in dynamic programming.

Consider how each approach solves subproblems and stores answers.

Practice

1. What is the main idea behind Tabulation Bottom Up DP in dynamic programming?

easy

A. Build a table from the smallest subproblems up to the main problem

B. Solve the problem by recursion without storing results

C. Use random guesses to find the solution

D. Start solving from the biggest problem and break it down

Solution

1. Step 1: Understand the DP approach

   Tabulation means filling a table starting from the smallest subproblems.

2. Step 2: Compare with other methods

   Unlike recursion without storage, tabulation avoids repeated work by building up solutions.

3. Final Answer:

   Build a table from the smallest subproblems up to the main problem -> Option A

4. Quick Check:

   Bottom Up DP = Build table from small to big [OK]

Hint: Think: solve small parts first, then bigger ones [OK]

Common Mistakes:

• Confusing tabulation with recursion without memoization
• Thinking tabulation starts from the biggest problem
• Assuming tabulation guesses solutions randomly

2. Which of the following is the correct way to declare a DP table for Fibonacci numbers using tabulation in C?

easy

A. int dp[] = {0};

B. int dp[100] = {0};

C. int dp = 100;

D. int dp(100);

Solution

1. Step 1: Check C array declaration syntax

   In C, arrays are declared with size and type, e.g., int dp[100];

2. Step 2: Identify correct initialization

   int dp[100] = {0}; declares an int array of size 100 initialized to zero, suitable for DP table.

3. Final Answer:

   int dp[100] = {0}; -> Option B

4. Quick Check:

   C arrays need size and type [OK]

Hint: Use int dp[size] = {0} for zero-initialized arrays [OK]

Common Mistakes:

• Using parentheses instead of brackets for arrays
• Not specifying array size
• Assigning integer to array variable

3. What will be the output of the following C code using bottom-up DP to find Fibonacci number for n=5?

#include 
int main() {
    int n = 5;
    int dp[6] = {0};
    dp[0] = 0;
    dp[1] = 1;
    for(int i = 2; i <= n; i++) {
        dp[i] = dp[i-1] + dp[i-2];
    }
    printf("%d", dp[n]);
    return 0;
}

medium

A. 13

B. 8

C. 3

D. 5

Solution

1. Step 1: Trace DP table filling

   dp[0]=0, dp[1]=1; then dp[2]=1, dp[3]=2, dp[4]=3, dp[5]=5.

2. Step 2: Identify dp[n] value

   dp[5] = 5, so output is 5.

3. Final Answer:

   5 -> Option D

4. Quick Check:

   Fibonacci(5) = 5 [OK]

Hint: Remember Fibonacci sequence: 0,1,1,2,3,5... [OK]

Common Mistakes:

• Off-by-one errors in loop range
• Confusing dp indices with Fibonacci numbers
• Printing dp[n-1] instead of dp[n]

4. Find the error in this bottom-up DP code snippet for Fibonacci numbers:

int fib(int n) {
    int dp[n+1];
    dp[0] = 0;
    dp[1] = 1;
    for(int i = 1; i <= n; i++) {
        dp[i] = dp[i-1] + dp[i-2];
    }
    return dp[n];
}

medium

A. Array dp is not initialized to zero

B. Return statement should be dp[n-1]

C. Loop starts at 1 instead of 2 causing invalid access

D. Function should return void

Solution

1. Step 1: Check loop start index

   Loop starts at i=1, but dp[i-2] for i=1 is dp[-1], invalid index.

2. Step 2: Correct loop start

   Loop should start at i=2 to avoid negative index access.

3. Final Answer:

   Loop starts at 1 instead of 2 causing invalid access -> Option C

4. Quick Check:

   Loop must start at 2 to access dp[i-2] safely [OK]

Hint: Start loop at 2 for Fibonacci tabulation [OK]

Common Mistakes:

• Starting loop at 1 causing negative index
• Ignoring array bounds in DP
• Returning wrong dp index

5. Given the problem to find the minimum number of coins to make amount 7 using coins {1, 3, 4} with bottom-up DP, what is the minimum number of coins needed? Use this approach:

int coins[] = {1, 3, 4};
int amount = 7;
int dp[8];
dp[0] = 0;
for (int i = 1; i <= amount; i++) {
    dp[i] = INT_MAX;
    for (int j = 0; j < 3; j++) {
        if (coins[j] <= i && dp[i - coins[j]] != INT_MAX) {
            dp[i] = dp[i] < dp[i - coins[j]] + 1 ? dp[i] : dp[i - coins[j]] + 1;
        }
    }
}
printf("%d", dp[amount]);

hard

A. 2

B. 4

C. 3

D. 5

Solution

1. Step 1: Calculate dp values step-by-step

   dp[0]=0 For i=1: dp[1]=1 (coin 1) For i=2: dp[2]=2 (1+1) For i=3: dp[3]=1 (coin 3) For i=4: dp[4]=1 (coin 4) For i=5: dp[5]=2 (4+1) For i=6: dp[6]=2 (3+3) For i=7: dp[7]=2 (3+4)

2. Step 2: Identify minimum coins for amount 7

   Minimum coins needed for 7 is 2.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Minimum coins for 7 = 2 [OK]

Hint: Build dp from 0 to amount, pick min coins each step [OK]

Common Mistakes:

• Not updating dp[i] with minimum value
• Ignoring INT_MAX checks causing wrong results
• Confusing coin values and indices