Tabulation Bottom Up DP in DSA Rust - 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 caches results to avoid repeated work.

beginner

In Tabulation, why do we start filling the table from the smallest subproblem?

Because bigger problems depend on smaller subproblems, starting from the smallest ensures all needed results are ready when solving bigger problems.

beginner

What is the typical data structure used in Tabulation?

Usually, an array or a 2D array (table) is used to store results of subproblems for quick lookup.

beginner

Show the base case initialization in Tabulation for the Fibonacci sequence.

For Fibonacci, initialize dp[0] = 0 and dp[1] = 1 because these are the smallest subproblems with known answers.

What is the main idea behind Tabulation in dynamic programming?

A Guess the answer and check correctness

B Solve subproblems iteratively from smallest to largest and store results

C Solve subproblems recursively and store results

D Use randomization to find solutions

Tabulation solves subproblems iteratively from smallest to largest and stores their results in a table.

Which of these is a benefit of Tabulation over Memoization?

A Slower runtime

B Uses more memory

C Avoids recursion and stack overflow

D Does not store subproblem results

Tabulation uses iteration, so it avoids recursion and potential stack overflow.

In Tabulation, what do you do after filling the table?

A Return the value for the largest problem from the table

B Start recursion again

C Discard the table

D Restart from base case

After filling the table, the answer to the original problem is stored in the table and can be returned directly.

What is the first step when using Tabulation for a problem?

A Define the table size and initialize base cases

B Write recursive calls

C Guess the answer

D Use random values

You first define the table size and initialize base cases before filling the table.

Which problem is a classic example to solve using Tabulation?

A Finding maximum in array

B Sorting an array

C Binary search

D Fibonacci sequence

Fibonacci sequence is a classic example where Tabulation is used to avoid repeated calculations.

Explain how Tabulation works in dynamic programming with an example.

Think about solving small problems first and using their answers to solve bigger problems.

Describe the steps to implement Tabulation for a problem.

Focus on how you prepare and fill the table step by step.

Practice

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

easy

A. Use random guesses to find the solution

B. Solve the problem by recursion without storing results

C. Start solving from the main problem and break it down recursively

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

Solution

1. Step 1: Understand bottom-up approach

   Tabulation means filling a table starting from the smallest subproblems, building up to the final answer.

2. Step 2: Compare with other methods

   Unlike recursion without memoization, bottom-up avoids repeated work by storing results in a table.

3. Final Answer:

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

4. Quick Check:

   Tabulation = Bottom-up DP [OK]

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

Common Mistakes:

• Confusing bottom-up with top-down recursion
• Ignoring the use of a table to store results
• Thinking tabulation uses random guesses

2. Which of the following Rust code snippets correctly initializes a vector for tabulation with size 5 and all zeros?

easy

A. let mut dp = vec![0; 5];

B. let dp = vec![0;];

C. let mut dp = Vec::new(5);

D. let dp = vec!(5, 0);

Solution

1. Step 1: Check Rust vector initialization syntax

   To create a vector of size 5 filled with zeros, use vec![0; 5].

2. Step 2: Analyze options

   let mut dp = vec![0; 5]; uses correct syntax. let dp = vec![0;]; is missing size. let mut dp = Vec::new(5); is invalid because Vec::new() takes no size. let dp = vec!(5, 0); uses wrong macro syntax.

3. Final Answer:

   let mut dp = vec![0; 5]; -> Option A

4. Quick Check:

   vec![value; size] creates vector [OK]

Hint: Use vec![value; size] to initialize vectors [OK]

Common Mistakes:

• Using Vec::new() with size argument
• Incorrect macro syntax with vec!()
• Omitting size in vector initialization

3. What will be the output of the following Rust code using bottom-up DP to compute Fibonacci numbers?

fn fib(n: usize) -> usize {
    let mut dp = vec![0; n + 1];
    dp[1] = 1;
    for i in 2..=n {
        dp[i] = dp[i - 1] + dp[i - 2];
    }
    dp[n]
}

fn main() {
    println!("{}", fib(6));
}

medium

A. 13

B. 5

C. 8

D. 21

Solution

1. Step 1: Trace Fibonacci calculation for n=6

   dp array fills as: dp[0]=0, dp[1]=1, dp[2]=1, dp[3]=2, dp[4]=3, dp[5]=5, dp[6]=8.

2. Step 2: Output dp[6]

   The function returns dp[6] which is 8.

3. Final Answer:

   8 -> Option C

4. Quick Check:

   fib(6) = 8 [OK]

Hint: Remember Fibonacci sequence: 0,1,1,2,3,5,8... [OK]

Common Mistakes:

• Starting dp[1] as 0 instead of 1
• Off-by-one errors in loop range
• Confusing dp indices with Fibonacci numbers

4. Identify the error in this Rust bottom-up DP code for computing factorial and select the fix:

fn factorial(n: usize) -> usize {
    let mut dp = vec![1; n + 1];
    for i in 1..=n {
        dp[i] = dp[i - 1] * i;
    }
    dp[n]
}

fn main() {
    println!("{}", factorial(5));
}

medium

A. Initialize dp with zeros instead of ones

B. Change loop to start from 2 instead of 1

C. Use dp[i] = dp[i] * i instead of dp[i - 1] * i

D. Return dp[n-1] instead of dp[n]

Solution

1. Step 1: Analyze initialization and loop

   dp[0] = 1 is correct for factorial base case. Loop starting at 1 causes dp[1] = dp[0] * 1 = 1, which is correct but redundant.

2. Step 2: Fix loop start to avoid redundant calculation

   Starting loop at 2 avoids unnecessary multiplication and matches factorial definition better.

3. Final Answer:

   Change loop to start from 2 instead of 1 -> Option B

4. Quick Check:

   Factorial loop starts at 2 [OK]

Hint: Factorial base case at dp[0], loop from 2 [OK]

Common Mistakes:

• Starting loop at 1 causing redundant calculation
• Initializing dp with zeros breaking multiplication
• Returning wrong dp index

5. You want to find the minimum number of coins needed to make amount 7 using coins [1, 3, 4] with bottom-up DP in Rust. Which code snippet correctly implements this?

hard

A. let mut dp = vec![usize::MAX; 8]; dp[0] = 0; for i in 1..=7 { for &coin in &[1,3,4] { if i >= coin && dp[i - coin] != usize::MAX { dp[i] = dp[i].min(dp[i - coin] + 1); } } } dp[7]

B. let mut dp = vec![0; 8]; for i in 1..=7 { for &coin in &[1,3,4] { dp[i] += dp[i - coin]; } } dp[7]

C. let mut dp = vec![usize::MAX; 7]; dp[0] = 0; for i in 1..7 { for &coin in &[1,3,4] { if i > coin { dp[i] = dp[i - coin] + 1; } } } dp[6]

D. let mut dp = vec![0; 7]; dp[0] = 1; for i in 1..7 { for &coin in &[1,3,4] { if i >= coin { dp[i] = dp[i - coin] + 1; } } } dp[6]

Solution

1. Step 1: Initialize dp with max values and base case

   dp size must be 8 for amounts 0 to 7. dp[0] = 0 means zero coins needed for amount 0.

2. Step 2: Update dp with minimum coins for each amount

   For each amount i, check all coins. If coin fits and previous dp is valid, update dp[i] with minimum coins.

3. Step 3: Return dp[7]

   dp[7] holds minimum coins needed for amount 7.

4. Final Answer:

   let mut dp = vec![usize::MAX; 8]; dp[0] = 0; for i in 1..=7 { for &coin in &[1,3,4] { if i >= coin && dp[i - coin] != usize::MAX { dp[i] = dp[i].min(dp[i - coin] + 1); } } } dp[7] -> Option A

5. Quick Check:

   Minimum coins DP uses min and checks valid previous states [OK]

Hint: Use usize::MAX and min() to track minimum coins [OK]

Common Mistakes:

• Incorrect dp size causing index errors
• Not checking if previous dp value is valid
• Adding instead of taking minimum coins