Overlapping Subproblems and Optimal Substructure in DSA Go - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to initialize the memoization map for Fibonacci.

DSA Go

memo := make(map[int]int)
result := fib([1], memo)

Drag options to blanks, or click blank then click option'

A10

Bn

C0

D1

Attempts:

3 left

💡 Hint

Common Mistakes

Passing 0 or 1 as input which returns base cases immediately.

Passing variable 'n' without defining it.

We call fib with 10 to compute the 10th Fibonacci number using memoization.

2fill in blank

medium

Complete the code to check if the Fibonacci value is already computed in memo.

DSA Go

if val, ok := memo[[1]]; ok {
    return val
}

Drag options to blanks, or click blank then click option'

Aresult

Bmemo

Cn

Dfib

Attempts:

3 left

💡 Hint

Common Mistakes

Using memo as key instead of n.

Using result or fib which are not keys.

We check if the current input n is already in the memo map to avoid recomputation.

3fill in blank

hard

Fix the error in the recursive Fibonacci call to use memoization correctly.

DSA Go

memo[[1]] = fib(n-1, memo) + fib(n-2, memo)

Drag options to blanks, or click blank then click option'

An

Bresult

Cval

Dmemo

Attempts:

3 left

💡 Hint

Common Mistakes

Using memo or val as keys instead of n.

Not storing the result in memo at all.

We store the computed Fibonacci value for n in memo[n] to reuse later.

4fill in blank

hard

Fill both blanks to complete the recursive Fibonacci function with memoization.

DSA Go

func fib(n int, memo map[int]int) int {
    if n <= 1 {
        return n
    }
    if val, ok := memo[[1]]; ok {
        return val
    }
    memo[[2]] = fib(n-1, memo) + fib(n-2, memo)
    return memo[n]
}

Drag options to blanks, or click blank then click option'

An

Bmemo

Cresult

Dval

Attempts:

3 left

💡 Hint

Common Mistakes

Using different keys for lookup and assignment.

Using variables like val or memo as keys.

Both blanks use n as the key to check and store Fibonacci values in memo.

5fill in blank

hard

Fill all three blanks to create a map comprehension that filters Fibonacci numbers greater than 5.

DSA Go

filtered := map[int]int{}
for k, v := range memo {
    if v [1] 5 {
        filtered[[2]] = [3]
    }
}

Drag options to blanks, or click blank then click option'

A>

Bk

Cv

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Using < instead of > in condition.

Swapping key and value in assignment.

We filter memo to keep entries where value v is greater than 5, copying key k and value v.

Practice

1. Which of the following best describes the Overlapping Subproblems property in dynamic programming?

easy

A. The problem requires sorting before solving.

B. The problem has a unique solution without repeating any subproblems.

C. The problem can be broken down into smaller subproblems which are solved multiple times.

D. The problem can only be solved using recursion without memoization.

Solution

1. Step 1: Understand Overlapping Subproblems

   Overlapping subproblems means the same smaller problems appear multiple times during the solution process.

2. Step 2: Match the definition with options

   The problem can be broken down into smaller subproblems which are solved multiple times. correctly states that subproblems are solved multiple times, which is the key idea behind overlapping subproblems.

3. Final Answer:

   The problem can be broken down into smaller subproblems which are solved multiple times. -> Option C

4. Quick Check:

   Overlapping Subproblems = repeated smaller problems [OK]

Hint: Look for repeated smaller problems solved multiple times [OK]

Common Mistakes:

• Confusing overlapping subproblems with unique subproblems
• Thinking sorting is required for overlapping subproblems
• Assuming recursion alone solves overlapping subproblems

2. Which Go code snippet correctly declares a memoization map for storing integer results keyed by integer inputs?

easy

A. memo := make([]int, 10)

B. memo := map[string]int{}

C. memo := []int{}

D. memo := make(map[int]int)

Solution

1. Step 1: Identify correct map declaration for memoization

   Memoization requires a map with keys and values. Here keys and values are integers.

2. Step 2: Check each option

   memo := make(map[int]int) uses make(map[int]int) which creates a map with int keys and int values, perfect for memoization.

3. Final Answer:

   memo := make(map[int]int) -> Option D

4. Quick Check:

   Map[int]int for memoization = memo := make(map[int]int) [OK]

Hint: Use make(map[int]int) for int-to-int memo maps [OK]

Common Mistakes:

• Using slices instead of maps for memoization
• Using wrong key types like string when int is needed
• Confusing slice initialization with map initialization

3. What is the output of the following Go code implementing Fibonacci with memoization?

package main
import "fmt"

func fib(n int, memo map[int]int) int {
    if n <= 1 {
        return n
    }
    if val, ok := memo[n]; ok {
        return val
    }
    memo[n] = fib(n-1, memo) + fib(n-2, memo)
    return memo[n]
}

func main() {
    memo := make(map[int]int)
    fmt.Println(fib(5, memo))
}

medium

A. 5

B. 8

C. 3

D. 13

Solution

1. Step 1: Understand Fibonacci calculation with memo

   fib(5) = fib(4) + fib(3), and so on, with memo storing results to avoid repeats.

2. Step 2: Calculate fib(5)

   fib(0)=0, fib(1)=1, fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib(5) = 5 [OK]

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

Common Mistakes:

• Confusing fib(5) with fib(6)
• Ignoring memoization effect and recalculating
• Mixing up base cases n<=1

4. Identify the bug in this Go code snippet for memoized Fibonacci:

func fib(n int, memo map[int]int) int {
    if n == 0 {
        return 0
    }
    if n == 1 {
        return 1
    }
    if val, ok := memo[n]; ok {
        return val
    }
    memo[n] = fib(n-1, memo) + fib(n-2, memo)
    return memo[n]
}

medium

A. Memo map is not initialized before use

B. Base case should be n <= 1 instead of separate checks

C. Base case for n == 0 is missing

D. Function does not return memo[n] after calculation

Solution

1. Step 1: Check base cases and memo usage

   Base cases for n==0 and n==1 are correctly handled. Memo is used properly inside function.

2. Step 2: Identify initialization issue

   The function expects memo map passed in, but if not initialized before calling, it causes runtime error.

3. Final Answer:

   Memo map is not initialized before use -> Option A

4. Quick Check:

   Uninitialized map causes runtime panic [OK]

Hint: Always initialize maps before passing to functions [OK]

Common Mistakes:

• Assuming base cases are wrong when they are correct
• Forgetting to initialize memo map before first call
• Missing return statement after memo assignment

5. You want to find the minimum cost path in a grid from top-left to bottom-right, moving only right or down. The cost grid is:

cost := [][]int{
  {1, 3, 5},
  {2, 1, 2},
  {4, 3, 1},
}

Which approach best uses Overlapping Subproblems and Optimal Substructure to solve this efficiently?

hard

A. Use recursion without memoization to explore all paths

B. Use dynamic programming with a 2D memo table storing minimum cost to each cell

C. Sort the grid and pick the smallest costs in order

D. Use a greedy approach picking the smallest cost neighbor at each step

Solution

1. Step 1: Understand problem properties

   Minimum cost path has optimal substructure: minimum cost to a cell depends on minimum cost to neighbors.

2. Step 2: Choose approach using overlapping subproblems

   Dynamic programming with a 2D memo table stores results of subproblems (cells), avoiding repeated calculations.

3. Final Answer:

   Use dynamic programming with a 2D memo table storing minimum cost to each cell -> Option B

4. Quick Check:

   DP with memo table = efficient solution [OK]

Hint: Use DP table to store subproblem results for grid paths [OK]

Common Mistakes:

• Trying recursion without memo causes exponential time
• Sorting grid ignores path constraints
• Greedy approach may miss global minimum path