Overlapping Subproblems and Optimal Substructure in DSA Go - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Fibonacci with Memoization

What is the output of the following Go code that calculates Fibonacci numbers using memoization?

DSA Go

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))
}

A 3

B 8

C 13

D 5

💡 Hint

Remember that Fibonacci sequence starts with 0, 1, 1, 2, 3, 5...

The function calculates Fibonacci numbers with memoization. fib(5) returns 5 because the sequence is 0,1,1,2,3,5...

❓ Predict Output

intermediate

Output of Recursive Coin Change Count

What is the output of this Go code that counts ways to make change using recursion with overlapping subproblems?

DSA Go

package main
import "fmt"

func countWays(coins []int, m int, n int) int {
    if n == 0 {
        return 1
    }
    if n < 0 || m <= 0 {
        return 0
    }
    return countWays(coins, m-1, n) + countWays(coins, m, n-coins[m-1])
}

func main() {
    coins := []int{1, 2, 3}
    fmt.Println(countWays(coins, len(coins), 4))
}

A 4

B 7

C 6

D 5

💡 Hint

Count all combinations of coins that sum to 4.

The function counts all ways to make 4 using coins 1,2,3. The total ways are 4.

🔧 Debug

advanced

Identify the error in memoized Fibonacci

What error does this Go code produce when run, and why?

DSA Go

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() {
    var memo map[int]int
    fmt.Println(fib(5, memo))
}

A panic: assignment to entry in nil map

B 0

C 5

D compile error: undefined variable memo

💡 Hint

Check how the map is initialized before use.

The map memo is declared but not initialized, so writing to it causes a runtime panic.

🚀 Application

advanced

Optimal Substructure in Longest Common Subsequence

Given two strings, which statement best describes the optimal substructure property used in the Longest Common Subsequence (LCS) problem?

A LCS length is always the sum of lengths of both strings

B If the last characters match, LCS length is 1 plus LCS of remaining strings; else max of LCS excluding last char of either string

C LCS length depends only on the first character of both strings

D LCS length is the minimum length of the two strings

💡 Hint

Think about how the problem breaks down into smaller subproblems.

The LCS problem uses optimal substructure by relating the solution to smaller subproblems depending on character matches.

🧠 Conceptual

expert

Why Overlapping Subproblems Matter in Dynamic Programming

Which of the following best explains why overlapping subproblems are important for dynamic programming?

A Because subproblems never repeat, so each must be solved independently

B Because overlapping subproblems mean the problem cannot be solved recursively

C Because solving the same subproblems multiple times wastes time, so storing results avoids recomputation

D Because overlapping subproblems increase the space complexity exponentially

💡 Hint

Think about how dynamic programming improves efficiency.

Dynamic programming stores results of subproblems to avoid repeated work, which is possible only if subproblems overlap.

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