Overlapping Subproblems and Optimal Substructure in DSA Python - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to return the Fibonacci number for n using recursion.

DSA Python

def fib(n):
    if n <= 1:
        return n
    return fib(n-1) [1] fib(n-2)

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

A+

B-

C*

D/

Attempts:

3 left

💡 Hint

Common Mistakes

Using subtraction instead of addition.

Using multiplication or division which is incorrect for Fibonacci.

The Fibonacci number is the sum of the two previous Fibonacci numbers: fib(n-1) + fib(n-2).

2fill in blank

medium

Complete the code to store computed Fibonacci values in a dictionary to avoid repeated calculations.

DSA Python

def fib_memo(n, memo={}):
    if n in memo:
        return memo[n]
    if n <= 1:
        return n
    memo[n] = fib_memo(n-1, memo) [1] fib_memo(n-2, memo)
    return memo[n]

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

A*

B-

C/

D+

Attempts:

3 left

💡 Hint

Common Mistakes

Not adding the two recursive calls.

Forgetting to store the result in memo.

We add the two previous Fibonacci numbers and store the result in memo to avoid repeated work.

3fill in blank

hard

Fix the error in the code to correctly implement the bottom-up dynamic programming approach for Fibonacci.

DSA Python

def fib_dp(n):
    dp = [0, 1] + [0] * (n-1)
    for i in range(2, n+1):
        dp[i] = dp[i-1] [1] dp[i-2]
    return dp[n]

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

A/

B+

C*

D-

Attempts:

3 left

💡 Hint

Common Mistakes

Using subtraction or multiplication instead of addition.

Incorrectly initializing the dp list size.

Each Fibonacci number is the sum of the two previous numbers in the dp list.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each number to its square only if the number is {{BLANK_1}} 5 and {{BLANK_2}} 10.

DSA Python

squares = {x: x**2 for x in range(1, 15) if x [1] 5 and x [2] 10}

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

A>

B<

C>=

D<=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '>=' or '<=' which includes boundary numbers.

Swapping the comparison operators.

We want numbers greater than 5 and less than 10 to be included in the dictionary.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps each word in words to its length only if the length is {{BLANK_1}} 3 and the word starts with {{BLANK_2}} 'a' or {{BLANK_3}} 'b'.

DSA Python

words = ['apple', 'bat', 'cat', 'ant', 'ball', 'dog']
result = {word: len(word) for word in words if len(word) [1] 3 and (word.startswith([2]) or word.startswith([3]))}

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

A==

B'a'

C'b'

D!=

Attempts:

3 left

💡 Hint

Common Mistakes

Using '!=' instead of '==' for length check.

Not quoting the letters in startswith.

We select words with length exactly 3 and starting with 'a' or 'b'.

Practice

1. Which of the following best describes the overlapping subproblems property in dynamic programming?

easy

A. The problem can be broken down into smaller subproblems that are reused multiple times.

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

C. The problem requires sorting before solving.

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 solving.

2. Step 2: Match definition to options

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

3. Final Answer:

   The problem can be broken down into smaller subproblems that are reused multiple times. -> Option A

4. Quick Check:

   Overlapping subproblems = reuse of subproblems [OK]

Hint: Look for repeated smaller problems in the main problem [OK]

Common Mistakes:

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

2. Which of the following Python function definitions correctly uses memoization to solve overlapping subproblems?

easy

A. def fib(n): if n <= 1: return n else: return fib(n-1) + fib(n-2)

B. def fib(n): return n * fib(n-1)

C. def fib(n): for i in range(n): print(i)

D. def fib(n, memo={}): if n in memo: return memo[n] if n <= 1: memo[n] = n else: memo[n] = fib(n-1, memo) + fib(n-2, memo) return memo[n]

Solution

1. Step 1: Identify memoization usage

   Memoization stores results of subproblems to avoid repeated work, usually with a dictionary.

2. Step 2: Check each option

   def fib(n, memo={}): if n in memo: return memo[n] if n <= 1: memo[n] = n else: memo[n] = fib(n-1, memo) + fib(n-2, memo) return memo[n] uses a dictionary 'memo' to save and reuse results, correctly implementing memoization.

3. Final Answer:

   def fib(n, memo={}): if n in memo: return memo[n] if n <= 1: memo[n] = n else: memo[n] = fib(n-1, memo) + fib(n-2, memo) return memo[n] -> Option D

4. Quick Check:

   Memoization = caching results [OK]

Hint: Look for a cache (dictionary) storing computed results [OK]

Common Mistakes:

• Missing memo dictionary parameter
• Not returning cached results
• Using loops without caching results

3. What is the output of the following code?

def fib(n, memo={}):
    if n in memo:
        return memo[n]
    if n <= 1:
        memo[n] = n
    else:
        memo[n] = fib(n-1, memo) + fib(n-2, memo)
    return memo[n]

print(fib(5))

medium

A. 13

B. 8

C. 5

D. 3

Solution

1. Step 1: Trace fib(5) calls with memoization

   fib(5) calls fib(4) and fib(3), which further call smaller fib values, caching 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 C

4. Quick Check:

   Fibonacci(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
• Miscounting base cases

4. The following code attempts to use memoization for Fibonacci numbers but has a bug. What is the error?

def fib(n, memo={}):
    if n <= 1:
        return n
    if n not in memo:
        memo[n] = fib(n-1) + fib(n-2)
    return memo[n]

print(fib(4))

medium

A. Missing memo argument in recursive calls causes infinite recursion.

B. Base case is incorrect for n <= 1.

C. Memo dictionary is not initialized.

D. Return statement is missing.

Solution

1. Step 1: Check recursive calls

   Recursive calls fib(n-1) and fib(n-2) do not pass 'memo', so memo is empty each time.

2. Step 2: Understand effect of missing memo argument

   Without passing memo, caching fails and recursion repeats infinitely, causing a stack overflow.

3. Final Answer:

   Missing memo argument in recursive calls causes infinite recursion. -> Option A

4. Quick Check:

   Pass memo in recursive calls to fix [OK]

Hint: Always pass memo/cache in recursive calls [OK]

Common Mistakes:

• Forgetting to pass memo in recursive calls
• Assuming base case alone stops recursion
• Not initializing memo outside function

5. You want to find the minimum cost to reach the top of stairs where each step has a cost. You can climb 1 or 2 steps at a time. Which approach best uses optimal substructure and overlapping subproblems to solve this?

hard

A. Use recursion without caching to try all paths.

B. Use dynamic programming to store minimum cost for each step and build up solution.

C. Sort the costs and pick the smallest steps first.

D. Use a greedy approach to always pick the next step with minimum cost.

Solution

1. Step 1: Understand problem structure

   Minimum cost to reach step i depends on minimum cost to reach steps i-1 and i-2 plus cost at i.

2. Step 2: Apply dynamic programming

   Store minimum costs for each step to avoid recomputing (overlapping subproblems) and use previous results (optimal substructure).

3. Final Answer:

   Use dynamic programming to store minimum cost for each step and build up solution. -> Option B

4. Quick Check:

   DP uses optimal substructure + overlapping subproblems [OK]

Hint: Build solution step-by-step storing results [OK]

Common Mistakes:

• Using recursion without memoization causing slow solution
• Trying greedy approach which may fail
• Sorting costs which is irrelevant