Memoization to Optimize Recursion in DSA C++ - Time & Space Complexity

Time Complexity: Memoization to Optimize Recursion

O(n)

Understanding Time Complexity

We want to see how using memoization changes the speed of recursive functions.

How does remembering past answers help us do less work?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

#include <iostream>
#include <vector>
using namespace std;

int fib(int n, vector<int>& memo) {
  if (n <= 1) return n;
  if (memo[n] != -1) return memo[n];
  memo[n] = fib(n - 1, memo) + fib(n - 2, memo);
  return memo[n];
}

int main() {
  int n = 10;
  vector<int> memo(n + 1, -1);
  cout << fib(n, memo) << endl;
  return 0;
}
    

This code calculates the nth Fibonacci number using recursion with memoization to store results.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls to fib with memo checks.
• How many times: Each unique fib(n) is computed once; repeated calls return stored results immediately.

How Execution Grows With Input

Each number from 0 to n is calculated once and saved, so work grows roughly in a straight line with n.

Input Size (n)	Approx. Operations
10	About 10 calculations
100	About 100 calculations
1000	About 1000 calculations

Pattern observation: Doubling n roughly doubles the work, showing linear growth.

Final Time Complexity

Time Complexity: O(n)

This means the time to finish grows directly with the input size, much faster than plain recursion.

Common Mistake

[X] Wrong: "Memoization still does all recursive calls like plain recursion."

[OK] Correct: Memoization saves results so repeated calls don't redo work, cutting down the total calls drastically.

Interview Connect

Understanding how memoization changes recursion speed is a key skill that shows you can improve algorithms smartly.

Self-Check

"What if we removed memoization and just used plain recursion? How would the time complexity change?"

Practice

1. What is the main purpose of using memoization in recursive functions?

easy

A. To make the function run slower by adding extra steps

B. To avoid using any variables inside the function

C. To convert recursion into iteration automatically

D. To store results of expensive function calls and reuse them

Solution

1. Step 1: Understand what memoization does

   Memoization saves the results of function calls so that if the same inputs occur again, the function can return the saved result instead of recalculating.

2. Step 2: Identify the effect on performance

   This reuse of results reduces repeated work and speeds up the function.

3. Final Answer:

   To store results of expensive function calls and reuse them -> Option D

4. Quick Check:

   Memoization = Store and reuse results [OK]

Hint: Memoization saves repeated results to speed recursion [OK]

Common Mistakes:

• Thinking memoization slows down the function
• Confusing memoization with converting recursion to loops
• Believing memoization removes all variables

2. Which of the following is the correct way to declare a memoization cache for a Fibonacci function in C++?

easy

A. std::unordered_map<int> memo;

B. std::vector<int> memo(100, -1);

C. int memo[];

D. map<int> memo;

Solution

1. Step 1: Identify suitable container for memoization

   We need a container that can store integer results indexed by integer keys. std::vector<int> initialized with -1 is common for fixed size memoization.

2. Step 2: Check syntax correctness

   std::vector<int> memo(100, -1); correctly declares a vector of size 100 with all values -1, which can be used to check if a value is computed.

3. Final Answer:

   std::vector<int> memo(100, -1); -> Option B

4. Quick Check:

   Vector with size and default value = Correct memo cache [OK]

Hint: Use vector with size and default -1 for memo cache [OK]

Common Mistakes:

• Using incomplete or wrong container syntax
• Not initializing memo with default values
• Using map without specifying key and value types

3. What will be the output of the following C++ code snippet?

int fib(int n, std::vector<int>& memo) {
  if (n <= 1) return n;
  if (memo[n] != -1) return memo[n];
  memo[n] = fib(n-1, memo) + fib(n-2, memo);
  return memo[n];
}

int main() {
  std::vector<int> memo(6, -1);
  std::cout << fib(5, memo);
  return 0;
}

medium

A. Error: vector index out of range

B. 8

C. 5

D. 3

Solution

1. Step 1: Understand Fibonacci calculation with memoization

   fib(5) calculates Fibonacci number at position 5 using memo to avoid repeated calls.

2. Step 2: Calculate fib(5)

   Fibonacci sequence: 0,1,1,2,3,5,... fib(5) = 5.

3. Final Answer:

   5 -> Option C

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Recall Fibonacci sequence: fib(5) = 5 [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Assuming vector size too small causes error
• Ignoring memoization effect on output

4. Consider this recursive function with memoization in C++:

int factorial(int n, std::vector<int>& memo) {
  if (n == 0) return 1;
  if (memo[n] != -1) return memo[n];
  memo[n] = n * factorial(n - 1, memo);
  return memo[n];
}

int main() {
  std::vector<int> memo(5, -1);
  std::cout << factorial(5, memo);
  return 0;
}

What is the error in this code?

medium

A. Vector size is too small causing out-of-range access

B. Base case condition is incorrect

C. Memoization check is missing

D. Factorial calculation formula is wrong

Solution

1. Step 1: Check vector size for memoization

   The vector memo has size 5, so valid indices are 0 to 4. But factorial(5) tries to access memo[5], which is out of range.

2. Step 2: Identify consequence of out-of-range access

   Accessing memo[5] causes undefined behavior or runtime error.

3. Final Answer:

   Vector size is too small causing out-of-range access -> Option A

4. Quick Check:

   Memo vector size < n causes error [OK]

Hint: Memo vector size must be >= n+1 to avoid errors [OK]

Common Mistakes:

• Ignoring vector size and indexing
• Thinking base case is wrong
• Assuming memo check is missing

5. You want to optimize a recursive function that calculates the number of ways to climb n stairs, taking 1 or 2 steps at a time. Which memoization approach below correctly applies to this problem in C++?

hard

A. Use a vector<int> memo(n+1, -1) and store results for each stair count

int ways(int n, std::vector<int>& memo) {
if (n == 0) return 1;
if (n < 0) return 0;
if (memo[n] != -1) return memo[n];
memo[n] = ways(n-1, memo) + ways(n-2, memo);
return memo[n];
}

B. Use a vector<int> memo(n, 0) and store results only for even stairs

int ways(int n, std::vector<int>& memo) {
if (n == 0) return 1;
if (n < 0) return 0;
if (memo[n] != 0) return memo[n];
memo[n] = ways(n-1, memo) + ways(n-2, memo);
return memo[n];
}

C. Use a map memo and store results only for odd stairs

int ways(int n, std::map& memo) {
if (n == 0) return 1;
if (n < 0) return 0;
if (memo.count(n)) return memo[n];
memo[n] = ways(n-1, memo) + ways(n-2, memo);
return memo[n];
}

D. Use a vector<int> memo(n+1, 0) but do not check memo before recursive calls

int ways(int n, std::vector<int>& memo) {
if (n == 0) return 1;
if (n < 0) return 0;
memo[n] = ways(n-1, memo) + ways(n-2, memo);
return memo[n];
}

Solution

1. Step 1: Understand memoization requirements

   Memo must store results for all n from 0 to n, so vector size n+1 with default -1 is correct.

2. Step 2: Check memo usage in recursion

   Before recursive calls, check if memo[n] is computed (not -1). If yes, return memo[n]. Otherwise, compute and store.

3. Step 3: Analyze options

   Use a vector<int> memo(n+1, -1) and store results for each stair count int ways(int n, std::vector<int>& memo) { if (n == 0) return 1; if (n < 0) return 0; if (memo[n] != -1) return memo[n]; memo[n] = ways(n-1, memo) + ways(n-2, memo); return memo[n]; } correctly implements memoization with vector size n+1, default -1, and memo check before recursion.

4. Final Answer:

   Option A code correctly applies memoization for climbing stairs problem -> Option A

5. Quick Check:

   Memo vector size n+1 and check before recursion = Correct [OK]

Hint: Memo vector size n+1 with -1 and check before recursion [OK]

Common Mistakes:

• Using wrong vector size causing out-of-range errors
• Not checking memo before recursive calls
• Storing results only for some stairs (odd/even)