Memoization to Optimize Recursion in DSA Go - Time & Space Complexity

Time Complexity: Memoization to Optimize Recursion

O(n)

Understanding Time Complexity

When we use recursion, some problems repeat the same work many times. Memoization helps us avoid repeating work by remembering results.

We want to know how using memoization changes the time it takes to run the recursive code.

Scenario Under Consideration

Analyze the time complexity of the following memoized recursive function.

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

This function calculates the nth Fibonacci number using recursion and stores results in a map to avoid repeated calculations.

Identify Repeating Operations

Look for repeated work in recursion and how memoization changes it.

• Primary operation: Recursive calls to fibMemo for smaller n.
• How many times: Without memoization, many calls repeat for the same n. With memoization, each n is computed once.

How Execution Grows With Input

With memoization, the function calculates each Fibonacci number from 0 to n once.

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

Pattern observation: The number of operations grows roughly in a straight line with n, not exponentially.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute grows directly with the input size, because each result is calculated once and reused.

Common Mistake

[X] Wrong: "Memoization still makes the recursion exponential because it calls itself twice each time."

[OK] Correct: Memoization stores results so each input is computed once, turning exponential calls into linear calls.

Interview Connect

Understanding how memoization changes recursion time is a key skill. It shows you can improve slow code by avoiding repeated work, a common real-world need.

Self-Check

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

Practice

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

easy

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

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

C. To remove recursion and use loops instead

D. To increase the memory usage without any benefit

Solution

1. Step 1: Understand what memoization does

   Memoization saves the results of function calls so they don't need to be recalculated.

2. Step 2: Identify the benefit of memoization

   By reusing saved results, the function runs faster and avoids repeated work.

3. Final Answer:

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

4. Quick Check:

   Memoization = Store and reuse results [OK]

Hint: Memoization saves repeated work by caching results [OK]

Common Mistakes:

• Thinking memoization slows down the function
• Confusing memoization with removing recursion
• Ignoring the memory benefit of caching

2. Which of the following Go code snippets correctly declares a memo map for memoization of integer keys and values?

easy

A. memo := new(map[int]int)

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

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

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

Solution

1. Step 1: Identify correct map declaration syntax

   In Go, make(map[int]int) creates a map with int keys and int values.

2. Step 2: Check other options

   memo := map[string]int{} uses string keys, not int; memo := make([]int, 10) creates a slice, not a map; memo := new(map[int]int) creates a pointer to a map but does not initialize it.

3. Final Answer:

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

4. Quick Check:

   Correct map syntax = make(map[int]int) [OK]

Hint: Use make(map[int]int) to create int-to-int map [OK]

Common Mistakes:

• Using slice instead of map for memo
• Using wrong key type in map
• Declaring pointer to map without initialization

3. What will be the output of the following Go code using memoization for Fibonacci numbers?

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. 8

B. Error

C. 3

D. 5

Solution

1. Step 1: Understand Fibonacci calculation with memo

   fib(5) = fib(4) + fib(3), and memo stores intermediate results to avoid repeats.

2. Step 2: Calculate fib(5) step-by-step

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

3. Final Answer:

   5 -> Option D

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Memoized fib(5) returns 5 quickly [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Ignoring memo and recalculating
• Expecting error due to map usage

4. Identify the error in this Go recursive function with memoization for factorial:

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

func main() {
    memo := make(map[int]int)
    fmt.Println(fact(-1, memo))
}

medium

A. Compilation error due to map usage

B. No error, output is 1

C. Infinite recursion for negative input

D. Returns 0 for negative input

Solution

1. Step 1: Analyze base case and input

   Base case is n == 0, but input is -1, so base case never reached.

2. Step 2: Understand recursion behavior

   For n = -1, function calls fact(-2), then fact(-3), causing infinite recursion.

3. Final Answer:

   Infinite recursion for negative input -> Option C

4. Quick Check:

   Negative input causes infinite recursion [OK]

Hint: Check base case covers all inputs to avoid infinite recursion [OK]

Common Mistakes:

• Assuming negative input returns 1
• Ignoring infinite recursion possibility
• Expecting compilation error incorrectly

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 using memoization in Go. Which approach below correctly applies memoization to avoid repeated calculations?

hard

A. Pass a map[int]int as a parameter to the function and store intermediate results in it

B. Use a slice of size n and fill it after recursion completes

C. Use a global variable to store results and update it inside the function

D. Call the recursive function twice without storing results

Solution

1. Step 1: Understand memoization with parameters

   Passing a map as a parameter allows storing and reusing results during recursion.

2. Step 2: Evaluate options for memoization

   Pass a map[int]int as a parameter to the function and store intermediate results in it correctly uses a map parameter; Use a global variable to store results and update it inside the function uses global state which is less safe; Use a slice of size n and fill it after recursion completes fills after recursion, not during; Call the recursive function twice without storing results repeats work without memo.

3. Final Answer:

   Pass a map[int]int as a parameter to the function and store intermediate results in it -> Option A

4. Quick Check:

   Memo map parameter stores intermediate results [OK]

Hint: Pass memo map as argument to cache results during recursion [OK]

Common Mistakes:

• Using global variables instead of parameters
• Filling memo after recursion ends
• Not storing results causing repeated work