Memoization to Optimize Recursion in DSA Java - Execution Trace

Concept Flow - Memoization to Optimize Recursion

Start Recursive Call

↓

Check Memo for Result?

Yes→Return Memoized Result

No↓

Compute Result Recursively

↓

Store Result in Memo

↓

Return Computed Result

The flow shows how recursion first checks if the answer is already saved (memo). If yes, it returns it immediately. Otherwise, it computes, saves, and returns the result.

Execution Sample

DSA Java

int fib(int n, Map<Integer, Integer> memo) {
  if (n <= 1) return n;
  if (memo.containsKey(n)) return memo.get(n);
  int result = fib(n-1, memo) + fib(n-2, memo);
  memo.put(n, result);
  return result;
}

This code calculates Fibonacci numbers using recursion with memoization to avoid repeated calculations.

Execution Table

Step	Operation	n	Memo Before	Recursive Calls	Memo After	Returned Value
1	fib(4) called	4	{}	[fib(3), fib(2)]	{}	?
2	fib(3) called	3	{}	[fib(2), fib(1)]	{}	?
3	fib(2) called	2	{}	[fib(1), fib(0)]	{}	?
4	fib(1) base case	1	{}	none	{}	1
5	fib(0) base case	0	{}	none	{}	0
6	fib(2) computed	2	{}	none	{2:1}	1
7	fib(1) base case	1	{2:1}	none	{2:1}	1
8	fib(3) computed	3	{2:1}	none	{2:1,3:2}	2
9	fib(2) memo hit	2	{2:1,3:2}	none	{2:1,3:2}	1
10	fib(4) computed	4	{2:1,3:2}	none	{2:1,3:2,4:3}	3
11	Return final result	4	{2:1,3:2,4:3}	none	{2:1,3:2,4:3}	3

💡 All recursive calls resolved; memo contains results for n=2,3,4; final fib(4) = 3 returned.

Variable Tracker

Variable	Start	After Step 4	After Step 6	After Step 8	After Step 10	Final
n	4	2	2	3	4	4
memo	{}	{}	{2:1}	{2:1,3:2}	{2:1,3:2,4:3}	{2:1,3:2,4:3}
Returned Value	?	1	1	2	3	3

Key Moments - 3 Insights

Why does the function check the memo before computing?

Because checking the memo (see Step 9) avoids repeating expensive recursive calls, saving time by returning stored results immediately.

What happens when the base case is reached?

At base cases (Steps 4 and 5), the function returns the known value directly without recursion, starting the memoization process.

How does memoization reduce the number of recursive calls?

Memoization stores results after first computation (Steps 6 and 8), so later calls (Step 9) reuse these results instead of recalculating.

Visual Quiz - 3 Questions

Look at the execution table, what is the returned value at Step 8?

A 1

B 2

C 3

D 0

💡 Hint

Check the 'Returned Value' column at Step 8 in the execution_table.

At which step does the memo first get updated with a value?

A Step 6

B Step 4

C Step 9

D Step 2

💡 Hint

Look at the 'Memo After' column to see when the first key-value pair appears.

If memoization was not used, how would the recursive calls change at Step 9?

A Fewer recursive calls would happen

B No change, memo is not used here

C More recursive calls would happen instead of returning memoized result

D The function would return 0 immediately

💡 Hint

Step 9 shows a memo hit avoiding recursion; without memo, recursion repeats.

Concept Snapshot

Memoization stores results of recursive calls in a map.
Before computing, check if result exists in memo.
If yes, return it immediately to save time.
If no, compute recursively, store result, then return.
This avoids repeated work and speeds up recursion.

Full Transcript

Memoization optimizes recursion by saving results of function calls in a map called memo. When the function is called with a value n, it first checks if the result is already in memo. If yes, it returns that result immediately, avoiding repeated calculations. If not, it computes the result recursively, stores it in memo, and then returns it. This process reduces the number of recursive calls drastically. For example, in the Fibonacci function, base cases return known values directly. Then, results for fib(2), fib(3), and fib(4) are computed and saved. Later calls reuse these saved results instead of recalculating. This makes recursion efficient and fast.

Practice

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

easy

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

B. To make the code shorter and simpler

C. To avoid using loops in recursion

D. To increase the number of recursive calls

Solution

1. Step 1: Understand recursion inefficiency

   Recursive functions often repeat the same calculations multiple times, which wastes time.

2. Step 2: Role of memoization

   Memoization saves the results of these repeated calculations so they can be reused instead of recalculated.

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 results to speed recursion [OK]

Common Mistakes:

• Thinking memoization makes code shorter
• Confusing memoization with iteration
• Believing memoization increases recursion calls

2. Which of the following Java code snippets correctly declares a memoization cache for storing integer results by input integer keys?

easy

A. Map memo = new HashMap<>();

B. List<Integer> memo = new ArrayList<>();

C. int[] memo = new int();

D. HashSet<Integer> memo = new HashSet<>();

Solution

1. Step 1: Identify suitable data structure for key-value pairs

   Memoization requires storing results by input keys, so a Map is ideal.

2. Step 2: Check Java syntax correctness

   Map memo = new HashMap<>(); uses Map with HashMap, which is correct syntax for memoization cache.

3. Final Answer:

   Map memo = new HashMap<>(); -> Option A

4. Quick Check:

   Memo cache = Map for key-value pairs [OK]

Hint: Use Map for key-value memo cache in Java [OK]

Common Mistakes:

• Using List instead of Map for memoization
• Incorrect array declaration syntax
• Using Set which stores only keys, no values

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

import java.util.*;
public class FibMemo {
  static Map memo = new HashMap<>();
  public static int fib(int n) {
    if (n <= 1) return n;
    if (memo.containsKey(n)) return memo.get(n);
    int result = fib(n-1) + fib(n-2);
    memo.put(n, result);
    return result;
  }
  public static void main(String[] args) {
    System.out.println(fib(5));
  }
}

medium

A. Error at runtime

B. 8

C. 5

D. 13

Solution

1. Step 1: Understand Fibonacci sequence calculation

   fib(5) = fib(4) + fib(3), and so on, with base cases fib(0)=0, fib(1)=1.

2. Step 2: Calculate fib(5) with memoization

   fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=5. Memoization avoids repeated calls but result is same.

3. Final Answer:

   5 -> Option C

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Recall Fibonacci(5) = 5, memoization doesn't change result [OK]

Common Mistakes:

• Confusing Fibonacci values for wrong n
• Assuming memoization changes output
• Thinking code causes runtime error

4. Identify the error in this Java memoized factorial function:

import java.util.*;
public class FactorialMemo {
  static Map memo = new HashMap<>();
  public static int fact(int n) {
    if (n == 0) return 1;
    if (memo.containsKey(n)) return memo.get(n);
    int result = n * fact(n - 1);
    memo.put(n, result);
    return result;
  }
  public static void main(String[] args) {
    System.out.println(fact(-1));
  }
}

medium

A. Memo map is not initialized properly

B. fact method should return long instead of int

C. Incorrect multiplication order in recursion

D. No base case for negative input causes infinite recursion

Solution

1. Step 1: Analyze base cases and input handling

   Base case handles n == 0 only, but negative n causes no stop condition.

2. Step 2: Effect of negative input

   Calling fact(-1) leads to infinite recursion because n never reaches 0 or memoized value.

3. Final Answer:

   No base case for negative input causes infinite recursion -> Option D

4. Quick Check:

   Missing negative base case = infinite recursion [OK]

Hint: Check base cases cover all input ranges [OK]

Common Mistakes:

• Ignoring negative input handling
• Assuming memo initialization error
• Thinking multiplication order causes error

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 implements this in Java?

hard

A. Use a List<Integer> to store results; always calculate recursively without checking cache.

B. Use a Map to store results; check cache before recursive calls; store results after calculation.

C. Use an int array without initializing values; store results only after all recursion ends.

D. Use a Set<Integer> to store visited stairs; return size of set as result.

Solution

1. Step 1: Understand problem and memoization needs

   We must store number of ways for each stair number to avoid repeated calculations.

2. Step 2: Evaluate memoization approaches

   Use a Map to store results; check cache before recursive calls; store results after calculation. uses Map to store and check results before recursion, which is correct memoization practice.

3. Final Answer:

   Use a Map to store results; check cache before recursive calls; store results after calculation. -> Option B

4. Quick Check:

   Memo = Map with cache check and store [OK]

Hint: Check cache before recursion, store after calculation [OK]

Common Mistakes:

• Not checking cache before recursive calls
• Using List or Set incorrectly for memoization
• Storing results too late or not at all