Overlapping Subproblems and Optimal Substructure in DSA Typescript - Build from Scratch

Understanding Overlapping Subproblems and Optimal Substructure with Fibonacci

📖 Scenario: Imagine you want to find the number of ways to climb stairs where you can take 1 or 2 steps at a time. This is a classic problem that can be solved using the Fibonacci sequence. To solve it efficiently, we use the ideas of overlapping subproblems and optimal substructure.

🎯 Goal: You will build a simple program to calculate Fibonacci numbers using a technique that remembers past answers to avoid repeated work. This shows how overlapping subproblems and optimal substructure help in solving problems efficiently.

📋 What You'll Learn

Create an array to store Fibonacci numbers for reuse

Set up a variable for the number of steps to climb

Write a function that uses stored results to calculate Fibonacci numbers

Print the Fibonacci number for the given steps

💡 Why This Matters

🌍 Real World

Many problems like calculating paths, costs, or sequences can be solved efficiently by remembering past answers and building solutions from smaller parts.

💼 Career

Understanding overlapping subproblems and optimal substructure is key for roles in software development, data science, and algorithm design where efficient problem solving is required.

1

Create the Fibonacci cache array

Create an array called fibCache with 11 elements, all initialized to -1.

DSA Typescript

# Your code here

Hint

Use new Array(11).fill(-1) to create an array of length 11 filled with -1.

2

Set the number of steps

Create a variable called steps and set it to 10.

DSA Typescript

const fibCache: number[] = new Array(11).fill(-1);
# Your code here

Hint

Use const steps: number = 10; to set the number of steps.

3

Write the Fibonacci function with memoization

Write a function called fib that takes a number n and returns the Fibonacci number using fibCache to store and reuse results. Use these rules:
- If n is 0 or 1, return 1.
- If fibCache[n] is not -1, return fibCache[n].
- Otherwise, calculate fib(n-1) + fib(n-2), store it in fibCache[n], and return it.

DSA Typescript

const fibCache: number[] = new Array(11).fill(-1);
const steps: number = 10;

# Your code here

Hint

Use a function with base cases for 0 and 1, check cache before computing, and store results in fibCache.

4

Print the Fibonacci number for the steps

Write a console.log statement to print the result of fib(steps).

DSA Typescript

const fibCache: number[] = new Array(11).fill(-1);
const steps: number = 10;

function fib(n: number): number {
    if (n === 0 || n === 1) {
        return 1;
    }
    if (fibCache[n] !== -1) {
        return fibCache[n];
    }
    fibCache[n] = fib(n - 1) + fib(n - 2);
    return fibCache[n];
}

# Your code here

Hint

Use console.log(fib(steps)); to print the Fibonacci number for 10 steps.

Practice

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

easy

A. The problem requires sorting elements before solving.

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

C. The problem can be broken down into smaller subproblems that 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 solving.

2. Step 2: Match definition with options

   The problem can be broken down into smaller subproblems that are solved multiple times. correctly states that subproblems repeat and can be reused.

3. Final Answer:

   The problem can be broken down into smaller subproblems that 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 TypeScript syntax correctly declares a memoization object for storing subproblem results?

easy

A. let memo = new Map();

B. const memo: number[] = [];

C. var memo = {};

D. const memo: string = '';

Solution

1. Step 1: Identify suitable data structure for memoization

   Memoization stores key-value pairs of subproblems and results, so Map is ideal.

2. Step 2: Check options for correct TypeScript syntax

   let memo = new Map(); uses Map which is correct for memoization.

3. Final Answer:

   let memo = new Map<number, number>(); -> Option A

4. Quick Check:

   Memoization uses Map for key-value storage [OK]

Hint: Use Map for flexible key-value memo storage in TypeScript [OK]

Common Mistakes:

• Using empty object without type safety
• Declaring memo as a string instead of a collection
• Using array when keys are not sequential indices

3. Consider this TypeScript function using memoization for Fibonacci numbers:

function fib(n: number, memo = new Map()): number {
  if (n <= 1) return n;
  if (memo.has(n)) return memo.get(n)!;
  const result = fib(n - 1, memo) + fib(n - 2, memo);
  memo.set(n, result);
  return result;
}
console.log(fib(5));

What is the output printed?

medium

A. 8

B. 5

C. 3

D. Error

Solution

1. Step 1: Understand Fibonacci calculation with memoization

   fib(5) = fib(4) + fib(3), recursively calculated and stored 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 B

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Recall Fibonacci sequence: 0,1,1,2,3,5... [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Forgetting base cases n <= 1
• Assuming memo is reset each call

4. The following TypeScript code attempts to compute Fibonacci numbers with memoization but has a bug:

function fib(n: number, memo = new Map()): number {
  if (n <= 1) return n;
  if (memo.has(n)) return memo.get(n)!;
  const result = fib(n - 1) + fib(n - 2);
  memo.set(n, result);
  return result;
}
console.log(fib(5));

What is the main issue causing incorrect behavior?

medium

A. Memo object is not passed in recursive calls, so memoization fails.

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

C. Map is not initialized properly.

D. Return statement is missing.

Solution

1. Step 1: Analyze recursive calls and memo usage

   Recursive calls fib(n - 1) and fib(n - 2) do not pass the memo Map, so each call creates a new Map.

2. Step 2: Understand impact on memoization

   Without passing memo, results are not stored or reused, causing repeated calculations and no optimization.

3. Final Answer:

   Memo object is not passed in recursive calls, so memoization fails. -> Option A

4. Quick Check:

   Pass memo in recursive calls to reuse stored results [OK]

Hint: Always pass memo object in recursive calls for memoization [OK]

Common Mistakes:

• Not passing memo in recursive calls
• Changing base case unnecessarily
• Assuming Map auto-shares across calls

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:

const cost = [
  [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 a greedy approach picking the smallest adjacent cost at each step.

B. Use recursion without memoization to explore all paths.

C. Sort the grid and pick the smallest values.

D. Use dynamic programming to store minimum cost for each cell based on previous cells.

Solution

1. Step 1: Identify problem properties

   Minimum cost path has optimal substructure: cost to reach a cell depends on minimum cost to reach adjacent cells.

2. Step 2: Choose approach using overlapping subproblems

   Dynamic programming stores results for each cell to avoid recomputing paths (overlapping subproblems).

3. Final Answer:

   Use dynamic programming to store minimum cost for each cell based on previous cells. -> Option D

4. Quick Check:

   DP uses overlapping subproblems + optimal substructure [OK]

Hint: Use DP to store and reuse subproblem results for grid paths [OK]

Common Mistakes:

• Using recursion without memoization causing slow performance
• Assuming greedy always finds minimum cost
• Sorting grid values unrelated to path costs