Rod Cutting Problem in DSA Typescript - Time & Space Complexity

Time Complexity: Rod Cutting Problem

O(2^n)

Understanding Time Complexity

We want to understand how the time needed to solve the rod cutting problem changes as the rod length grows.

Specifically, how the number of steps grows when we try all possible cuts to find the best price.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function rodCutting(prices: number[], n: number): number {
  if (n <= 0) return 0;
  let maxVal = -Infinity;
  for (let i = 1; i <= n; i++) {
    maxVal = Math.max(maxVal, prices[i - 1] + rodCutting(prices, n - i));
  }
  return maxVal;
}
    

This code tries every possible first cut and recursively solves the smaller rod to find the maximum price.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The for-loop tries all cut lengths from 1 to n.
• How many times: For each call with rod length n, it makes up to n recursive calls for smaller rods.

How Execution Grows With Input

Each rod length n calls itself up to n times with smaller lengths, leading to many repeated calls.

Input Size (n)	Approx. Operations
10	More than 1000 calls
100	More than 10^14 calls (very large)
1000	Extremely large, practically impossible without optimization

Pattern observation: The number of calls grows very fast, much faster than just doubling.

Final Time Complexity

Time Complexity: O(2^n)

This means the time needed doubles roughly with each additional unit of rod length, making it very slow for large n.

Common Mistake

[X] Wrong: "The recursive solution runs in linear time because it just loops from 1 to n once."

[OK] Correct: Each recursive call itself makes many more calls, so the total work multiplies quickly, not just once.

Interview Connect

Understanding this problem helps you see why recursion alone can be slow and why adding memory to save results is important.

Self-Check

"What if we store results of smaller rods to avoid repeated work? How would the time complexity change?"

Practice

1. What is the main goal of the Rod Cutting Problem in algorithms?

easy

A. To count the total number of ways to cut the rod

B. To minimize the number of cuts made on the rod

C. To maximize the total profit by cutting the rod into pieces

D. To find the shortest rod length possible

Solution

1. Step 1: Understand the problem objective

   The rod cutting problem aims to cut a rod into pieces to get the maximum total price from selling those pieces.

2. Step 2: Identify the goal

   The goal is to maximize profit, not minimize cuts or count ways.

3. Final Answer:

   To maximize the total profit by cutting the rod into pieces -> Option C

4. Quick Check:

   Maximize profit = maximize total price [OK]

Hint: Focus on maximizing profit from pieces, not cuts or counts [OK]

Common Mistakes:

• Thinking the goal is to minimize cuts
• Confusing with counting ways to cut
• Assuming shortest rod length is the goal

2. Which of the following TypeScript function signatures correctly represents the rod cutting problem function?

easy

A. function rodCutting(prices: string[], length: number): number

B. function rodCutting(length: number, prices: number[]): void

C. function rodCutting(prices: number[], length: number): string

D. function rodCutting(prices: number[], length: number): number

Solution

1. Step 1: Analyze function parameters and return type

   The function should take an array of prices (number[]) and the rod length (number), returning the maximum profit (number).

2. Step 2: Match correct signature

   function rodCutting(prices: number[], length: number): number matches this: prices array first, length second, returns number.

3. Final Answer:

   function rodCutting(prices: number[], length: number): number -> Option D

4. Quick Check:

   Input: prices array, length; Output: number [OK]

Hint: Function returns max profit as number, inputs are prices array and length [OK]

Common Mistakes:

• Swapping parameter order
• Returning void or string instead of number
• Using wrong parameter types like string[] for prices

3. What is the output of the following TypeScript code for rod cutting?

const prices = [1, 5, 8, 9];
const length = 4;
function rodCutting(prices: number[], n: number): number {
  if (n === 0) return 0;
  let maxVal = -Infinity;
  for (let i = 1; i <= n; i++) {
    maxVal = Math.max(maxVal, prices[i - 1] + rodCutting(prices, n - i));
  }
  return maxVal;
}
console.log(rodCutting(prices, length));

medium

A. 10

B. 9

C. 11

D. 8

Solution

1. Step 1: Understand recursive calls and prices

   Prices for lengths 1 to 4 are [1,5,8,9]. The function tries all cuts and sums prices recursively.

2. Step 2: Calculate max profit for length 4

   Possible cuts:
   - 1 + rodCutting(3) = 1 + 8 = 9
   - 5 + rodCutting(2) = 5 + 5 = 10
   - 8 + rodCutting(1) = 8 + 1 = 9
   - 9 + rodCutting(0) = 9 + 0 = 9
   rodCutting(2) = max(1+rodCutting(1),5+rodCutting(0)) = max(1+1,5+0)=5
   rodCutting(3) = max(1+rodCutting(2),5+rodCutting(1),8+rodCutting(0)) = max(1+5,5+1,8+0)= max(6,6,8)=8
   So max is 10.

3. Final Answer:

   10 -> Option A

4. Quick Check:

   Max profit for length 4 = 10 [OK]

Hint: Try all cuts recursively, pick max sum [OK]

Common Mistakes:

• Ignoring recursive calls
• Using wrong indices for prices
• Assuming max is price of full length only

4. Identify the bug in this TypeScript rod cutting function:

function rodCutting(prices: number[], n: number): number {
  if (n <= 0) return 0;
  let maxVal = 0;
  for (let i = 1; i < n; i++) {
    maxVal = Math.max(maxVal, prices[i - 1] + rodCutting(prices, n - i));
  }
  return maxVal;
}

medium

A. Loop should run until i <= n, not i < n

B. Base case should return -Infinity instead of 0

C. prices array indexing is off by one

D. Function should return void, not number

Solution

1. Step 1: Check loop range correctness

   The loop runs from i = 1 to i < n, so it misses the case when i = n (full length cut).

2. Step 2: Fix loop to include full length cut

   Changing loop condition to i <= n ensures all cuts including full length are considered.

3. Final Answer:

   Loop should run until i <= n, not i < n -> Option A

4. Quick Check:

   Include full length cut by i <= n [OK]

Hint: Loop must include full length cut (i <= n) [OK]

Common Mistakes:

• Using i < n misses full rod cut
• Changing base case incorrectly
• Misindexing prices array

5. Given a rod of length 5 and prices array [2, 5, 7, 8, 10], what is the maximum profit using dynamic programming approach?

hard

A. 13

B. 12

C. 14

D. 15

Solution

1. Step 1: Use bottom-up DP to find max profit

   Prices: length 1=2, 2=5, 3=7, 4=8, 5=10.
   Calculate max profit for each length:
   dp[0]=0
   dp[1]=max(2)=2
   dp[2]=max(2+dp[1],5)=max(2+2,5)=5
   dp[3]=max(2+dp[2],5+dp[1],7)=max(2+5,5+2,7)=7
   dp[4]=max(2+dp[3],5+dp[2],7+dp[1],8)=max(2+7,5+5,7+2,8)=max(9,10,9,8)=10
   dp[5]=max(2+dp[4],5+dp[3],7+dp[2],8+dp[1],10)=max(2+10,5+7,7+5,8+2,10)=max(12,12,12,10,10)=12
   Therefore, correct answer is 12.

2. Final Answer:

   12 -> Option B

3. Quick Check:

   DP max profit for length 5 = 12 [OK]

Hint: Use bottom-up DP, check all cuts for max profit [OK]

Common Mistakes:

• Ignoring smaller cuts combinations
• Miscomputing dp array values
• Assuming price of full length is max