Rod Cutting Problem in DSA Typescript - Execution Trace

Concept Flow - Rod Cutting Problem

Start with full rod length N

↓

Try all cut lengths i from 1 to N

↓

Calculate price for cut i + best price for remaining length N-i

↓

Choose max price among all cuts

↓

Store max price for length N

↓

Repeat for smaller lengths until length 0

↓

Final max price for full rod length

We try all possible first cuts, calculate prices recursively, and pick the best total price for each rod length.

Execution Sample

DSA Typescript

function rodCut(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] + rodCut(prices, n - i));
  }
  return maxVal;
}

This code finds the maximum price by trying all cuts recursively for rod length n.

Execution Table

Step	Operation	Rod Length (n)	Cut Length (i)	Price for cut i	Remaining length	Recursive call result	Max price so far	Visual State (max prices array)
1	Start	4	-	-	-	-	-Infinity	[0, -, -, -, -]
2	Try cut i=1	4	1	2	3	7	9	[0, -, -, -, -]
3	Try cut i=2	4	2	5	2	5	10	[0, -, -, -, -]
4	Try cut i=3	4	3	7	1	2	10	[0, -, -, -, -]
5	Try cut i=4	4	4	8	0	0	10	[0, -, -, -, -]
6	Finish length 4	4	-	-	-	-	10	[0, 2, 5, 7, 10]
7	Finish length 3	3	-	-	-	-	7	[0, 2, 5, 7, 10]
8	Finish length 2	2	-	-	-	-	5	[0, 2, 5, 7, 10]
9	Finish length 1	1	-	-	-	-	2	[0, 2, 5, 7, 10]
10	Finish length 0	0	-	-	-	-	0	[0, -, -, -, -]
11	End	-	-	-	-	-	10	[0, 2, 5, 7, 10]

💡 All rod lengths from 0 to 4 computed, max price for length 4 is 10

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	Final
n	4	4	4	4	4	4	4
i	-	1	2	3	4	-	-
maxVal	-Infinity	9	10	10	10	10	10
prices	[2,5,7,8]	[2,5,7,8]	[2,5,7,8]	[2,5,7,8]	[2,5,7,8]	[2,5,7,8]	[2,5,7,8]
maxPricesArray	[0,-,-,-,-]	[0,-,-,-,-]	[0,-,-,-,-]	[0,-,-,-,-]	[0, -, -, -, -]	[0, 2, 5, 7, 10]	[0, 2, 5, 7, 10]

Key Moments - 3 Insights

Why do we try all cut lengths from 1 to n instead of just one cut?

Because the best price might come from cutting the rod into multiple pieces, not just one. The execution_table rows 2 to 5 show trying all cuts and picking the max price.

Why does the recursion stop when rod length n is 0?

Because a rod of length 0 has no value, so the base case returns 0. This is shown in execution_table row 10 where n=0 returns 0.

How does the max price get updated during recursion?

At each step, maxVal stores the highest price found so far by comparing current cut price plus recursive result. See execution_table rows 2 to 5 where maxVal increases.

Visual Quiz - 3 Questions

Look at the execution_table at Step 4, what is the max price so far for rod length 4?

A 7

B 12

C 10

D 9

💡 Hint

Check the 'Max price so far' column at Step 4 in execution_table

At which step does the recursion base case (n=0) return 0?

A Step 6

B Step 10

C Step 2

D Step 11

💡 Hint

Look for rod length 0 in the 'Rod Length (n)' column in execution_table

If the price for cut length 3 was increased to 10, how would max price for rod length 4 change at Step 4?

A It would become 12

B It would stay 10

C It would become 17

D It would become 10

💡 Hint

Calculate prices[i-1] + recursive result for i=3 with new price 10 and check maxVal update in execution_table

Concept Snapshot

Rod Cutting Problem:
- Given rod length n and prices for each length
- Try all cuts i=1 to n
- Max price = max(prices[i-1] + max price for n-i)
- Use recursion or DP to store max prices
- Base case: length 0 price = 0
- Goal: maximize total price by cutting rod optimally

Full Transcript

The Rod Cutting Problem finds the best way to cut a rod to maximize price. We try all cut lengths from 1 to n. For each cut, we add the price of that cut length to the best price for the remaining rod length. We keep track of the maximum price found. The recursion stops when the rod length is zero, returning zero price. The execution table shows step-by-step how max prices are computed for rod lengths from 0 to 4. Variable tracker shows how maxVal and cut length i change during execution. Key moments clarify why all cuts are tried, why recursion stops at zero length, and how max price updates. The visual quiz tests understanding of max price updates, base case, and effect of price changes. This approach helps beginners see how the problem breaks down into smaller subproblems and builds up the solution.

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