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

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Choose max price among all cuts

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Store max price for length N

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Repeat for smaller lengths until length 0

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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.