Rod Cutting Problem in DSA Go - 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 grows as the rod length increases.

How does the number of steps change when the rod gets longer?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

func RodCutting(prices []int, n int) int {
    if n == 0 {
        return 0
    }
    maxVal := 0
    for i := 1; i <= n; i++ {
        val := prices[i-1] + RodCutting(prices, n-i)
        if val > maxVal {
            maxVal = val
        }
    }
    return maxVal
}
    

This code tries all ways to cut a rod of length n to get the best price by checking every possible first cut.

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 length n, it makes n recursive calls with smaller lengths.

How Execution Grows With Input

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

Input Size (n)	Approx. Operations
10	Thousands of calls
100	Millions of calls
1000	Billions of calls

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 roughly doubles with each increase in rod length, making it very slow for large n.

Common Mistake

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

[OK] Correct: Each loop iteration calls the function again, causing many repeated calls and exponential growth.

Interview Connect

Understanding this helps you see why naive recursion can be slow and why dynamic programming is needed to improve efficiency.

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 maximize the total profit by cutting the rod into pieces

C. To find the shortest rod length possible

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

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 B

4. Quick Check:

   Maximize profit = To maximize the total profit by cutting the rod into pieces [OK]

Hint: Focus on maximizing profit, not minimizing cuts or counting ways [OK]

Common Mistakes:

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

2. Which of the following Go function signatures correctly represents a rod cutting function that takes rod length and price slice as input and returns maximum profit as int?

easy

A. func rodCutting(prices int, length []int) int

B. func rodCutting(prices []int, length int) bool

C. func rodCutting(length int, prices []int) string

D. func rodCutting(length int, prices []int) int

Solution

1. Step 1: Analyze parameter types and order

   The rod length should be an int, prices should be a slice of ints ([]int), and return type should be int for max profit.

2. Step 2: Match correct signature

   func rodCutting(length int, prices []int) int matches: func rodCutting(length int, prices []int) int

3. Final Answer:

   func rodCutting(length int, prices []int) int -> Option D

4. Quick Check:

   Correct parameter types and return int = func rodCutting(length int, prices []int) int [OK]

Hint: Remember: length int, prices []int, return int for profit [OK]

Common Mistakes:

• Swapping parameter order
• Wrong return type like string or bool
• Using incorrect parameter types

3. What will be the output of the following Go code snippet for rod length 4 and prices [1, 5, 8, 9]?

package main

import "fmt"

func rodCutting(n int, prices []int) int {
    dp := make([]int, n+1)
    for i := 1; i <= n; i++ {
        maxVal := 0
        for j := 1; j <= i; j++ {
            val := prices[j-1] + dp[i-j]
            if val > maxVal {
                maxVal = val
            }
        }
        dp[i] = maxVal
    }
    return dp[n]
}

func main() {
    prices := []int{1, 5, 8, 9}
    result := rodCutting(4, prices)
    fmt.Println(result)
}

medium

A. 8

B. 9

C. 10

D. 11

Solution

1. Step 1: Understand the dp array calculation

   dp[i] stores max profit for rod length i. For length 4, check all cuts: - Cut 1: price 1 + dp[3] = 1 + 8 = 9 - Cut 2: price 5 + dp[2] = 5 + 5 = 10 - Cut 3: price 8 + dp[1] = 8 + 1 = 9 - Cut 4: price 9 + dp[0] = 9 + 0 = 9

2. Step 2: Calculate dp values stepwise

   dp[1]=1, dp[2]=5, dp[3]=8, dp[4]=max(9,10,9,9)=10 But re-check dp[3]: - Cut 1: 1 + dp[2]=1+5=6 - Cut 2: 5 + dp[1]=5+1=6 - Cut 3: 8 + dp[0]=8+0=8 So dp[3]=8 Now dp[4]: - Cut 1: 1 + dp[3]=1+8=9 - Cut 2: 5 + dp[2]=5+5=10 - Cut 3: 8 + dp[1]=8+1=9 - Cut 4: 9 + dp[0]=9+0=9 Max is 10

3. Final Answer:

   10 -> Option C

4. Quick Check:

   Max profit for length 4 = 10 [OK]

Hint: Calculate dp array stepwise checking all cuts [OK]

Common Mistakes:

• Ignoring dp values for smaller lengths
• Choosing wrong max profit among cuts
• Confusing indices in prices slice

4. Identify the bug in the following Go code for rod cutting and select the correct fix:

func rodCutting(n int, prices []int) int {
    dp := make([]int, n)
    for i := 1; i <= n; i++ {
        maxVal := 0
        for j := 1; j <= i; j++ {
            val := prices[j-1] + dp[i-j]
            if val > maxVal {
                maxVal = val
            }
        }
        dp[i] = maxVal
    }
    return dp[n]
}

medium

A. Change dp size to n+1 and access dp[i] safely

B. Change loop to start from i := 0 instead of 1

C. Remove inner loop and use only prices[i-1]

D. Change return to dp[n-1] instead of dp[n]

Solution

1. Step 1: Identify index out of range error

   dp slice is size n but code accesses dp[i] where i can be n, causing out of range error.

2. Step 2: Fix dp size to n+1

   Make dp size n+1 to safely store dp[n] and avoid index errors.

3. Final Answer:

   Change dp size to n+1 and access dp[i] safely -> Option A

4. Quick Check:

   dp size must be n+1 for dp[n] access [OK]

Hint: dp array size must be n+1 to avoid index errors [OK]

Common Mistakes:

• Using dp size n causes index out of range
• Changing loop start incorrectly
• Removing inner loop breaks logic

5. Given a rod of length 5 and prices [2, 5, 7, 8, 10], what is the maximum profit and the cuts to achieve it using the rod cutting problem approach?

hard

A. Max profit 12 by cuts [2,3]

B. Max profit 11 by cuts [1,4]

C. Max profit 13 by cuts [1,1,3]

D. Max profit 10 by cuts [5]

Solution

1. Step 1: Calculate max profit dp array

   dp[0] = 0
   dp[1] = max(prices[0] + dp[0]) = 2 + 0 = 2
   dp[2] = max(2 + dp[1] = 4, 5 + dp[0] = 5) = 5
   dp[3] = max(2 + dp[2] = 7, 5 + dp[1] = 7, 7 + dp[0] = 7) = 7
   dp[4] = max(2 + dp[3] = 9, 5 + dp[2] = 10, 7 + dp[1] = 9, 8 + dp[0] = 8) = 10
   dp[5] = max(2 + dp[4] = 12, 5 + dp[3] = 12, 7 + dp[2] = 12, 8 + dp[1] = 10, 10 + dp[0] = 10) = 12

2. Step 2: Identify cuts for max profit 12

   Max profit 12 achieved by cuts [2,3] or [3,2]. Prices 5 + 7 = 12.

3. Final Answer:

   Max profit 12 by cuts [2,3] -> Option A

4. Quick Check:

   Max profit 12 with cuts 2 and 3 [OK]

Hint: Calculate dp and track cuts for max profit [OK]

Common Mistakes:

• Not checking all cut combinations
• Assuming single cut is best
• Ignoring dp values for smaller lengths