Dynamic Programming – Rod Cutting Problem

Objective: Given a rod of length n inches and a table of prices pi, i=1,2,…,n, write an algorithm to find the maximum revenue rn obtainable by cutting up the rod and selling the pieces.

This is very good basic problem after fibonacci sequence if you are new to Dynamic programming. We will see how the dynamic programming is used to overcome the issues with recursion(Time Complexity).

Given: Rod lengths are integers and For i=1,2,…,n we know the price pi of a rod of length i inches

Example:

Length 1 2 3 4 5 6 7 8 9 10
Price 1 5 8 9 10 17 17 20 24 30
for rod of length: 4
Ways to sell :
•	selling length : 4  ( no cuts) , Price: 9
•	selling length : 1,1,1,1  ( 3 cuts) , Price: 4
•	selling length : 1,1,2  ( 2 cuts) , Price: 7
•	selling length : 2,2  ( 1 cut) , Price: 10
•	selling length : 3, 1  ( 1 cut) , Price: 9

Best Price for rod of length 4: 10

Approach:

Naive Approach : Recursion

  • There can be n-1 cuts can be made in the rod of length n, so there are 2n-1 ways to cut the rod.
  • So for every length we have 2 options either we cut it or not. we will consider both the options and choose the optimal out of it.

Code:

Time Complexity: O(2^n-1)
But this time complexity is very high since we are solving many sub problems repeatedly.

Rod Cutting Problem - Overlapping Sub problems

Rod Cutting Problem – Overlapping Sub problems

Dynamic Programming: Bottom-Up

Instead of solving the sub problems repeatedly we can store the results of it in an array and use it further rather than solving it again.

See the Code for better explanation:

Code:

Result: Max profit for length is 5:11

Rod Cutting Problem - Solution Matrix

Reference: Link

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If you find anything incorrect or you feel that there is any better approach to solve the above problem, please write comment.
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  • Valentin

    I was just about to burst laughing any second while the professor was explaining all the details of “rod cutting” in the class. Seriously, why did it have to be a “rod”?

  • Eugen Daroczy

    In the DP solution, since there’s the option of not cutting the rod/rope at any given point, then we should not initialize:
    int max = -1;

    but use instead the formula below, so we start off using the value of the entire substring. If the cuts add up to more value, that will be the partial solution.
    int max = value[i-1];

    Similarly, for the recursive problem we need so initialize as:
    int max = value[length-1];

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