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Is there an existing algorithm that can solve the following problem?

Problem

Given an list N comprised of integers of various values, shard the list into k buckets such that:

  1. The sum of the integers in each bucket are roughly evenly divided (as much as possible
  2. Only consecutive integers are placed in the same bucket

Example:

Given k (the number of buckets) = 4

List N = [4 1 2 6 19 1 10 3 2 2]

A Valid Solution

[4 1 2] [6] [19] [1 10 3 2 2]

(Since the bucket weights add up to 7, 6, 19, 18)

An Invalid Solution

[1 1 2 2 2 3] [1 4 6] [10] [19]

(Where the bucket weights would add up to 11, 11, 10, 19)

It's invalid because the buckets no longer contain only numbers that were adjacent in the original list.

Thanks!

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Yes, this can be solved as a straightforward application of dynamic programming, once you define an objective function to measure how "evenly divided" the buckets are. Let $A[j]$ denote the value of the objective function for the best way to split integers 1..j. Then you can compute the value of $A[j]$ given the values of $A[1],\dots,A[j-1]$ (you just consider all possibilities for how many integers go in the last bucket), so there is a dynamic programming algorithm with running time at most $O(n^2)$.

This problem is sometimes called the Linear Partition Problem, and a solution can be found in Section 8.5 of Skiena's "Algorithm Design Manual".

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  • $\begingroup$ Thanks. Turns out that this problem is called the Linear Partition Problem, with an exact solution provided in Skiena's "Algorithm Design Manual" (in section 8.5) $\endgroup$ – Zain Rizvi Jan 17 '18 at 0:44

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