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Recently, I was trying to solve this dynamic programing problem, but somehow not getting the approach to solve it.

The editorial to the problem seems very confusing too and i would understand how to think properly to approach this problem.

Thanks.

The problem

F. Yet Another Minimization Problem: Time limit per test: 2 seconds | Memory limit per test: 256 megabytes | Input: standard input | Output: standard output

You are given an array of n integers a1... an. The cost of a subsegment is the number of unordered pairs of distinct indices within the subsegment that contain equal elements. Split the given array into k non-intersecting non-empty subsegments so that the sum of their costs is minimum possible. Each element should be present in exactly one subsegment.

Input

The first line contains two integers n and k (2 ≤ n ≤ 105, 2 ≤ k ≤ min (n, 20)) - the length of the array and the number of segments you need to split the array into.

The next line contains n integers a1, a2, ..., an (1 ≤ ai ≤ n) - the elements of the array.

Output

Print single integer: the minimum possible total cost of resulting subsegments.

Example 1

input
7 3
1 1 3 3 3 2 1
output
1

Example 2

input
10 2
1 2 1 2 1 2 1 2 1 2
output
8

Example 3

input
13 3
1 2 2 2 1 2 1 1 1 2 2 1 1
output
9

Note:

In the first example it's optimal to split the sequence into the following three subsegments: [1], [1, 3], [3, 3, 2, 1]. The costs are 0, 0 and 1, thus the answer is 1.

In the second example it's optimal to split the sequence in two equal halves. The cost for each half is 4.

In the third example it's optimal to split the sequence in the following way: [1, 2, 2, 2, 1], [2, 1, 1, 1, 2], [2, 1, 1]. The costs are 4, 4, 1.

pirs
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Ham
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1 Answers1

2

Understanding the problem

You just need to count the number of pairs in the serie of numbers (line 2) divised by k (line 1, second number)

From the example 2:

10 2 
1 2 1 2 1 2 1 2 1 2

[numbers length] [divider]
[serie of numbers]

Take a look about the serie of numbers: 1 2 1 2 1 2 1 2 1 2

You need to divide by 2 the serie as an minimum total cost by pairs [1,2,1,2,1] [2,1,2,1,2]

Then count the numbers of pairs for each number:

[1,2,1,2,1]
1 => [2,1,2,1] = 2 pairs
2 => [1,2,1] = 1 pair
1 => [2,1] = 1 pair
2 => [1] = 0
1 => [] = 0
Sum = 4 pairs

[2,1,2,1,2]
2 => [1,2,1,2] = 2 pairs
1 => [2,1,2] = 1 pairs
2 => [1,2] = 1 pairs
1 => [2] = 0
2 => [] = 0
Sum = 4 pairs

Total = 8

How to find the best segment cost (Finally wrong..)

This method is totally experimental and seems to be correct (but could be totally wrong too ... for now i tested some series and it worked)

I took an other example:

10 3
7 7 2 4 6 9 7 4 1 1

Subsegment 1: I count the number of pairs in all the serie, then i sum each paired value by her key in the array (starting to 0), divide this result by the number of pairs, then round (ceil?) at the closest integrer to find the position (which starting to 1).

enter image description here

Subsegment 2 and 3: I do exactly the same but i excluded the first sub segment.

enter image description here

Total: [7,7,2,4,6,9] [7,4,1] [1] = 1 + 0 + 0 = 1

Testing

I did a little script to generate random values relative to the problem:

var n = document.getElementById('n')
var k = document.getElementById('k')
var s = document.getElementById('s')
var btn = document.getElementById('btn')

// apply random values
function random() {
  n.value = Math.ceil(Math.random() * (Math.random()*20)+2)
  k.value = Math.ceil(Math.random() * (n.value/3)+1)
  var serie = []
  for(var i = 0; i<n.value; i++){
    serie.push(Math.ceil(Math.random() * (n.value-1)))
  }
  s.value = serie.join(' ')
}

window.onload = function() { random() }
btn.onclick = function() { random() }
<input id="n" size="2" value="14">
<input id="k" size="2" value="3">
<input id="s" size="40">
<button id="btn">Random</button>

Hope it will help.

pirs
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