Hello everyone,

finding the diameter is one of the most frequent ways to solve problems about trees. In this tutorial we will see how to find the diameter and some of its properties, and we will use them to solve some problems of increasing difficulty. I tried to put a lot of examples to make the understanding easier.

The first part of the tutorial is quite basic, so feel free to skip it and jump to the problems if you already know the concepts.

Target: rating $$$[1400, 2100]$$$ on CF

Prerequisites: basic graph theory, greedy

## The diameter

Let's define $$$\text{dist}(a, b) =$$$ the number of edges in the simple path $$$a \rightarrow b$$$.

*You are given a permutation $$$a$$$ of length $$$n$$$. In one move, you can swap two elements in adjacent positions. What's the minimum number of moves required to sort the array?*

#### Claim

The result $$$k$$$ is equal to the number of inversions, i.e. the pairs $$$(i, j)$$$ ($$$1 \leq i < j \leq n$$$) such that $$$a_i > a_j$$$.

#### Proof 1

Let $$$f(x)$$$ be the number of inversions after $$$x$$$ moves.

In one move, if you swap the values on positions $$$i, i + 1$$$, $$$f(x)$$$ either increases by $$$1$$$ or decreases by $$$1$$$. This is because the only pair $$$(a_i, a_j)$$$ whose relative order changed is $$$(a_i, a_{i+1})$$$. Since the sorted array has $$$0$$$ inversions, you need at least $$$k$$$ moves to sort the array.

For example, if you have the permutation $$$[2, 3, 7, 8, 6, 9, 1, 4, 5]$$$ ($$$16$$$ inversions) and you swap two adjacent elements such that $$$a_i > a_{i+1}$$$ (getting, for example, $$$[2, 3, 7, 6, 8, 9, 1, 4, 5]$$$), the resulting array has $$$15$$$ inversions, and if you swap two adjacent elements such that $$$a_i < a_{i+1}$$$ (getting, for example, $$$[3, 2, 7, 8, 6, 9, 1, 4, 5]$$$), the resulting array has $$$17$$$ inversions.

On the other hand, if the array is not sorted you can always find an $$$i$$$ such that $$$a_i > a_{i+1}$$$, so you can sort the array in $$$k$$$ moves.

#### Proof 2

For each $$$x$$$, let $$$f(x)$$$ be the number of inversions if you consider only the elements from $$$1$$$ to $$$x$$$ in the permutation.

First, let's put $$$x$$$ at the end of the permutation: this requires $$$x - \text{pos}(x)$$$ moves. That's optimal (the actual proof is similar to Proof 1; in an intuitive way, if you put the last element to the end of the array, it doesn't interfere anymore with the other swaps).

For example, if you have the permutation $$$[2, 3, 7, 8, 6, 9, 1, 4, 5]$$$ and you move the $$$9$$$ to the end, you get $$$[2, 3, 7, 8, 6, 1, 4, 5, 9]$$$ and now you need to sort $$$[2, 3, 7, 8, 6, 1, 4, 5]$$$. Hence, $$$f(x) = f(x-1) + x - \text{pos}(x)$$$. For each $$$x$$$, $$$x - \text{pos}(x)$$$ is actually the number of pairs $$$(i, j)$$$ ($$$1 \leq i < j \leq x$$$) such that $$$x = a_i > a_j$$$. So $$$f(x)$$$ is equal to the number of inversions.

#### Counting inversions in $$$O(n \log n)$$$

You can use a Fenwick tree (or a segment tree). There are other solutions (for example, using divide & conquer + merge sort), but they are usually harder to generalize.

For each $$$j$$$, calculate the number of $$$i < j$$$ such that $$$a_i > a_j$$$.

The Fenwick tree should contain the frequency of each value in $$$[1, n]$$$ in the prefix $$$[1, j - 1]$$$ of the array.

So, for each $$$j$$$, the queries look like

- $$$res := res + \text{range_sum}(a_j + 1, n)$$$
- add $$$1$$$ in the position $$$a_j$$$ of the Fenwick tree

#### Observations / slight variations of the problem

By using a Fenwick tree, you are actually calculating the number of inversions for each prefix of the array.

You can calculate the number of swaps required to sort an array (not necessarily a permutation, but for now let's assume that its elements are distinct) by compressing the values of the array. For example, the array $$$[13, 18, 34, 38, 28, 41, 5, 29, 30]$$$ becomes $$$[2, 3, 7, 8, 6, 9, 1, 4, 5]$$$.

You can also calculate the number of swaps required to get an array $$$b$$$ (for now let's assume that its elements are distinct) starting from $$$a$$$, by renaming the values. For example,

$$$a = [2, 3, 7, 8, 6, 9, 1, 4, 5], b = [9, 8, 5, 2, 1, 4, 7, 3, 6]$$$

is equivalent to

$$$a = [4, 8, 7, 2, 9, 1, 5, 6, 3], b = [1, 2, 3, 4, 5, 6, 7, 8, 9]$$$

$$$a^{-1}$$$ (a permutation such that $$$(a^{-1})_{a_x} = x$$$, i.e. $$$(a^{-1})_x$$$ is equal to the position of $$$x$$$ in $$$a$$$) has the same number of inversions as $$$a$$$. For example, $$$[2, 3, 7, 8, 6, 9, 1, 4, 5]$$$ and $$$[7, 1, 2, 8, 9, 5, 3, 4, 6]$$$ have both $$$16$$$ inversions. Sketch of a proof: note that, when you swap two elements in adjacent positions in $$$a$$$, you are swapping two adjacent values in $$$a^{-1}$$$, and the number of inversions in $$$a^{-1}$$$ also increases by $$$1$$$ or decreases by $$$1$$$ (like in Proof 1).

## 1430E - String Reversal (rating: 1900)

**Hint 1**

**Hint 2**

**Hint 3**

**Solution**

## 103148B - Luna Likes Love (EGOI 2021/2)

**Hint 1**

**Hint 2**

**Hint 3**

**Hint 4**

**Solution**

## arc088_e (rating: 2231)

**Hint 1**

**Hint 2**

**Hint 3**

**Hint 4**

**Solution**

## arc097_e (rating: 2247)

**Hint 1**

**Hint 2**

**Hint 3**

**Hint 4**

**Solution**

## Other problems

IOI 2019/1

arc120_c (suggested by Ghassane)

Hackerearth — Swapping numbers (Inferno03)

Hackerearth — Make the strings equal (Inferno03)

1526D - Kill Anton (somil_jain_120)

JOI 2021/3 (Final Round) (you can submit here)

## Conclusions

We've seen that a lot of problems where you have to swap adjacent elements can be tackled with greedy observations, such as looking at the optimal relative positions of the values in the final array; then, a lot of these problems can be reduced to "find the number of inversions" or similar.

Of course, suggestions/corrections are welcome. In particular, please share in the comments other problems where you have to swap adjacent elements.

I hope you enjoyed the blog!