Hi, I need help understanding this question solution. The problem is https://leetcode.com/problems/find-the-minimum-cost-array-permutation/

Basically we are given a permutation of 0 to n and have to construct another permutation $$$res$$$ that minimizes the function:

Can anyone explain to me the general idea of the solution? I see union find data structure being used and I assume:

1. start with 0 in the permutation because of cyclic property (any optimal answer can be cyclically rotated to have 0 at start)

2. ans[i] = element to follow i in permutation

We find the relation between nums[ans[i]] and i and try to 'merge' elements to minimize the absolute difference But i don't understand the check code. What is the dir and need arrays and why do we check !merge(i,i+1). Why is two adjacent nodes having a connection in nums array disqualify the candidate for next value (i.e ans[i]) ? Why are the ranges chosen while calling work() for x < i and x > i ?

What is the use of merging blocks and what is the significance?

class Solution {
public:
    struct DSU {
    std::vector<int> f, siz;
    DSU(int n) : f(n), siz(n, 1) { std::iota(f.begin(), f.end(), 0); }
    int leader(int x) {
        while (x != f[x]) x = f[x] = f[f[x]];
        return x;
    }
    bool same(int x, int y) { return leader(x) == leader(y); }
    bool merge(int x, int y) {
        x = leader(x);
        y = leader(y);
        if (x == y) return false;
        siz[x] += siz[y];
        f[y] = x;
        return true;
    }
};
    vector<int> findPermutation(vector<int>& nums) {
        int n = nums.size();
        vector<int> ans(n, -1), res(n);
        DSU dsu(n);
        std::vector<bool> vis(n);
        for (int i = 0; i < n; i++) {
            if (vis[i]) {
                continue;
            }
            for (int j = i; !vis[j]; j = nums[j]) {
                vis[j] = true;
                dsu.merge(j, i);
            }
        }
        
        auto check = [&]() {
            DSU g = dsu;
            
            std::vector<bool> need(n - 1);
            for (int i = 0; i < n; i++) {
                if (ans[i] != -1) {
                    int x = nums[ans[i]];
                    if (x < i) {
                        for (int j = x; j < i; j++) {
                            need[j] = true;
                        }
                    } else {
                        for (int j = i; j < x; j++) {
                            need[j] = true;
                        }
                    }
                }
            }
            
            for (int i = 0; i < n - 1; i++) {
                if (need[i] && !g.merge(i, i + 1)) {
                    return false;
                }
            }
            
            std::vector<int> dir(n - 2, -1);
            std::vector<bool> cant(n - 1);
            
            auto work = [&](int x, int d) {
                if (dir[x] >= 0 && dir[x] != d) {
                    dir[x] = 0;
                } 
                else {
                    dir[x] = d;
                }
            };
                
            for (int i = 0; i < n; i++) {
                if (ans[i] != -1) {
                    int x = nums[ans[i]];
                    if (x < i) {
                        for (int j = x; j + 1 < i; j++) {
                            work(j, 1);
                        }
                        if (x > 0) {
                            work(x - 1, 2);
                        }
                        if (i + 1 < n) {
                            work(i - 1, 2);
                        }
                    } else if (x > i) {
                        for (int j = x-1; j > i ; j--) {
                            work(j-1 , 2);
                        }
                        if (x + 1 < n) {
                            work(x - 1, 1);
                        }
                        if (i > 0) {
                            work(i - 1, 1);
                        }
                    } else {
                        if (x + 1 < n) {
                            cant[x] = true;
                        }
                        if (x > 0) {
                            cant[x - 1] = true;
                        }
                    }
                }
            }
                
            for (int i = 0; i + 2 < n; i++) {
                if (need[i] && need[i + 1] && dir[i] == 0) {
                    return false;
                }
            }
            for (int i = 0; i + 1 < n; i++) {
                if (need[i] && cant[i]) {
                    return false;
                }
            }
            
            for (int i = 0; i < n - 1; i++) {
                if (need[i]) {
                    continue;
                }
                if (cant[i]) {
                    continue;
                }
                if (i > 0 && dir[i - 1] == 0) {
                    continue;
                }
                if (i < n - 2 && dir[i] == 0) {
                    continue;
                }
                g.merge(i, i + 1);
            }
            for (int i = 0; i < n; i++) {
                if (!g.same(0, i)) {
                    return false;
                }
            }
            
            return true;
        };
        
        std::vector<bool> cyc(n);
        int cnt = 0;
        int j = 0;
        for (int i = 0; ; i = ans[i]) {
            res[j] = i;
            j++;
            cyc[i] = true;
            cnt++;
            ans[i] = 0;
            while ((cnt < n && cyc[ans[i]]) || !check()) {
                ans[i]++;
            }
            if(ans[i] == 0) break;
        }
        return res;
    }
};