Problem:
Score : 400 points

Problem Statement

Snuke lives in another world, where slimes are real creatures and kept by some people.
Slimes come in N colors. Those colors are conveniently numbered 1 through N.
Snuke currently has no slime. His objective is to have slimes of all the colors together.

Snuke can perform the following two actions:

Select a color i (1≤i≤N), such that he does not currently have a slime in color i, and catch a slime in color i. This action takes him a_i seconds.

Cast a spell, which changes the color of all the slimes that he currently has. The color of a slime in color i (1≤i≤N-1) will become color i+1, and the color of a slime in color N will become color 1. This action takes him x seconds.

Find the minimum time that Snuke needs to have slimes in all N colors.

Constraints

2≤N≤2,000

a_i are integers.

1≤a_i≤10^9

x is an integer.

1≤x≤10^9

Input

The input is given from Standard Input in the following format:

N x
a_1 a_2 ... a_N

Output

Find the minimum time that Snuke needs to have slimes in all N colors.

Sample Input 1

2 10
1 100

Sample Output 1

12

Snuke can act as follows:

Catch a slime in color 1. This takes 1 second.

Cast the spell. The color of the slime changes: 1 → 2. This takes 10 seconds.

Catch a slime in color 1. This takes 1 second.

Sample Input 2

3 10
100 1 100

Sample Output 2

23

Snuke can act as follows:

Catch a slime in color 2. This takes 1 second.

Cast the spell. The color of the slime changes: 2 → 3. This takes 10 seconds.

Catch a slime in color 2. This takes 1 second.

Cast the soell. The color of each slime changes: 3 → 1, 2 → 3. This takes 10 seconds.

Catch a slime in color 2. This takes 1 second.

Sample Input 3

4 10
1 2 3 4

Sample Output 3

10

Snuke can act as follows:

Catch a slime in color 1. This takes 1 second.

Catch a slime in color 2. This takes 2 seconds.

Catch a slime in color 3. This takes 3 seconds.

Catch a slime in color 4. This takes 4 seconds.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
using i64 = int64_t;
using vi = vector<i64>;
using vvi = vector<vi>;

struct SlideMin {
    // data[i]: mininum over [max(0, i + 1 - width), i + 1)
    int n;
    vi data;
    SlideMin(const vi &as, int width, bool minimum = true) : n(as.size()), data(as.size()) {
        using ii = pair<i64, int>;
        deque<ii> deq;
        auto comp = [&](ii &e, i64 v) {
            return minimum ? e.first >= v : e.first <= v;
        };
        for (int i = 0; i < n; i++) {
            if (deq.size() && deq.front().second <= i - width) {
                deq.pop_front();
            }
            while (deq.size()) {
                if (comp(deq.back(), as[i])) deq.pop_back();
                else break;
            }
            deq.push_back(ii(as[i], i));
            data[i] = deq.front().first;
        }
    }
};

int main() {
    int n, x;
    cin >> n >> x;
    vi as(2 * n);
    for (int i = 0; i < n; i++) {
        int a;
        cin >> a;
        as[i] = as[i + n] = a;
    }
    i64 ans = 1e18;
    for (int i = 1; i <= n; i++) {
        SlideMin smin(as, i);
        i64 now = 1ll * x * (i - 1);
        for (int j = n; j < 2 * n; j++) {
            now += smin.data[j];
        }
        ans = min(ans, now);
    }

    cout << ans << endl;
}