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#include <bits/stdc++.h>

using namespace std;

typedef unsigned uint;
typedef long long ll;
typedef unsigned long long ull;
typedef long double ldbl;
typedef pair<int, int> pii;
typedef pair<uint, uint> puu;
typedef pair<ll, ll> pll;
typedef pair<ull, ull> pull;
typedef pair<double, double> pdd;
typedef vector<int> vi;
typedef vector<uint> vu;
typedef vector<ll> vll;
typedef vector<ull> vull;
typedef vector<pii> vpii;
typedef vector<puu> vpuu;
typedef vector<pll> vpll;
typedef vector<pull> vpull;
typedef vector<string> vstr;
typedef vector<double> vdbl;
typedef vector<ldbl> vldbl;
#define pb push_back
#define ppb pop_back
#define pfr push_front
#define ppfr pop_front
#define emp emplace
#define empb emplace_back
#define be begin
#define rbe rbegin
#define all(x) (x).be(), (x).end()
#define rall(x) (x).rbe(), (x).rend()
#define fir first
#define sec second
#define mkp make_pair
#define brif(cond) if (cond) break
#define ctif(cond) if (cond) continue
#define retif(cond) if (cond) return
void canhazfast() {ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);}
template<typename T> T gcd(T a, T b) {return b ? gcd(b, a%b) : a;}
template<typename T> T extgcd(T a, T b, T &x, T &y)
{
    T x0 = 1, y0 = 0, x1 = 0, y1 = 1;
    while (b) {
        T q = a/b; a %= b; swap(a, b);
        x0 -= q*x1; swap(x0, x1);
        y0 -= q*y1; swap(y0, y1);
    }
    x = x0; y = y0; return a;
}
int ctz(uint x) {return __builtin_ctz(x);}
int ctzll(ull x) {return __builtin_ctzll(x);}
int clz(uint x) {return __builtin_clz(x);}
int clzll(ull x) {return __builtin_clzll(x);}
int popcnt(uint x) {return __builtin_popcount(x);}
int popcntll(ull x) {return __builtin_popcountll(x);}
int bsr(uint x) {return 31^clz(x);}
int bsrll(ull x) {return 63^clzll(x);}

#define N 100016
#define INF ((ll)2e10)

char a[N];
int ps[N], f[N];
ll dp[2][N];

int main()
{
    canhazfast();

    int n, k, m, lo;
    vi cuts(1, 0);
    ll *pre = dp[0];
    ll *nxt = dp[1];
    ll cnt = 0;

    cin >> n >> k >> a;
    ///retif(k > n/2) cout << "0\n", 0;
    for (int i = 0; i < n; ++i) {
        ps[i+1] = ps[i];
        if (a[i] == '(') ++ps[i+1];
        else --ps[i+1];
    }
    lo = *min_element(ps, ps+n+1);
    for (int i = 0; i <= n; ++i) ps[i] -= lo;
    for (int i = 1; i < n; ++i) {
        if (ps[i] > max(ps[i-1], ps[i+1])) cuts.pb(i);
    }
    m = cuts.size();
    retif(k >= m) cout << "0\n", 0;
    cuts.pb(n);
    f[ps[0]] = 1;
    for (int i = 0; i < m; ++i) {
        for (int j = cuts[i]+1; j <= cuts[i+1]; ++j) {
            cnt += f[ps[j]]++;
            f[ps[j]+1] = 0;
        }
        pre[i+1] = cnt;
    }
    for (int lvl = 1; lvl < k; ++lvl) {
        fill_n(nxt, m+1, INF);
        for (int st = lvl; st < m; ++st) {
            fill_n(f, ps[cuts[st]], 0);
            f[ps[cuts[st]]] = 1;
            cnt = 0;
            for (int i = st; i < m; ++i) {
                for (int j = cuts[i]+1; j <= cuts[i+1]; ++j) {
                    cnt += f[ps[j]]++;
                    f[ps[j]+1] = 0;
                }
                nxt[i+1] = min(nxt[i+1], pre[st]+cnt);
            }
        }
        swap(pre, nxt);
    }
    cout << pre[m] << '\n';

    return 0;
}