#include <cstdio>
#include <iostream>
#include <algorithm>
#include <cstring>
#include <vector>
#include <set>
#include <map>
#include <array>
#include <random>
#include <cmath>
#include <chrono>
#include <list>
#include <ctime>
#include <sstream>
#include <queue>
#include <climits>
#include <stack>
#include <valarray>
#include <random>
#include <bitset>
#include <numeric>
#include <iomanip>
#include <cassert>
using namespace std;
typedef vector<int> vi;
typedef pair<int,int> pii;
typedef long long ll;
#define rep(x, b, e) for(int x=(b); x<(e); ++x)
#define trav(a, x) for(auto& a : x)
#define ford(x, b, e) for(int x=((int)(b))-1; x>=(e); --x)
#define all(c) c.begin(),c.end()
#define sz(x) ((int)((x).size()))
#define pb push_back
#define eb emplace_back
#define st first
#define nd second
#define mp(x,y) make_pair(x,y)
typedef short int sint;
template<typename T> bool ckmin(T& a, const T& b){return b<a?a=b,1:0;}
template<typename T> bool ckmax(T& a, const T& b){return b>a?a=b,1:0;}


template <typename A, typename B>
string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string& s) {
  return '"' + s + '"';
}

string to_string(char c) {
  return string(1, c);
}

string to_string(const char* s) {
  return to_string((string) s);
}

string to_string(bool b) {
  return (b ? "true" : "false");
}

string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}

template <size_t N>
string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}

template <typename A>
string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto &x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}

template <typename A, typename B>
string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}

#ifdef LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif


// #include <atcoder/modint>
// using namespace atcoder;
// using mint = modint998244353; // modint1000000007;
// typedef vector<mint> vmi;

mt19937 rng(chrono::steady_clock::now().time_since_epoch().count()); // use rng() to get unsigned int
// mt19937_64 for random long longs

const ll inf = 1e18;

vector<ll> dp, ndp;
vi c;
vi activeOpen, activeClosed; // number of closed in current region
int ga, gb;
ll val;
string x;

// that is added: [0; 0]
void update(int goalA, int goalB) {
  while (ga > goalA) {
    if (c[ga - 1] != -1) {
      int v = c[ga - 1];
      if (x[ga - 1] == '(') {
        val += activeClosed[v];
        ++activeOpen[v];
      } else {
        ++activeClosed[v];
      }
    }
    --ga;
    debug("move a left", ga, gb, val);
  }

  while (gb < goalB) { // dodajemy!
    if (c[gb + 1] != -1) {
      int v = c[gb + 1];
      if (x[gb + 1] == '(') {
        ++activeOpen[v];
      } else {
        val += activeOpen[v];
        ++activeClosed[v];
      }
    }
    ++gb;
    debug("move b right", ga, gb, val);
  }

  while (ga < goalA) {
    // need to remove from beginning
    if (c[ga] != -1) {
      int v = c[ga];
      if (x[ga] == '(') {
        val -= activeClosed[v];
        --activeOpen[v];
      } else {
        --activeClosed[v];
      }
    }
    ga++;
    debug("move a right", ga, gb, val);
  }

  while (gb > goalB) {
    if (c[gb] != -1) {
      int v = c[gb];
      if (x[gb] == '(') {
        --activeOpen[v];
      } else {
        val -= activeOpen[v];
        --activeClosed[v];
      }
    }
    gb--;
    debug("move b left", ga, gb, val);
  }

}

void compute(int l, int r, int optl, int optr) {
  if (l > r) return;

  int mid = (l + r) / 2;
  pair<ll, int> best = {inf, -1};

  for (int k = optl; k <= min(mid, optr); ++k) {
    update(k, mid);
    ckmin(best, { (k ? dp[k-1] : 0) + val, k });
  }

  ndp[mid] = best.first;
  int opt = best.second;
  compute(l, mid - 1, optl, opt);
  compute(mid + 1, r, opt, optr);
}

void solve() {
  int n, k;
  cin >> n >> k;
  cin >> x;
  x = '#' + x;
  map<int, int> wys; // kiedy ostatnio dana suma wystepowala
  vi a(n + 1, -1); // pointer to previous
  c.assign(n + 1, -1);
  vi kt(n + 1);
  activeOpen.resize(n + 1);
  activeClosed.resize(n + 1);
  int nr = 0;
  int s = 0;
  wys[s] = 0;
  rep(i, 1, n + 1) {
    if (x[i] == '(') {
      ++s; 
    } else {
      --s;
      if (wys.count(s)) {
        // we can put a pointer
        int pop = wys[s];
        a[i] = pop;
        if (c[pop] != -1) {
          c[i] = c[pop];
          kt[i] = kt[pop] + 1;
        } else {
          c[i] = nr++;
          kt[i] = 1;
        }
        a[pop + 1] = pop + 1;
        c[pop + 1] = c[i];
      }
    }
    wys[s] = i;
  }
  debug(a);
  debug(c);
  debug(kt);

  debug(C);
  update(0, 0);
  dp.resize(n + 1);
  rep(i, 1, n + 1) {
    update(1, i);
    dp[i] = val;
  }
  debug(dp);
  ndp.resize(n + 1);
  rep(i, 1, k) {
    update(0, 0);
    debug(val);
    assert(val == 0);
    compute(1, n, 1, n);
    dp = ndp;
  }
  cout << dp[n] << endl;
}

int main() {
  ios_base::sync_with_stdio(false); cin.tie(0);
  int t;
  // cin >> t;
  t = 1;
  while (t--) {
    solve();
  }
}