백준25546 가채점

작성일 | In

문제 링크

  • https://icpc.me/25546

사용 알고리즘

  • Suffix Array
  • Persistent Segment Tree

풀이

각 패턴 $P_i$에 대해, $S$에서 $P_i$가 $K_i$번째로 등장하는 위치를 구하는 문제입니다.
문자열 $T = S#P_1#P_2#\cdots#P_M$를 생각해 봅시다. 패턴 $P_i$가 등장하는 위치는 $T$의 접미사 배열 상에서 구간 형태로 나타낼 수 있고, LCP 배열과 RMQ를 이용해 파라메트릭 서치를 하면 각 패턴마다 $O(\log N)$ 시간에 구간을 구할 수 있습니다.

이 과정을 거치면 구간에서 $K_i$번째 수를 찾는 문제로 바뀌는데, 이건 PST를 사용하면 $O(\log N)$에 구할 수 있습니다.

전체 코드

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#include <bits/stdc++.h>
using namespace std;
using ll = long long;

vector<int> internal_sa_is(const vector<int> &s, int upper){
    int n = s.size(), m = 0;
    if(n == 0) return {};
    if(n == 1) return {0};
    if(n == 2) return s[0] < s[1] ? vector<int>{0, 1} : vector<int>{1, 0};

    vector<int> sa(n), ls(n), sum_l(upper+1), sum_s(upper+1);
    for(int i=n-2; i>=0; i--) ls[i] = s[i] != s[i+1] ? s[i] < s[i+1] : ls[i+1];
    for(int i=0; i<n; i++) (!ls[i] ? sum_s[s[i]] : sum_l[s[i]+1])++;
    for(int i=0; i<upper; i++) sum_s[i] += sum_l[i], sum_l[i+1] += sum_s[i];
    sum_s[upper] += sum_l[upper];

    auto induce = [&](const vector<int> &lms) -> void {
        fill(sa.begin(), sa.end(), -1);
        auto buf = sum_s;
        for(auto i : lms) if(i != n) sa[buf[s[i]]++] = i;
        buf = sum_l; sa[buf[s[n-1]]++] = n-1;
        for(int i=0; i<n; i++) if(sa[i] >= 1 && !ls[sa[i]-1]) sa[buf[s[sa[i]-1]]++] = sa[i]-1;
        buf = sum_l;
        for(int i=n-1; i>=0; i--) if(sa[i] >= 1 && ls[sa[i]-1]) sa[--buf[s[sa[i]-1]+1]] = sa[i]-1;
    };

    vector<int> lms_map(n+1, -1);
    for(int i=1; i<n; i++) if(!ls[i-1] && ls[i]) lms_map[i] = m++;
    std::vector<int> lms; lms.reserve(m);
    for(int i=1; i<n; i++) if(!ls[i-1] && ls[i]) lms.push_back(i);
    induce(lms);
    if(!m) return sa;

    int rec_upper = 0;
    vector<int> sorted_lms, rec_s(m); sorted_lms.reserve(m);
    for(const auto i : sa) if(lms_map[i] != -1) sorted_lms.push_back(i);
    rec_s[lms_map[sorted_lms[0]]] = 0;
    for(int i=1; i<m; i++){
        int l = sorted_lms[i-1], r = sorted_lms[i];
        int el = lms_map[l] + 1 < m ? lms[lms_map[l]+1] : n;
        int er = lms_map[r] + 1 < m ? lms[lms_map[r]+1] : n;
        bool flag = true;
        if(el - l != er - r) flag = false;
        else{
            for(; l<el; l++, r++) if(s[l] != s[r]) break;
            if(l == n || s[l] != s[r]) flag = false;
        }
        if(!flag) rec_upper++;
        rec_s[lms_map[sorted_lms[i]]] = rec_upper;
    }
    auto rec_sa = internal_sa_is(rec_s, rec_upper);
    for(int i=0; i<m; i++) sorted_lms[i] = lms[rec_sa[i]];
    induce(sorted_lms);
    return sa;
}

template<typename Container>
pair<vector<int>, vector<int>> SuffixArray(const Container &s){
    int n = s.size(), ch = 0;
    vector<int> idx(n), s2(n);
    iota(idx.begin(), idx.end(), 0);
    sort(idx.begin(), idx.end(), [&](int a, int b){ return s[a] < s[b]; });
    for(int i=0; i<n; i++) s2[idx[i]] = (ch += i && s[idx[i-1]] != s[idx[i]]);

    auto sa = internal_sa_is(s2, ch);
    vector<int> pos(n), lcp(n);
    for(int i=0; i<n; i++) pos[sa[i]] = i;
    for(int i=0, j=0; i<n; i++, j=max(j-1, 0)){
        if(!pos[i]) continue;
        while(sa[pos[i]-1]+j < n && sa[pos[i]]+j < n && s[sa[pos[i]-1]+j] == s[sa[pos[i]]+j]) j++;
        lcp[pos[i]] = j;
    }
    return { sa, lcp };
}

template<typename T> struct RMQ {
    vector<vector<T>> st;
    vector<int> lg;
    RMQ() = default;
    RMQ(const vector<T> &a){
        int n = a.size();
        st = vector<vector<T>>(__lg(n)+1, vector<T>(n));
        for(int i=0; i<n; i++) st[0][i] = a[i];
        for(int i=1; i<st.size(); i++) for(int j=0; j<n; j++) if(j + (1<<i) - 1 < n) st[i][j] = min(st[i-1][j], st[i-1][j+(1<<(i-1))]);
        lg.resize(n);
        for(int i=0; i<st.size(); i++) lg[1<<i] = i;
        for(int i=1; i<n; i++) if(!lg[i]) lg[i] = lg[i-1];
    }
    T query(int l, int r){
        if(l > r) return 0x3f3f3f3f;
        int k = lg[r-l+1];
        return min(st[k][l], st[k][r-(1<<k)+1]);
    }
};

constexpr int SZ = 1 << 21;
struct PST{
    struct Node{
        int l, r, v;
        Node(){ l = r = v = 0; }
    } T[SZ * 24];
    int Root[SZ*2], Count;
    PST(){ memset(Root, 0, sizeof Root); Count = 0; }
    void Init(int node, int s, int e){
        if(s == e) return;
        int m = (s + e) / 2;
        T[node].l = ++Count; T[node].r = ++Count;
        Init(T[node].l, s, m);
        Init(T[node].r, m+1, e);
    }
    void Update(int prv, int now, int s, int e, int x, int v){
        if(s == e){ T[now].v = prv ? T[prv].v + v : v; return; }
        int m = (s + e) / 2;
        if(x <= m){
            T[now].l = ++Count; T[now].r = T[prv].r;
            Update(T[prv].l, T[now].l, s, m, x, v);
        }
        else{
            T[now].r = ++Count; T[now].l = T[prv].l;
            Update(T[prv].r, T[now].r, m+1, e, x, v);
        }
        T[now].v = T[T[now].l].v + T[T[now].r].v;
    }
    int Query(int prv, int now, int s, int e, int k){
        if(s == e) return s;
        int diff = T[T[now].l].v - T[T[prv].l].v;
        int m = (s + e) / 2;
        if(k <= diff) return Query(T[prv].l, T[now].l, s, m, k);
        else return Query(T[prv].r, T[now].r, m+1, e, k-diff);
    }
} T;

int N, M; ll A, B[101010];
vector<string> V;
vector<int> S, C, CC;
vector<int> L, R;
vector<int> Pos;
vector<vector<int>> Idx;
RMQ<int> Q;

int FindL(int suf, int len){
    int l = 0, r = suf;
    while(l < r){
        int m = (l + r) / 2;
        if(Q.query(m+1, suf) >= len) r = m;
        else l = m + 1;
    }
    return r;
}

int FindR(int suf, int len){
    int l = suf, r = S.size() - 1;
    while(l < r){
        int m = (l + r + 1) / 2;
        if(Q.query(suf+1, m) >= len) l = m;
        else r = m - 1;
    }
    return l;
}

int main(){
    ios_base::sync_with_stdio(false); cin.tie(nullptr);
    cin >> N >> M >> A; V.resize(M+1);
    cin >> V[0];
    for(int i=1,t; i<=M; i++) cin >> t >> B[i] >> V[i];

    for(int i=0; i<V.size(); i++){
        L.push_back(S.size());
        for(auto c : V[i]) S.push_back(c);
        R.push_back(S.size()-1);
        S.push_back(-i-1);
    }
    auto [SA,LCP] = SuffixArray(S);
    Pos.resize(S.size());
    for(int i=0; i<S.size(); i++) Pos[SA[i]] = i;
    Idx.resize(V.size());
    for(int i=0; i<V.size(); i++){
        for(int j=L[i]; j<=R[i]; j++) Idx[i].push_back(Pos[j]);
        sort(Idx[i].begin(), Idx[i].end());
    }

    Q = RMQ<int>(LCP);
    C = vector<int>(S.size(), -1);
    CC = vector<int>(S.size(), 0);
    for(int i=L[0]; i<=R[0]; i++) C[Pos[i]] = i, CC[Pos[i]] = 1;
    partial_sum(CC.begin(), CC.end(), CC.begin());

    int cnt = 0;
    T.Init(T.Root[0]=++T.Count, 0, SZ-1);
    for(int i=1; i<S.size(); i++){
        if(C[i] == -1){ T.Root[i] = T.Root[i-1]; continue; }
        T.Root[i] = ++T.Count;
        T.Update(T.Root[i-1], T.Root[i], 0, SZ-1, C[i], 1);
    }

    int res = 0x3f3f3f3f;
    for(int i=1; i<=M; i++){
        int now = Pos[L[i]];
        int le = FindL(now, V[i].size());
        int ri = FindR(now, V[i].size());
        ll k = (A + B[i] - 1) / B[i];
        if(CC[ri] - CC[le-1] < k) continue;
        res = min<int>(res, T.Query(T.Root[le-1], T.Root[ri], 0, SZ-1, k) + V[i].size());
    }
    if(res == 0x3f3f3f3f) cout << "Fail";
    else cout << res;
}