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Allowed Letters Hall定理

題意:

給你一個字串,你可以任意交換字元,現在要求某些下標只能是一個字元集合裡面的字元,問字典序最小是多少?

字元在a - f之間。

題解:

把所有原來的字母當成二分圖左邊部分, 右邊部分是所有下標的字元集合, 如果字元出現在字元集合中就連邊,一種方案就是一種完美匹配。

現在要貪心的求字典序最小,我們只要列舉每個位置選哪個字元,然後用hall定理判斷選之後還是否能構成完美匹配即可。

程式碼:

#include <bits/stdc++.h>
#ifdef LOCAL
#define debug(x) cout<<#x<<" = "<<(x)<<endl;
#else
#define debug(x) 1;
#endif

#define chmax(x,y) x=max(x,y)
#define chmin(x,y) x=min(x,y)
#define lson id<<1,l,mid
#define rson id<<1|1,mid+1,r
#define lowbit(x) x&-x
#define mp make_pair
#define pb push_back
#define fir first
#define sec second
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;

const int MOD = 1e9 + 7;
const double PI = acos (-1.);
const double eps = 1e-10;
const int INF = 0x3f3f3f3f;
const ll INFLL = 0x3f3f3f3f3f3f3f3f;
const int MAXN = 5e5 + 5;

int cnt[MAXN][1 << 6];
char s[MAXN], t[MAXN];
int cnt2[6];
int val[MAXN];
char ans[MAXN];

int main() {
#ifdef LOCAL
    freopen ("input.txt", "r", stdin);
#endif
    int m;
    scanf ("%s %d", s + 1, &m);
    int n = strlen (s + 1);
    for (int i = 1; i <= n; i++) cnt2[s[i] - 'a']++;
    for (int i = 1; i <= m; i++) {
        int id;
        scanf ("%d %s", &id, t);
        for (int j = 0; t[j]; j++) val[id] |= (1 << (t[j] - 'a') );
    }
    for (int i = n; i >= 1; i--) {
        if (val[i] == 0) val[i] = (1 << 6) - 1;
        for (int j = 0; j < (1 << 6); j++) {
            cnt[i][j] = cnt[i + 1][j];
            if (j & val[i]) cnt[i][j]++;
        }
    }
    for (int i = 1; i <= n; i++) {
        int can = 0;
        for (int j = 0; j < 6; j++) {
            if (cnt2[j] && (val[i] >> j & 1)) {
                cnt2[j]--;
                int ok = 1;
                for (int k = 0; k < (1 << 6); k++) {
                    int num1 = 0;
                    for (int l = 0; l < 6; l++)
                        if (k >> l & 1) num1 += cnt2[l];
                    if (num1 > cnt[i + 1][k]) ok = 0;
                }
                if (ok) {
                    ans[i] = j + 'a';
                    can = 1;
                    break;
                }
                cnt2[j]++;
            }
        }
        if (!can) return  0 * puts ("Impossible");
    }
    printf ("%s\n", ans + 1);
    return 0;
}