Subsequence
题目背景:
分析:DP + 复杂度分析
这个题感觉难点在复杂度分析,看似总状态数n * m转移复杂度k,但是实际上,可以证明得到答案数与k的乘积是小于(n + m)的,证明:假设当前匹配到A的第j位,B的第i位,那么第j + 1位到第j + k - 1位,那么一定有一个数没有出现过,那么选择这个数就会让i + j增加k,所以每一次操作至少能增加k,所以最后的答案不超过(n + m) / k,所以直接按照O(n3)的方式DP就可以了,实际复杂度为O(n2),定义dp[i][j]表示,当前枚举第i位,在A串中匹配到j,dp[i][j]表示在B串中匹配到的最后的位置。预处理next_a[i][j]表示第i位之后第一个j出现的位置,最后结尾设为n + 1,那么当dp[i][n + 1] = m + 1时,说明i就是可行的长度了。
Source:
/* created by scarlyw */ #include <cstdio> #include <string> #include <algorithm> #include <cstring> #include <iostream> #include <cmath> #include <cctype> #include <vector> #include <set> #include <queue> #include <ctime> #include <bitset> inline char read() { static const int IN_LEN = 1024 * 1024; static char buf[IN_LEN], *s, *t; if (s == t) { t = (s = buf) + fread(buf, 1, IN_LEN, stdin); if (s == t) return -1; } return *s++; } ///* template<class T> inline void R(T &x) { static char c; static bool iosig; for (c = read(), iosig = false; !isdigit(c); c = read()) { if (c == -1) return ; if (c == '-') iosig = true; } for (x = 0; isdigit(c); c = read()) x = ((x << 2) + x << 1) + (c ^ '0'); if (iosig) x = -x; } //*/ const int OUT_LEN = 1024 * 1024; char obuf[OUT_LEN], *oh = obuf; inline void write_char(char c) { if (oh == obuf + OUT_LEN) fwrite(obuf, 1, OUT_LEN, stdout), oh = obuf; *oh++ = c; } template<class T> inline void W(T x) { static int buf[30], cnt; if (x == 0) write_char('0'); else { if (x < 0) write_char('-'), x = -x; for (cnt = 0; x; x /= 10) buf[++cnt] = x % 10 + 48; while (cnt) write_char(buf[cnt--]); } } inline void flush() { fwrite(obuf, 1, oh - obuf, stdout); } /* template<class T> inline void R(T &x) { static char c; static bool iosig; for (c = getchar(), iosig = false; !isdigit(c); c = getchar()) if (c == '-') iosig = true; for (x = 0; isdigit(c); c = getchar()) x = ((x << 2) + x << 1) + (c ^ '0'); if (iosig) x = -x; } //*/ const int MAXN = 4000 + 10; int n, m, k; int dp[MAXN][MAXN], a[MAXN], b[MAXN]; int next_a[MAXN][MAXN], next_b[MAXN][MAXN]; inline void read_in() { R(n), R(m), R(k); for (int i = 1; i <= n; ++i) R(a[i]); for (int i = 1; i <= m; ++i) R(b[i]); for (int i = 1; i <= k; ++i) { next_a[n][i] = next_a[n + 1][i] = n + 1; next_b[m][i] = next_b[m + 1][i] = m + 1; } for (int i = n - 1; i >= 0; --i) { for (int j = 1; j <= k; ++j) next_a[i][j] = next_a[i + 1][j]; next_a[i][a[i + 1]] = i + 1; } for (int i = m - 1; i >= 0; --i) { for (int j = 1; j <= k; ++j) next_b[i][j] = next_b[i + 1][j]; next_b[i][b[i + 1]] = i + 1; } } inline void solve() { memset(dp, -1, sizeof(dp)); dp[0][0] = 0; for (int i = 0; ; ++i) { for (int j = 0; j <= n + 1; ++j) { if (dp[i][j] == -1) continue ; if (j == n + 1 && dp[i][j] == m + 1) std::cout << i, exit(0); for (int c = 1; c <= k; ++c) dp[i + 1][next_a[j][c]] = std::max(dp[i + 1][next_a[j][c]], next_b[dp[i][j]][c]); } } } int main() { freopen("subsequence.in", "r", stdin); freopen("subsequence.out", "w", stdout); read_in(); solve(); return 0; }