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  • 51Nod 1376 最长递增子序列的数量 (DP+BIT)

    题意:略。

    析:dp[i] 表示以第 i 个数结尾的LIS的长度和数量,状态方程很好转移,先说长度 dp[i] = max { dp[j] + 1 | a[i] > a[j] && j < i },然后是数量,dp[i] = sigma(dp[j]) if dp[i] == dp[j] + 1。

    如果普通转移时间复杂度很高,达不到要求,由于有个求和的操作,可以考虑用BIT优化,先把每个数离散化,然后对每个数只要求小于它的数,并且长度最长的就好了,数量也是,如果长度一样就进行合并,否则不合并,或者更新长度。

    代码如下:

    #pragma comment(linker, "/STACK:1024000000,1024000000")
    #include <cstdio>
    #include <string>
    #include <cstdlib>
    #include <cmath>
    #include <iostream>
    #include <cstring>
    #include <set>
    #include <queue>
    #include <algorithm>
    #include <vector>
    #include <map>
    #include <cctype>
    #include <cmath>
    #include <stack>
    #include <sstream>
    #include <list>
    #include <assert.h>
    #include <bitset>
    #include <numeric>
    #define debug() puts("++++")
    #define gcd(a, b) __gcd(a, b)
    #define lson l,m,rt<<1
    #define rson m+1,r,rt<<1|1
    #define fi first
    #define se second
    #define pb push_back
    #define sqr(x) ((x)*(x))
    #define ms(a,b) memset(a, b, sizeof a)
    #define sz size()
    #define pu push_up
    #define pd push_down
    #define cl clear()
    #define all 1,n,1
    #define FOR(i,x,n)  for(int i = (x); i < (n); ++i)
    #define freopenr freopen("in.txt", "r", stdin)
    #define freopenw freopen("out.txt", "w", stdout)
    using namespace std;
    
    typedef long long LL;
    typedef unsigned long long ULL;
    typedef pair<LL, int> P;
    const int INF = 0x3f3f3f3f;
    const LL LNF = 1e17;
    const double inf = 1e20;
    const double PI = acos(-1.0);
    const double eps = 1e-8;
    const int maxn = 5e4 + 10;
    const int maxm = 1e6 + 5;
    const int mod = 1000000007;
    const int dr[] = {-1, 0, 1, 0};
    const int dc[] = {0, -1, 0, 1};
    const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};
    int n, m;
    const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};
    const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};
    inline bool is_in(int r, int c) {
      return r >= 0 && r < n && c >= 0 && c < m;
    }
    
    void update(P &a, P b){
      if(a.fi < b.fi)  a = b;
      else if(a.fi == b.fi){
        a.se += b.se;
        if(a.se >= mod)  a.se -= mod;
      }
    }
    
    P sum[maxn];
    
    inline int lowbit(int x){ return -x&x; }
    
    void add(int x, P c){
      while(x <= m){
        update(sum[x], c);
        x += lowbit(x);
      }
    }
    
    P query(int x){
      P ans(0, 1);
      while(x){
        update(ans, sum[x]);
        x -= lowbit(x);
      }
      return ans;
    }
    int a[maxn];
    vector<int> v;
    
    int getpos(int x){ return lower_bound(v.begin(), v.end(), x) - v.begin() + 1; }
    
    int main(){
      scanf("%d", &n);
      v.resize(n);
      for(int i = 0; i < n; ++i){
        scanf("%d", a+i);
        v[i] = a[i];
      }
      sort(v.begin(), v.end());
      v.resize(unique(v.begin(), v.end()) - v.begin());
      m = v.sz;
      P ans(0, 1);
      for(int i = 0; i < n; ++i){
        int pos = getpos(a[i]);
        P tmp = query(pos - 1);
        ++tmp.fi;
        update(ans, tmp);
        add(pos, tmp);
      }
      printf("%d
    ", ans.se);
      return 0;
    }
    

     分治:

    #pragma comment(linker, "/STACK:1024000000,1024000000")
    #include <cstdio>
    #include <string>
    #include <cstdlib>
    #include <cmath>
    #include <iostream>
    #include <cstring>
    #include <set>
    #include <queue>
    #include <algorithm>
    #include <vector>
    #include <map>
    #include <cctype>
    #include <cmath>
    #include <stack>
    #include <sstream>
    #include <list>
    #include <assert.h>
    #include <bitset>
    #include <numeric>
    #define debug() puts("++++")
    #define gcd(a, b) __gcd(a, b)
    #define lson l,m,rt<<1
    #define rson m+1,r,rt<<1|1
    #define fi first
    #define se second
    #define pb push_back
    #define sqr(x) ((x)*(x))
    #define ms(a,b) memset(a, b, sizeof a)
    #define sz size()
    #define pu push_up
    #define pd push_down
    #define cl clear()
    #define all 1,n,1
    #define FOR(i,x,n)  for(int i = (x); i < (n); ++i)
    #define freopenr freopen("in.txt", "r", stdin)
    #define freopenw freopen("out.txt", "w", stdout)
    using namespace std;
    
    typedef long long LL;
    typedef unsigned long long ULL;
    typedef pair<LL, int> P;
    const int INF = 0x3f3f3f3f;
    const LL LNF = 1e17;
    const double inf = 1e20;
    const double PI = acos(-1.0);
    const double eps = 1e-8;
    const int maxn = 5e4 + 10;
    const int maxm = 1e6 + 5;
    const int mod = 1000000007;
    const int dr[] = {-1, 0, 1, 0};
    const int dc[] = {0, -1, 0, 1};
    const char *de[] = {"0000", "0001", "0010", "0011", "0100", "0101", "0110", "0111", "1000", "1001", "1010", "1011", "1100", "1101", "1110", "1111"};
    int n, m;
    const int mon[] = {0, 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};
    const int monn[] = {0, 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31};
    inline bool is_in(int r, int c) {
      return r >= 0 && r < n && c >= 0 && c < m;
    }
    
    void update(P &a, P b){
      if(a.fi < b.fi)  a = b;
      else if(a.fi == b.fi){
        a.se += b.se;
        if(a.se >= mod)  a.se -= mod;
      }
    }
    P dp[maxn], res;
    
    int id[maxn], a[maxn];
    
    inline bool cmp(int x, int y){ return a[x] == a[y] ? x > y : a[x] < a[y];                                                                                              }
    
    void dfs(int l, int r){
      if(l == r){ update(dp[l], P(1, 1));  return ; }
      int m = l + r >> 1;
      dfs(l, m);
      for(int i = l; i <= r; ++i)  id[i] = i;
      sort(id+l, id+r+1, cmp);
    
      P ans = P(0, 0);
      for(int i = l; i <= r; ++i){
        int idx = id[i];
        if(idx <= m)  update(ans, dp[idx]);
        else{
          P tmp = ans;
          ++tmp.fi;
          update(dp[idx], tmp);
          update(res, tmp);
        }
      }
      dfs(m+1, r);
    }
    
    int main(){
      scanf("%d", &n);
      for(int i = 0; i < n; ++i)  scanf("%d", a+i);
      dfs(0, n-1);
      printf("%d
    ", res.se);
      return 0;
    }
    

      

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  • 原文地址:https://www.cnblogs.com/dwtfukgv/p/7749741.html
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