【HDOJ】5632 Rikka with Array

1. 题目描述
$A[i]$表示二级制表示的$i$的数字之和。求$1 \le i < j \le n$并且$A[i]>A[j]$的$(i,j)$的总对数。

2. 基本思路
$n \le 10^300$。$n$这么大，显然只能用数位DP来做，我们可以预先处理一下将$n$表示成二进制，然后再进行DP。
$dp[i][j][k]$表示长度为i，两者$A$的差为$j$，状态为$k$的总数。
不妨令$|n| = l$，因此$j \in [-l, l]$，因此需要$+l$，将$j$映射到$[0,l*2]$上。
在考虑$k$有多少种情况？不妨令$(x,y)， x<y$表示一对可行解。
(0) $Pref(x) < Pref(y), Pref(y) < Pref(n)$;
(1) $Pref(x) < Pref(y), Pref(y) == Pref(n)$;
(2) $Pref(x) == Pref(y), Pref(y) < Pref(n)$;
(3) $Pref(x) == Pref(y), Pref(y) == Pref(n)$;
上面4中情况分别对应$k \in [0, 3]$，剩下的就是状态转移就好了，还是挺简单的。总对数就是
\[\sum_{j = l+1}^{l*2}{dp[l][j][0]+dp[l][j][1]}\]
可以使用滚动数组优化，其实也可以不使用。

3. 代码

 /* 5632 */
 #include <iostream>
 #include <sstream>
 #include <string>
 #include <map>
 #include <queue>
 #include <set>
 #include <stack>
 #include <vector>
 #include <deque>
 #include <bitset>
 #include <algorithm>
 #include <cstdio>
 #include <cmath>
 #include <ctime>
 #include <cstring>
 #include <climits>
 #include <cctype>
 #include <cassert>
 #include <functional>
 #include <iterator>
 #include <iomanip>
 using namespace std;
 //#pragma comment(linker,"/STACK:102400000,1024000")
 
 #define sti                set<int>
 #define stpii            set<pair<int, int> >
 #define mpii            map<int,int>
 #define vi                vector<int>
 #define pii                pair<int,int>
 #define vpii            vector<pair<int,int> >
 #define rep(i, a, n)     for (int i=a;i<n;++i)
 #define per(i, a, n)     for (int i=n-1;i>=a;--i)
 #define clr                clear
 #define pb                 push_back
 #define mp                 make_pair
 #define fir                first
 #define sec                second
 #define all(x)             (x).begin(),(x).end()
 #define SZ(x)             ((int)(x).size())
 #define lson            l, mid, rt<<1
 #define rson            mid+1, r, rt<<1|1
 
 const int mod = ;
 const int maxl = ;
 const int maxn = ;
 char ss[maxl];
 int a[maxn];
 int dp[][maxn<<][];
 
 void solve() {
     int l = , tmp;
     int len = strlen(ss);
 
     rep(i, , len)
         ss[i] -= '';
 
     int b = ;
 
     while (b<len && ss[b]==)
         ++b;
     if (b >= len) {
         puts("");
         return ;
     }
 
     while () {
         a[l++] = ss[len-] & ;
         tmp = ;
         rep(i, b, len) {
             if (ss[i] & ) {
                 ss[i] = (tmp+ss[i])>>;
                 tmp = ;
             } else {
                 ss[i] = (tmp+ss[i])>>;
                 tmp = ;
             }
         }
         while (b<len && ss[b]==)
             ++b;
         if (b >= len)
             break;
     }
 
     reverse(a, a+l);
 
     int l2 = l + l;
     int p = , q = ;
 
     memset(dp, , sizeof(dp));
 
     rep(ii, , a[]+) {
         rep(jj, , a[]+) {
             if (ii > jj)
                 continue;
 
             int nj = ii - jj + l;
             int nk = (ii==jj) ? (jj==a[])| : (jj==a[]);
             ++dp[p][nj][nk];
         }
     }
 
     rep(i, , l) {
         rep(j, , l2+) {
             // i < j
             rep(k, , ) {
                 if (!dp[p][j][k])
                     continue;
 
                 int mn1, mn2, nj, nk;
 
                 mn1 = ;
                 mn2 = (k&) ? a[i]:;
 
                 rep(ii, , mn1+) {
                     rep(jj, , mn2+) {
                         nj = j + ii - jj;
                         nk = (k==) && (jj==a[i]);
                         if (nj >= )
                             dp[q][nj][nk] = (dp[q][nj][nk] + dp[p][j][k]) % mod;
                     }
                 }
             }
             // i = j
             rep(k, , ) {
                 if (!dp[p][j][k])
                     continue;
 
                 int mn, nj, nk;
 
                 mn = (k&) ? a[i]:;
                 rep(ii, , mn+) {
                     rep(jj, , mn+) {
                         if (ii > jj)
                             continue;
 
                         nj = j + (ii==) - (jj==);
                         if (k == ) {
                             nk = (ii<jj) ? :;
                         } else {
                             nk = (ii<jj) ? (jj==a[i]) : (jj==a[i])|;
                         }
                         if (nj >= )
                             dp[q][nj][nk] = (dp[q][nj][nk] + dp[p][j][k]) % mod;
                     }
                 }
             }
         }
         p ^= ;
         q ^= ;
         memset(dp[q], , sizeof(dp[q]));
     }
 
     int ans = ;
 
     rep(j, l+, l2+)
         rep(k, , )
             ans = (ans + dp[p][j][k]) % mod;
 
     printf("%d\n", ans);
 }
 
 int main() {
     ios::sync_with_stdio(false);
     #ifndef ONLINE_JUDGE
         freopen("data.in", "r", stdin);
         freopen("data.out", "w", stdout);
     #endif
 
     int t;
 
     scanf("%d", &t);
     while (t--) {
         scanf("%s", ss);
         solve();
     }
 
     #ifndef ONLINE_JUDGE
         printf("time = %d.\n", (int)clock());
     #endif
 
     return ;
 }

4. 数据生成器

 import sys
 import string
 from random import randint, shuffle
 
 def GenData(fileName):
     with open(fileName, "w") as fout:
         t = 10
         fout.write("%d\n" % (t))
         ld = string.digits
         for tt in xrange(t):
             length = randint(200, 300)
             L = [0] * length
             for i in xrange(length):
                 L[i] = randint(0, 9)
             L[0] = randint(1, 9)
             fout.write("".join(map(str, L)) + "\n")
 
 def MovData(srcFileName, desFileName):
     with open(srcFileName, "r") as fin:
         lines = fin.readlines()
     with open(desFileName, "w") as fout:
         fout.write("".join(lines))
 
 def CompData():
     print "comp"
     srcFileName = "F:\Qt_prj\hdoj\data.out"
     desFileName = "F:\workspace\cpp_hdoj\data.out"
     srcLines = []
     desLines = []
     with open(srcFileName, "r") as fin:
         srcLines = fin.readlines()
     with open(desFileName, "r") as fin:
         desLines = fin.readlines()
     n = min(len(srcLines), len(desLines))-1
     for i in xrange(n):
         ans2 = int(desLines[i])
         ans1 = int(srcLines[i])
         if ans1 > ans2:
             print "%d: wrong" % i
 
 if __name__ == "__main__":
     srcFileName = "F:\Qt_prj\hdoj\data.in"
     desFileName = "F:\workspace\cpp_hdoj\data.in"
     GenData(srcFileName)
     MovData(srcFileName, desFileName)