【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 = 998244353;
const int maxl = 305;
const int maxn = 1205;
char ss[maxl];
int a[maxn];
int dp[2][maxn<<1][4];

void solve() {
int l = 0, tmp;
int len = strlen(ss);

rep(i, 0, len)
ss[i] -= '0';

int b = 0;

while (b<len && ss[b]==0)
++b;
if (b >= len) {
puts("0");
return ;
}

while (1) {
a[l++] = ss[len-1] & 1;
tmp = 0;
rep(i, b, len) {
if (ss[i] & 1) {
ss[i] = (tmp+ss[i])>>1;
tmp = 10;
} else {
ss[i] = (tmp+ss[i])>>1;
tmp = 0;
}
}
while (b<len && ss[b]==0)
++b;
if (b >= len)
break;
}

reverse(a, a+l);

int l2 = l + l;
int p = 0, q = 1;

memset(dp, 0, sizeof(dp));

rep(ii, 0, a[0]+1) {
rep(jj, 0, a[0]+1) {
if (ii > jj)
continue;

int nj = ii - jj + l;
int nk = (ii==jj) ? (jj==a[0])|2 : (jj==a[0]);
++dp[p][nj][nk];
}
}

rep(i, 1, l) {
rep(j, 0, l2+1) {
// i < j
rep(k, 0, 2) {
if (!dp[p][j][k])
continue;

int mn1, mn2, nj, nk;

mn1 = 1;
mn2 = (k&1) ? a[i]:1;

rep(ii, 0, mn1+1) {
rep(jj, 0, mn2+1) {
nj = j + ii - jj;
nk = (k==1) && (jj==a[i]);
if (nj >= 0)
dp[q][nj][nk] = (dp[q][nj][nk] + dp[p][j][k]) % mod;
}
}
}
// i = j
rep(k, 2, 4) {
if (!dp[p][j][k])
continue;

int mn, nj, nk;

mn = (k&1) ? a[i]:1;
rep(ii, 0, mn+1) {
rep(jj, 0, mn+1) {
if (ii > jj)
continue;

nj = j + (ii==1) - (jj==1);
if (k == 2) {
nk = (ii<jj) ? 0:2;
} else {
nk = (ii<jj) ? (jj==a[i]) : (jj==a[i])|2;
}
if (nj >= 0)
dp[q][nj][nk] = (dp[q][nj][nk] + dp[p][j][k]) % mod;
}
}
}
}
p ^= 1;
q ^= 1;
memset(dp[q], 0, sizeof(dp[q]));
}

int ans = 0;

rep(j, l+1, l2+1)
rep(k, 0, 2)
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 0;
}

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)