【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
", 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.
", (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
" % (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)) + "
")
            
            
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_prjhdojdata.out"
    desFileName = "F:workspacecpp_hdojdata.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_prjhdojdata.in"
    desFileName = "F:workspacecpp_hdojdata.in"
    GenData(srcFileName)
    MovData(srcFileName, desFileName)