HDU 5179 beautiful number 数位dp

题目链接：

hdu: http://acm.hdu.edu.cn/showproblem.php?pid=5179

bc(中文): http://bestcoder.hdu.edu.cn/contests/contest_chineseproblem.php?cid=569&pid=1002

题解：

1、数位dp

dp[i][j]表示第i位的数值为j的时候所有合法的情况，转移方程为dp[i][j]+=dp[i-1][k](j%k==0)数位最多为10位，可以离线处理出来。

计算1到x（x十进制按位存储在arr[]里面）的所有合法情况：

对于第i位，另前面几位等于arr[j](j>i)，第i位为k<arr[i]，则满足条件arr[i+1]%k==0的都是合法的,累加起来即可，然后考虑完i位之后，继续讨论i-1位，一直做下去。

0要单独讨论，假设x的长度为tot，则考虑最高位dp[tot-1][0]（位数为第0位到第tot-1位）就是所有位数比tot小的合法数的总数，所以只要考虑最高位的0就可以了，其他位都不用考虑0。

区间[L,R]可以考虑为[1,R]-[1,L-1]。

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;

const int maxn = 100000 + 10;
typedef long long LL;

int a, b;

LL dp[11][11];
void get_dp() {
    memset(dp, 0, sizeof(dp));
    dp[0][0] = 0;
    for (int i = 1; i < 10; i++) dp[0][i] = 1;
    for (int i = 1; i < 10; i++) {
        for (int j = 0; j < 10; j++) {
            for (int k = 1; k <= 9; k++) {
                if (j%k == 0) {
                    dp[i][j] += dp[i - 1][k];
                }
            }
            if (j == 0) dp[i][j] += dp[i - 1][j];
        }
    }
}

int arr[11];
LL solve(int x) {
    if (x == 0) return 0;
    LL ret = 0;
    int tot = 0;
    memset(arr, 0, sizeof(arr));
    while (x) {
        arr[tot++] = x % 10;
        x /= 10;
    }
    ret += dp[tot - 1][0];
    for (int i = tot - 1; i >= 0; i--) {
        if (arr[i] == 0) break;
        //表示从前几位不变第i位开始变小的所有合法数字
        for (int j = arr[i] - 1; j >= 1; j--) {
            if (arr[i + 1] % j == 0) {
                ret += dp[i][j];
            }
        }
        //最后一个数可以相等了，收尾了
        if (i == 0 && arr[i + 1] % arr[i] == 0) ret += dp[i][arr[i]];
        //说明第i位不变的话就不可能合法了，不用继续做下去。
        if (arr[i + 1] % arr[i] != 0) break;
    }
    return ret;
}

int main() {
    get_dp();
    int tc;
    scanf("%d", &tc);
    while (tc--) {
        scanf("%d%d", &a, &b);
        LL ans = solve(b) - solve(a - 1);
        printf("%lld
", ans);
    }
    return 0;
}

 2、由于合法数不多，可以讨论离线暴力出一张表来，再二分答案。

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>
using namespace std;

const int maxn = 100000 + 10;
typedef long long LL;

const int tab[] = {
    1,2,3,4,5,6,7,8,9,11,21,22,31,33,41,42,44,51,55,61,62,63,66,71,77,81,82,84,88,91,93,99,111,211,221,222,311,331,333,411,421,422,441,442,444,511,551,555,611,621,622,631,633,661,662,663,666,711,771,777,811,821,822,841,842,844,881,882,884,888,911,931,933,991,993,999,1111,2111,2211,2221,2222,3111,3311,3331,3333,4111,4211,4221,4222,4411,4421,4422,4441,4442,4444,5111,5511,5551,5555,6111,6211,6221,6222,6311,6331,6333,6611,6621,6622,6631,6633,6661,6662,6663,6666,7111,7711,7771,7777,8111,8211,8221,8222,8411,8421,8422,8441,8442,8444,8811,8821,8822,8841,8842,8844,8881,8882,8884,8888,9111,9311,9331,9333,9911,9931,9933,9991,9993,9999,11111,21111,22111,22211,22221,22222,31111,33111,33311,33331,33333,41111,42111,42211,42221,42222,44111,44211,44221,44222,44411,44421,44422,44441,44442,44444,51111,55111,55511,55551,55555,61111,62111,62211,62221,62222,63111,63311,63331,63333,66111,66211,66221,66222,66311,66331,66333,66611,66621,66622,66631,66633,66661,66662,66663,66666,71111,77111,77711,77771,77777,81111,82111,82211,82221,82222,84111,84211,84221,84222,84411,84421,84422,84441,84442,84444,88111,88211,88221,88222,88411,88421,88422,88441,88442,88444,88811,88821,88822,88841,88842,88844,88881,88882,88884,88888,91111,93111,93311,93331,93333,99111,99311,99331,99333,99911,99931,99933,99991,99993,99999,111111,211111,221111,222111,222211,222221,222222,311111,331111,333111,333311,333331,333333,411111,421111,422111,422211,422221,422222,441111,442111,442211,442221,442222,444111,444211,444221,444222,444411,444421,444422,444441,444442,444444,511111,551111,555111,555511,555551,555555,611111,621111,622111,622211,622221,622222,631111,633111,633311,633331,633333,661111,662111,662211,662221,662222,663111,663311,663331,663333,666111,666211,666221,666222,666311,666331,666333,666611,666621,666622,666631,666633,666661,666662,666663,666666,711111,771111,777111,777711,777771,777777,811111,821111,822111,822211,822221,822222,841111,842111,842211,842221,842222,844111,844211,844221,844222,844411,844421,844422,844441,844442,844444,881111,882111,882211,882221,882222,884111,884211,884221,884222,884411,884421,884422,884441,884442,884444,888111,888211,888221,888222,888411,888421,888422,888441,888442,888444,888811,888821,888822,888841,888842,888844,888881,888882,888884,888888,911111,931111,933111,933311,933331,933333,991111,993111,993311,993331,993333,999111,999311,999331,999333,999911,999931,999933,999991,999993,999999,1111111,2111111,2211111,2221111,2222111,2222211,2222221,2222222,3111111,3311111,3331111,3333111,3333311,3333331,3333333,4111111,4211111,4221111,4222111,4222211,4222221,4222222,4411111,4421111,4422111,4422211,4422221,4422222,4441111,4442111,4442211,4442221,4442222,4444111,4444211,4444221,4444222,4444411,4444421,4444422,4444441,4444442,4444444,5111111,5511111,5551111,5555111,5555511,5555551,5555555,6111111,6211111,6221111,6222111,6222211,6222221,6222222,6311111,6331111,6333111,6333311,6333331,6333333,6611111,6621111,6622111,6622211,6622221,6622222,6631111,6633111,6633311,6633331,6633333,6661111,6662111,6662211,6662221,6662222,6663111,6663311,6663331,6663333,6666111,6666211,6666221,6666222,6666311,6666331,6666333,6666611,6666621,6666622,6666631,6666633,6666661,6666662,6666663,6666666,7111111,7711111,7771111,7777111,7777711,7777771,7777777,8111111,8211111,8221111,8222111,8222211,8222221,8222222,8411111,8421111,8422111,8422211,8422221,8422222,8441111,8442111,8442211,8442221,8442222,8444111,8444211,8444221,8444222,8444411,8444421,8444422,8444441,8444442,8444444,8811111,8821111,8822111,8822211,8822221,8822222,8841111,8842111,8842211,8842221,8842222,8844111,8844211,8844221,8844222,8844411,8844421,8844422,8844441,8844442,8844444,8881111,8882111,8882211,8882221,8882222,8884111,8884211,8884221,8884222,8884411,8884421,8884422,8884441,8884442,8884444,8888111,8888211,8888221,8888222,8888411,8888421,8888422,8888441,8888442,8888444,8888811,8888821,8888822,8888841,8888842,8888844,8888881,8888882,8888884,8888888,9111111,9311111,9331111,9333111,9333311,9333331,9333333,9911111,9931111,9933111,9933311,9933331,9933333,9991111,9993111,9993311,9993331,9993333,9999111,9999311,9999331,9999333,9999911,9999931,9999933,9999991,9999993,9999999,11111111,21111111,22111111,22211111,22221111,22222111,22222211,22222221,22222222,31111111,33111111,33311111,33331111,33333111,33333311,33333331,33333333,41111111,42111111,42211111,42221111,42222111,42222211,42222221,42222222,44111111,44211111,44221111,44222111,44222211,44222221,44222222,44411111,44421111,44422111,44422211,44422221,44422222,44441111,44442111,44442211,44442221,44442222,44444111,44444211,44444221,44444222,44444411,44444421,44444422,44444441,44444442,44444444,51111111,55111111,55511111,55551111,55555111,55555511,55555551,55555555,61111111,62111111,62211111,62221111,62222111,62222211,62222221,62222222,63111111,63311111,63331111,63333111,63333311,63333331,63333333,66111111,66211111,66221111,66222111,66222211,66222221,66222222,66311111,66331111,66333111,66333311,66333331,66333333,66611111,66621111,66622111,66622211,66622221,66622222,66631111,66633111,66633311,66633331,66633333,66661111,66662111,66662211,66662221,66662222,66663111,66663311,66663331,66663333,66666111,66666211,66666221,66666222,66666311,66666331,66666333,66666611,66666621,66666622,66666631,66666633,66666661,66666662,66666663,66666666,71111111,77111111,77711111,77771111,77777111,77777711,77777771,77777777,81111111,82111111,82211111,82221111,82222111,82222211,82222221,82222222,84111111,84211111,84221111,84222111,84222211,84222221,84222222,84411111,84421111,84422111,84422211,84422221,84422222,84441111,84442111,84442211,84442221,84442222,84444111,84444211,84444221,84444222,84444411,84444421,84444422,84444441,84444442,84444444,88111111,88211111,88221111,88222111,88222211,88222221,88222222,88411111,88421111,88422111,88422211,88422221,88422222,88441111,88442111,88442211,88442221,88442222,88444111,88444211,88444221,88444222,88444411,88444421,88444422,88444441,88444442,88444444,88811111,88821111,88822111,88822211,88822221,88822222,88841111,88842111,88842211,88842221,88842222,88844111,88844211,88844221,88844222,88844411,88844421,88844422,88844441,88844442,88844444,88881111,88882111,88882211,88882221,88882222,88884111,88884211,88884221,88884222,88884411,88884421,88884422,88884441,88884442,88884444,88888111,88888211,88888221,88888222,88888411,88888421,88888422,88888441,88888442,88888444,88888811,88888821,88888822,88888841,88888842,88888844,88888881,88888882,88888884,88888888,91111111,93111111,93311111,93331111,93333111,93333311,93333331,93333333,99111111,99311111,99331111,99333111,99333311,99333331,99333333,99911111,99931111,99933111,99933311,99933331,99933333,99991111,99993111,99993311,99993331,99993333,99999111,99999311,99999331,99999333,99999911,99999931,99999933,99999991,99999993,99999999,111111111,211111111,221111111,222111111,222211111,222221111,222222111,222222211,222222221,222222222,311111111,331111111,333111111,333311111,333331111,333333111,333333311,333333331,333333333,411111111,421111111,422111111,422211111,422221111,422222111,422222211,422222221,422222222,441111111,442111111,442211111,442221111,442222111,442222211,442222221,442222222,444111111,444211111,444221111,444222111,444222211,444222221,444222222,444411111,444421111,444422111,444422211,444422221,444422222,444441111,444442111,444442211,444442221,444442222,444444111,444444211,444444221,444444222,444444411,444444421,444444422,444444441,444444442,444444444,511111111,551111111,555111111,555511111,555551111,555555111,555555511,555555551,555555555,611111111,621111111,622111111,622211111,622221111,622222111,622222211,622222221,622222222,631111111,633111111,633311111,633331111,633333111,633333311,633333331,633333333,661111111,662111111,662211111,662221111,662222111,662222211,662222221,662222222,663111111,663311111,663331111,663333111,663333311,663333331,663333333,666111111,666211111,666221111,666222111,666222211,666222221,666222222,666311111,666331111,666333111,666333311,666333331,666333333,666611111,666621111,666622111,666622211,666622221,666622222,666631111,666633111,666633311,666633331,666633333,666661111,666662111,666662211,666662221,666662222,666663111,666663311,666663331,666663333,666666111,666666211,666666221,666666222,666666311,666666331,666666333,666666611,666666621,666666622,666666631,666666633,666666661,666666662,666666663,666666666,711111111,771111111,777111111,777711111,777771111,777777111,777777711,777777771,777777777,811111111,821111111,822111111,822211111,822221111,822222111,822222211,822222221,822222222,841111111,842111111,842211111,842221111,842222111,842222211,842222221,842222222,844111111,844211111,844221111,844222111,844222211,844222221,844222222,844411111,844421111,844422111,844422211,844422221,844422222,844441111,844442111,844442211,844442221,844442222,844444111,844444211,844444221,844444222,844444411,844444421,844444422,844444441,844444442,844444444,881111111,882111111,882211111,882221111,882222111,882222211,882222221,882222222,884111111,884211111,884221111,884222111,884222211,884222221,884222222,884411111,884421111,884422111,884422211,884422221,884422222,884441111,884442111,884442211,884442221,884442222,884444111,884444211,884444221,884444222,884444411,884444421,884444422,884444441,884444442,884444444,888111111,888211111,888221111,888222111,888222211,888222221,888222222,888411111,888421111,888422111,888422211,888422221,888422222,888441111,888442111,888442211,888442221,888442222,888444111,888444211,888444221,888444222,888444411,888444421,888444422,888444441,888444442,888444444,888811111,888821111,888822111,888822211,888822221,888822222,888841111,888842111,888842211,888842221,888842222,888844111,888844211,888844221,888844222,888844411,888844421,888844422,888844441,888844442,888844444,888881111,888882111,888882211,888882221,888882222,888884111,888884211,888884221,888884222,888884411,888884421,888884422,888884441,888884442,888884444,888888111,888888211,888888221,888888222,888888411,888888421,888888422,888888441,888888442,888888444,888888811,888888821,888888822,888888841,888888842,888888844,888888881,888888882,888888884,888888888,911111111,931111111,933111111,933311111,933331111,933333111,933333311,933333331,933333333,991111111,993111111,993311111,993331111,993333111,993333311,993333331,993333333,999111111,999311111,999331111,999333111,999333311,999333331,999333333,999911111,999931111,999933111,999933311,999933331,999933333,999991111,999993111,999993311,999993331,999993333,999999111,999999311,999999331,999999333,999999911,999999931,999999933,999999991,999999993,999999999
};



int solve(int x) {
    if (x == 0) return 0;
    return upper_bound(tab, tab+1299, x)-tab;
}

int a, b;

int main() {
    int tc;
    scanf("%d", &tc);
    while (tc--) {
        scanf("%d%d", &a, &b);
        int ans = solve(b) - solve(a - 1);
        printf("%d
", ans);
    }
    return 0;
}

 3、上一发正规模板化数位dp

#include<map>
#include<set>
#include<cmath>
#include<queue>
#include<stack>
#include<ctime>
#include<vector>
#include<cstdio>
#include<string>
#include<bitset>
#include<cstdlib>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<functional>
using namespace std;
#define X first
#define Y second
#define mkp make_pair
#define lson (o<<1)
#define rson ((o<<1)|1)
#define mid (l+(r-l)/2)
#define sz() size()
#define pb(v) push_back(v)
#define all(o) (o).begin(),(o).end()
#define clr(a,v) memset(a,v,sizeof(a))
#define bug(a) cout<<#a<<" = "<<a<<endl
#define rep(i,a,b) for(int i=a;i<(b);i++)
#define scf scanf
#define prf printf

typedef int LL;
typedef vector<int> VI;
typedef pair<int,int> PII;
typedef vector<pair<int,int> > VPII;

const int INF=0x3f3f3f3f;
const LL INFL=10000000000000000LL;
const double eps=1e-9;

const double PI = acos(-1.0);

//start----------------------------------------------------------------------

LL dp[22][11];
int arr[22],tot;
LL dfs(int len,int j, bool ismax,bool iszer) {
    if (len == 0) {
        return 1LL;
    }
    if (!ismax&&dp[len][j]>=0) return dp[len][j];

    LL res = 0;
    int ed = ismax ? arr[len] : 9;
    for (int i = 0; i <= ed; i++) {
        if(iszer&&i==0) {
            res+=dfs(len-1,10,ismax&&i==ed,iszer&&i==0);
        } else {
            if(i==0) continue;
            if(j==10) {
                res+=dfs(len-1,i,ismax&&i==ed,iszer&&i==0);
            } else {
                if(j>=i&&j%i==0) {
                    res+=dfs(len-1,i,ismax&&i==ed,iszer&&i==0);
                }
            }
        }

    }
    return ismax ? res : dp[len][j] = res;
}

LL solve(LL x) {
    tot = 0;
    while (x) {
        arr[++tot] = x % 10;
        x /= 10;
    }
    return dfs(tot,10,true,true);
}

int main() {
    clr(dp,-1);
    int tc,kase=0;
    scf("%d",&tc);
    while(tc--) {
        LL l,r;
        scf("%d%d",&l,&r);
        prf("%d
",solve(r)-solve(l-1));
    }
    return 0;
}

//end-----------------------------------------------------------------------