The journey of Ural dynamic programming

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我是靠谱客的博主发嗲飞鸟，最近开发中收集的这篇文章主要介绍The journey of Ural dynamic programming，觉得挺不错的，现在分享给大家，希望可以做个参考。

概述

一句话概括Ural的dp，输出路径+滚动数组。下文中，#是注意点，$是小优化。3Q for

YY’s Code.

一、经典题：

l 1146联系大白书P55

l 1017 Sol1:dp[i][j]=sum{dp[i-j][k]} ;(0<=k<j)

Sol2:dp[i+j]+=dp[j]; (0<=j<n) 视为背包问题来理解

l 1303 代码贪心，甚至不需排序。

l 1223朱晨光：《优化，再优化！——从《鹰蛋》一题浅析对动态规划算法的优化》

l 1513 f[i]=sum{f[i-1-j]}(+B+j个L)(0<=j<=m)

l 1238 区间dp

l 1276 类似noip2009乌龟棋。#1：火车第一节车厢的头也要和最后一节车厢的尾相匹配。

二、难题：

l 1260 dp[n]=dp[n-1](12……)+ dp[n-3](132……)+1(1357……,根据n奇偶性略有不同)

l 1627 周冬：《生成树的计数及其应用》

l 1171 http://blog.csdn.net/chm517/article/details/18889199

l 1459 代码插头DP+矩阵快速幂

三、好题：

l 1117 代码一般读入[a,b]就可以考虑[1,b]-[1,a-1]

l 1310 代码（类似的题有Spoj 2319）

l 1427 贪心略有坑，写着渣的代码会在n=1时跪了。

l 1900 dp[i][j] = max(dp[i][j],dp[g][j-1]+o[i][i+1]-o[g][i+1])

l 1577 代码If in your current state s1[i]==s2[j], you should not assume states i+1,j and i,j+1. Only i+1,j+1.test：aa/ac

l 1696 代码dp[N][最大值][出现在最大值后的严格小于最大值的尽可能大的值]

l 1552 代码通过27-10进制转化成功优化了一维，代码还很优雅

l 1745 http://blog.csdn.net/chm517/article/details/21652557

l 1526 代码关键在于状态的设计，有点区间的味道，但只需考虑区间长度，还有一维是状压

l 1476预处理的优化http://blog.csdn.net/chm517/article/details/21751089

l 1655代码#1：初始位置那个点的限制时间为inf

l 1739代码

l 1895代码$1:第一维的1000可以降为50*2; $2:代码中两个break。

四、概率题

l 1776代码

l 1716 http://blog.csdn.net/chm517/article/details/21655187 (类似的有Codeforces 148D Bag of mice)

l 1887 基本同hdu 4336

//ural 1303
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
using namespace std;
#define N 100005
int n,m,o;
int ans[N],r[N];
void doit()
{
memset(r,0,sizeof(r));
int x,y,ori,t;
for (int i=1;;i++)
{scanf("%d%d",&x,&y);
if (x<0) t=0;else t=x;
if (y>r[t]) {r[t]=y; if (t==0) ori=x;}
if (x==0&&y==0) {n=i-1; break;}
}
o=0;
int L=0,R=0,i=0,id;
while (L<m)
{
while (i<=L)
{
if (R<r[i]){R=r[i];id=i;}
i++;
}
if (L==R) {printf("No solutionn"); return;}
ans[++o]=id;
L=R;
}
printf("%dn",o);
for (int i=1;i<=o;i++)
if (ans[i]==0)printf("%d %dn",ori,r[0]);
else printf("%d %dn",ans[i],r[ans[i]]);
}
int main()
{
while (scanf("%d",&m)!=EOF) doit();
}

//ural 1459
#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#define MAX 22
typedef long long LL;
using namespace std;
const int HASH=4096;
const int INF=0x3f3f3f3f;
const LL mod=1000000000;
int n,m;
int id[HASH];
int jz[6];
int ex,ey,o;
int HA;
class Mat{
public:
LL s[MAX][MAX];
int sizei,sizej;
public:
Mat(int a=MAX,int b=MAX,int type=0);
void clear(int type = 0);
void ReSize(int a = MAX,int b=MAX,int type=0);
void Add(const Mat& B);
void Sub(const Mat& B);
void CopyMat(const Mat& B);
void Multiply(const Mat &B);
void Er_work(int n);
void solve(int p) ;
};
Mat::Mat(int a,int b,int type):sizei(a),sizej(b)
{
memset(s,0,sizeof(s));
if(type==1)
for(int i=0;i<min(a,b);i++)
s[i][i]=1;
else if(type == 2)
for(int i=0;i<a;i++)
for(int j=0;j<b;j++)
s[i][j]=1;
}
void Mat::clear(int type)
{
memset(s,0,sizeof(s));
if(type==1)
for(int i=0;i<min(sizei,sizej);i++)
s[i][i]=1;
else if(type == 2)
for(int i=0;i<sizei;i++)
for(int j=0;j<sizej;j++)
s[i][j]=1;
}
void Mat::ReSize(int a,int b,int type)
{
sizei=a; sizej=b;
memset(s,0,sizeof(s));
if(type==1)
for(int i=0;i<min(sizei,sizej);i++)
s[i][i]=1;
else if(type == 2)
for(int i=0;i<sizei;i++)
for(int j=0;j<sizej;j++)
s[i][j]=1;
}
void Mat::Add(const Mat& B)
{
for(int i=0;i<sizei;i++)
for(int j=0;j<sizej;j++)
{s[i][j]=s[i][j]+B.s[i][j];
s[i][j]%=mod;
}
}
void Mat::Sub(const Mat& B)
{
for(int i=0;i<sizei;i++)
for(int j=0;j<sizej;j++)
{s[i][j]=s[i][j]-B.s[i][j];
s[i][j]=(s[i][j]+mod)%mod;
}
}
void Mat::CopyMat(const Mat& B)
{
sizei=B.sizei; sizej=B.sizej;
for(int i=0;i<sizei;i++)
for(int j=0;j<sizej;j++)
s[i][j]=B.s[i][j];
}
void Mat::Multiply(const Mat &B)
{
Mat c(sizei,B.sizej,0) ;
for(int i=0;i<sizei;i++)
for(int k=0;k<sizej;k++)
if (s[i][k]>0)
for(int j=0;j<B.sizej;j++)
{c.s[i][j]+=s[i][k]*B.s[k][j];
c.s[i][j]%=mod ;
}
*this = c;
}
void Mat::Er_work(int n)
{
Mat e(sizei,sizej,1);
while (n)
{
if(n&1)
e.Multiply(*this);
n>>=1;
Multiply(*this);
}
*this = e;
}
void Mat::solve(int p)
{
Mat temp1 = *this, temp2 = *this;
Mat e(sizei, sizej, 1);
if(p == 1)
return;
else if(p&1)
{
temp1.Er_work(p);
temp2.solve(p-1);
temp1.Add(temp2);
}
else
{
temp1.Er_work(p>>1);
temp1.Add(e);
temp2.solve(p>>1);
temp1.Multiply(temp2);
}
*this = temp1;
};
Mat ee,ans;
struct HASHMAP
{
int head[HASH],next[HASH],size;
int state[HASH];
LL f[HASH];
void init()
{
size=0;
memset(head,-1,sizeof(head));
}
void push(int st,LL ans)
{
int h=st%HA;
if(head[h]!=-1)
{
f[head[h]]+=ans;
f[head[h]]%=mod;
return;
}
state[size]=h;
f[size]=ans;
next[size]=head[h];
head[h]=size++;
}
}hm[2];
void shift(int cur)
{
for(int i=0;i<hm[cur].size;i++)
{hm[cur].state[i]<<=2;
hm[cur].state[i]%=HA;
}
}
inline int cp(int p,int i)
{
return p&(~(3<<jz[i]));
}
inline int cp(int p,int i,int j)
{
return p&(~(3<<jz[i]))&(~(3<<jz[j]));
}
inline int cp(int p,int i,int j,int k)
{
return p&(~(3<<jz[i]))&(~(3<<jz[j]))&(~(3<<jz[k]));
}
inline int getp(int p,int i)
{
return 3&(p>>jz[i]);
}
inline int pp(int i,int k)
{
return k<<jz[i];
}
inline int lp(int p,int j)
{
for (int it=j-2,cnt=1;;it--)
{
int pi=getp(p,it);
if (pi==1) cnt--;
else if (pi==2) cnt++;
if (cnt==0) return it;
}
}
inline int rp(int p,int j)
{
for (int it=j+1,cnt=1;;it++)
{
int pi=getp(p,it);
if (pi==2) cnt--;
else if (pi==1) cnt++;
if (cnt==0) return it;
}
}
void dp(int i,int j,int cur)
{
for(int k=0;k<hm[cur].size;k++)
{int s=hm[cur].state[k];
LL data=hm[cur].f[k];
int left=getp(s,j);
int up=getp(s,j+1);
if (left==0&&up==0)
{if (j+1<m)
{
hm[cur^1].push(s|pp(j,1)|pp(j+1,2),data);
}
continue;
}
if (left==0||up==0)
{
int w=left+up;
int tmp=cp(s,j,j+1);
if (j+1<m) hm[cur^1].push(tmp|pp(j+1,w),data);
hm[cur^1].push(tmp|pp(j,w),data);
continue;
}
if (left==up)
{
int it=(left==1)?rp(s,j+1):lp(s,j+1);
hm[cur^1].push(cp(s,j,j+1,it)|pp(it,left),data);
continue;
}
//
if (left==1&&up==2)
//
{
if (i==ex&&j==ey) ans+=data;
//
continue;
//
}
if (left==2&&up==1)
{
hm[cur^1].push(cp(s,j,j+1),data);
continue;
}
}
}
void init()
{
ex=n-1; ey=m-1;
}
void solve()
{
ee.ReSize(o,o,0);
int cur;
for(int i=0;i<HA;i++)
if (id[i]!=-1)
{cur=0;
hm[cur].init();
hm[cur].push(i,1);
for(int j=0;j<m;j++)
{
hm[cur^1].init();
dp(i,j,cur);
cur^=1;
}
shift(cur);
for(int ii=0;ii<hm[cur].size;ii++)
ee.s[id[i]][id[hm[cur].state[ii]]]=hm[cur].f[ii];
}
ee.Er_work(n);
ans.ReSize(1,o,0);
ans.s[0][0]=1;
ans.Multiply(ee);
LL anss=0;
for (int i=0;i<HA;i++)
if (id[i]!=-1)
{
if ((1<<jz[m])*2+(1<<jz[m-1])==i)
{anss+=ans.s[0][id[i]];
anss%=mod;
//printf("~n");
}
}
printf("%I64dn",anss*2%mod);
}
int tmp[6];
void prepared_jinzhi()
{
for (int i=0;i<=m;i++)
jz[i]=i*2;
memset(id,255,sizeof(id)); o=0;
for (int i=0;i<HA;i++)
{
int f=1;
for (int j=0;j<=m;j++)
{
tmp[j]=3&(i>>jz[j]);
if (tmp[j]==3) {f=0; break;}
}
if (!f) continue;
if (tmp[0]) continue;
int s=0;
for (int j=1;j<=m;j++)
{
if (tmp[j]==1) s++;
if (tmp[j]==2) s--;
if (s<0) f=0;
}
if (s) f=0;
if (f) id[i]=o++;
}
}
int main()
{
while(scanf("%d%d",&m,&n)!=EOF)
{
init();
HA=1<<(m*2+2);
prepared_jinzhi();
if (m*n%2==1) {printf("%dn",0);continue;}
solve();
}
return 0;
}

//ural 1117
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cstdio>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
typedef long long LL;
using namespace std;
LL s[41],l[41],a,b;
void pre()
{
l[0]=0; s[0]=0;
for (int i=1;i<=40;i++)
{
l[i]=(l[i-1]+s[i-1])*2;
s[i]=s[i-1]+1;
}
}
LL solve(LL x)
{
LL ans=0;
while (x>1)
{
LL p=1,t=-1;
while (p<=x) {p<<=1; t++;}
p>>=1; t--;
x-=p;
if (x)ans+=(l[t]+s[t]*2);else ans+=(l[t]+s[t]);
}
return ans;
}
int main()
{
pre();
while (scanf("%I64d%I64d",&a,&b)!=EOF)
printf("%I64dn",abs(solve(b)-solve(a)));
}

//ural 1310
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <queue>
#include <vector>
#include <iostream>
#include <string>
#include <cstdlib>
#include <map>
#define pb push_back
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define MS(x, y) memset(x, y, sizeof(x))
#define FOR(i, x, y) for(int i=x; i<=y; i++)
#define rFOR(i, x, y) for(int i=x; i>=y; i--)
using namespace std;
const int maxn = 175;
struct bigint
{
int s[maxn];
bigint() {MS(s, 0);}
bigint(int num) {*this = num;}
bigint(const char* num){*this = num;}
bigint operator = (const char* num)
{
MS(s, 0);
if(num[0] == '-')
{
num = num + 1;
s[0] = -1;
}
else s[0] = 1;
while(num[0] == '0') num = num + 1;
s[0] = s[0] * strlen(num);
int len = abs(s[0]);
FOR(i, 1, len) s[i] = num[len - i] - 48;
return *this;
}
bigint operator = (int num)
{
char s[maxn];
sprintf(s, "%d", num);
*this = s;
return *this;
}
string str() const
{
string res = "";
FOR(i, 1, abs(s[0])) res = (char)(s[i] + 48) + res;
if(res == "") return res = "0";
if(s[0] < 0) res = '-' + res;
return res;
}
bool operator < (const bigint& b) const
{
if(s[0] != b.s[0]) return s[0] < b.s[0];
int len = abs(s[0]);
rFOR(i, len, 1)
if(s[i] != b.s[i])
return (s[i] < b.s[i]) ^ (s[0] < 0);
return false;
}
bool operator > (const bigint& b) const{return b < *this;}
bool operator <= (const bigint& b) const{return !(b < *this);}
bool operator >= (const bigint& b) const{return !(*this < b);}
bool operator != (const bigint& b) const{return b < *this || *this < b;}
bool operator == (const bigint& b) const{return !(b < *this) && !(*this < b);}
friend bigint abs(bigint b)
{
b.s[0] = abs(b.s[0]);
return b;
}
friend bigint _add(const bigint& a, const bigint& b)
{
bigint c;
c.s[0] = max(a.s[0], b.s[0]);
FOR(i, 1, c.s[0]) c.s[i] = a.s[i] + b.s[i];
FOR(i, 1, c.s[0])
if(c.s[i] >= 10)
{
c.s[i+1]++;
c.s[i]-=10;
}
if(c.s[c.s[0]+1]) ++c.s[0];
return c;
}
friend bigint _sub(const bigint& a, const bigint& b)
{
bigint c;
c.s[0] = a.s[0];
FOR(i, 1, c.s[0]) c.s[i] = a.s[i] - b.s[i];
FOR(i, 1, c.s[0])
if(c.s[i] < 0)
{
c.s[i+1]--;
c.s[i] += 10;
}
while(!c.s[c.s[0]] && c.s[0]) --c.s[0];
return c;
}
bigint operator + (const bigint& b) const
{
if(s[0] >= 0 && b.s[0] >= 0) return _add(*this, b);
if(b.s[0] < 0) return *this - abs(b);
if(s[0] < 0) return b - abs(*this);
}
bigint operator - (const bigint& b) const
{
if(s[0] >= 0 && b.s[0] >= 0)
{
bigint c;
if(*this >= b) return _sub(*this, b);
c = _sub(b, *this);
c.s[0] = -c.s[0];
return c;
}
if(b.s[0] < 0) return *this + abs(b);
if(s[0] < 0)
{
bigint c;
c = abs(*this) + b;
c.s[0] = -c.s[0];
return c;
}
}
bigint operator * (const bigint& b) const
{
bigint c;
c.s[0] = abs(s[0]) + abs(b.s[0]);
FOR(i, 1, abs(s[0]))
FOR(j, 1, abs(b.s[0]))
c.s[i + j - 1] += s[i] * b.s[j];
FOR(i, 1, c.s[0])
{
c.s[i+1] += c.s[i] / 10;
c.s[i] %= 10;
}
while(!c.s[c.s[0]] && c.s[0]) --c.s[0];
if((s[0] > 0) != (b.s[0] > 0)) c.s[0] = -c.s[0];
return c;
}
bigint operator / (const bigint& _b) const
{
bigint c, t;
bigint b = abs(_b);
c.s[0] = abs(s[0]);
rFOR(i, c.s[0], 1)
{
rFOR(j, t.s[0], 1) t.s[j + 1] = t.s[j];
t.s[1] = s[i];
if (t.s[t.s[0]+1])t.s[0]++;
while(t >= b)
{
++c.s[i];
t -= b;
}
}
while(!c.s[c.s[0]] && c.s[0]) --c.s[0];
if((s[0] > 0) != (b.s[0] > 0)) c.s[0] = -c.s[0];
return c;
}
bigint operator % (const bigint& _b) const
{
bigint c, t;
bigint b = abs(_b);
rFOR(i, abs(s[0]), 1)
{
rFOR(j, t.s[0], 1) t.s[j + 1] = t.s[j];
t.s[1] = s[i];
if (t.s[t.s[0]+1])t.s[0]++;
while(t >= b) t -= b;
}
if((s[0] < 0)) t.s[0] = -t.s[0];
return t;
}
bigint operator += (const bigint& b) {*this = *this + b;return *this;}
bigint operator -= (const bigint& b) {*this = *this - b;return *this;}
bigint operator *= (const bigint& b) {*this = *this * b;return *this;}
bigint operator /= (const bigint& b) {*this = *this / b;return *this;}
bigint operator %= (const bigint& b) {*this = *this % b;return *this;}
friend bigint operator + (int& a, const bigint& b){return (bigint)a + b;}
friend bigint operator - (int& a, const bigint& b){return (bigint)a - b;}
friend bigint operator * (int& a, const bigint& b){return (bigint)a * b;}
friend bigint operator / (int& a, const bigint& b){return (bigint)a / b;}
friend bigint operator % (int& a, const bigint& b){return (bigint)a % b;}
friend bigint operator <
(int& a, const bigint& b){return (bigint)a < b;}
friend bigint operator <= (int& a, const bigint& b){return (bigint)a <=b;}
friend bigint operator >
(int& a, const bigint& b){return (bigint)a > b;}
friend bigint operator >= (int& a, const bigint& b){return (bigint)a >=b;}
friend bigint operator == (int& a, const bigint& b){return (bigint)a ==b;}
friend bigint operator != (int& a, const bigint& b){return (bigint)a !=b;}
};
istream& operator >> (istream &in, bigint& x)
{
string s;
in >> s;
x = s.c_str();
return in;
}
ostream& operator << (ostream &out, const bigint& x)
{
out << x.str();
return out;
}
using namespace std;
int l,m,k;
bigint x;
bigint dp[101][50];
void doit()
{
cin>>x;x+=1;
for (int z=0;z<k;z++)
dp[0][z]=0;
dp[0][0]=1;
for (int i=1;i<=l;i++)
{
for (int z=0;z<k;z++)
dp[i][z]=0;
for (int j=1;j<=m;j++)
for (int z=0;z<k;z++)
dp[i][(z+j)%k]+=dp[i-1][z];
}
int w=l,i,p=0;
bool have=0;
bigint tmp;
while (w)
{
for (i=1;i<=m;i++)
{
tmp=dp[w-1][((p-i)%k+k)%k];
if (x<=tmp) break;else x-=tmp;
}
if (have) printf(" "); have=1;
printf("%d",i);
w--;
p=((p-i)%k+k)%k;
}
printf("n");
}
int main()
{
while (scanf("%d%d%d",&l,&m,&k)!=EOF) doit();
}
/*
3 10 4
213
*/

//ural 1577
//CONTINUE!
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
const long long INF=1LL<<50;
using namespace std;
typedef long long LL;
typedef pair <int,double> PII;
int mod,n,m,t,tot;
#define N 2005
#define e 1000000007
int s[N][N],f[N][N];
char s1[N],s2[N];
void doit()
{
int la=strlen(s1),lb=strlen(s2);
memset(f,0x3f,sizeof(f));
f[0][0]=0;
s[0][0]=1;
for (int i=0;i<=la;i++)
for (int j=0;j<=lb;j++)
{
int tmp;
if (i&&j&&s1[i-1]==s2[j-1])
{
if (f[i][j]>f[i-1][j-1]+1)
{f[i][j]=f[i-1][j-1]+1; s[i][j]=s[i-1][j-1];}else
if (f[i][j]==f[i-1][j-1]+1) {s[i][j]=(s[i][j]+s[i-1][j-1])%e;}
continue; //!
}
if (i)
{if (f[i][j]>f[i-1][j]+1)
{f[i][j]=f[i-1][j]+1; s[i][j]=s[i-1][j];}else
if (f[i][j]==f[i-1][j]+1) {s[i][j]=(s[i][j]+s[i-1][j])%e;}
}
if (j)
{if (f[i][j]>f[i][j-1]+1)
{f[i][j]=f[i][j-1]+1; s[i][j]=s[i][j-1];}else
if (f[i][j]==f[i][j-1]+1) {s[i][j]=(s[i][j]+s[i][j-1])%e;}
}
//printf("%d %d %d %dnnn",i,j,f[i][j],s[i][j]);
}
printf("%dn",s[la][lb]);
}
int main()
{
while (scanf("%s%s",s1,s2)!=EOF) doit();
}
/*
aa
ba
aa
ab
*/

//ural 1696
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
const long long INF=1LL<<50;
using namespace std;
typedef long long LL;
typedef pair <int,double> PII;
#define N 202
int dp[2][N][N],a[N][N],b[N][N];
int n,m,mod;
void doit()
{
memset(dp,0,sizeof(dp));
memset(a,0,sizeof(a));
memset(b,0,sizeof(b));
for (int i=1;i<=m;i++)
dp[1][i][0]=1;
for (int i=1;i<n;i++)
{memset(dp[(i&1)^1],0,sizeof(dp[(i&1)^1]));
int o=i&1;
for (int j=1;j<=m;j++)
a[j][0]=dp[o][j][0];
for (int j=1;j<=m;j++)
for (int k=1;k<=j-1;k++)a[j][k]=(dp[o][j][k]+a[j][k-1])%mod;
for (int k=0;k<=m-1;k++)
b[k][k]=0;
for (int k=0;k<=m-1;k++)
for (int j=k+1;j<=m;j++)
b[j][k]=(dp[o][j][k]+b[j-1][k])%mod;
for (int j=1;j<=m;j++)
for (int k=0;k<=j-1;k++)
{
dp[o^1][j][k]=(dp[o^1][j][k]+b[j][k])%mod;
if (j-k!=1) dp[o^1][j][k]=(dp[o^1][j][k]+a[j][k])%mod;
}
}
int ans=0;
for (int j=1;j<=m;j++)
for (int k=0;k<=j-1;k++)
ans=(ans+dp[n&1][j][k])%mod;
printf("%dn",ans+1);
}
int main()
{
while (scanf("%d%d%d",&n,&m,&mod)!=EOF) doit();
}

//ural 1552
#include<algorithm>
#include<cstring>
#include<cstdio>
#include<cstdlib>
const int inf=0x3f3f3f3f;
using namespace std;
const int N=27*27*27;
#define M 51
int f[M][4][N];
int dp[M][4][N];
char s[M];
int num[M+1],n;
int a[4];
inline int get(int a,int b)
{
if (a==0) a-=96;
if (b==0) b-=96;
return abs(a-b);
}
bool miniz(int &a,int b)
{
if (b>a) return 0;
a=b;
return 1;
}
void print(int i,int j,int k)
{
if (!i) {for (int i=1;i<=num[0]+96;i++) printf("+"); printf("."); return;}
int t1=f[i][j][k]/N,t2=f[i][j][k]%N;
print(i-1,t1,t2);
int x=t2;
for (int p=3;p>=0;p--)
if (p==t1) a[p]=num[i-1];
else {a[p]=x%27; x/=27;}
if (t1<j) for (int jj=1;jj<=j-t1;jj++) printf(">");
if (t1>j) for (int jj=1;jj<=t1-j;jj++) printf("<");
if (a[j]<num[i])
for (int jj=1;jj<=get(a[j],num[i]);jj++) printf("+");
if (a[j]>num[i])
for (int jj=1;jj<=get(a[j],num[i]);jj++) printf("-");
printf(".");
}
void doit()
{
for (int i=0;i<n;i++)
num[i]=s[i]-'a'+1;
memset(dp,0x3f,sizeof(dp));
for (int i=0;i<4;i++)
dp[0][i][0]=num[0]+96;
for (int i=1;i<n;i++)
for (int j=0;j<4;j++)
for (int k=0;k<N;k++)
if (dp[i-1][j][k]!=inf)
{
int x=k;
for (int p=3;p>=0;p--)
if (p==j) a[p]=num[i-1];
else {a[p]=x%27; x/=27;}
for (int nj=0;nj<4;nj++)
{
x=0;
for (int p=0;p<4;p++)
if (p!=nj) x=x*27+a[p];
if (miniz(dp[i][nj][x],dp[i-1][j][k]+abs(j-nj)+get(a[nj],num[i])))
{
//printf("%d %d %d ->%d %d %dn",i-1,j,k,i,nj,x);
f[i][nj][x]=j*N+k;
}
}
}
int ans=inf;
for (int j=0;j<4;j++)
for (int k=0;k<N;k++)
ans=min(ans,dp[n-1][j][k]);
for (int j=0;j<4;j++)
for (int k=0;k<N;k++)
if (ans==dp[n-1][j][k]) {print(n-1,j,k);
return;}
}
int main()
{
while (scanf("%s",s)!=EOF) {n=strlen(s);doit();}
}
//azazazazazazazazazazazazazazazazazazazaz

//ural 1655
#include<cstdio>
#include<iostream>
#include<cstring>
#include<algorithm>
#include<cmath>
#define N 505
#define eps 1e-6
#define MAX 1e7
typedef long long LL;
using namespace std;
const double tt=21600.0;
int n,nn,res;
double w,aa,tot;
double dp[N][N][2];
bool vis[N][N][2];
int from[N][N][2];
//dp[x][y][op];
//x,(x+1)%nn,(x+2)%nn,...,y
//op==0 end in x;
//op==1 end in y;
struct node{
double qt,dt;
int id;
bool operator<(const node& oth)const
{
return qt<oth.qt;
}
}a[N];
double get(int from,int to)
{
if (from>to) return tot-a[from].qt+a[to].qt;
else return a[to].qt-a[from].qt;
}
void print(int x,int y,int op)
{
if (x==res&&y==res) return;
int rr=from[x][y][op]/2,ii=from[x][y][op]%2;
if (rr==0)
{
print((x+1)%nn,y,ii);
printf("%dn",a[x].id);
}
if (rr==1)
{
print(x,(y-1+nn)%nn,ii);
printf("%dn",a[y].id);
}
}
double f(int x,int y,int op)
{
if (vis[x][y][op]) return dp[x][y][op];
double &ans=dp[x][y][op];
int &fr=from[x][y][op];
vis[x][y][op]=1;
ans=MAX;
int next;
double ad1,ad2,ad3;
next=(y-1+nn)%nn;
ad3=get(next,y);
if (op==0) ad1=get(x,y); else ad1=0;
for (int i=0;i<2;i++)
{if (i==0) ad2=get(x,next); else ad2=0;
double tmp=f(x,next,i);
if (tmp+ad2+ad3<=a[y].dt&&ans>tmp+ad1+ad2+ad3)
{ans=tmp+ad1+ad2+ad3;
fr=i+2;
}
}
next=(x+1)%nn;
ad3=get(x,next);
if (op==1) ad1=get(x,y); else ad1=0;
for (int i=0;i<2;i++)
{if (i==1) ad2=get(next,y); else ad2=0;
double tmp=f(next,y,i);
if (tmp+ad2+ad3<=a[x].dt&&ans>tmp+ad1+ad2+ad3)
{ans=tmp+ad1+ad2+ad3;
fr=i;
}
}
return ans;
}
void doit()
{
double x,y,z;
w*=tt;
for (int i=1;i<=n;i++)
{
scanf("%lf%lf%lf",&x,&y,&z);
a[i].qt=x/w;
a[i].dt=(y-1)/z+eps;
a[i].id=i;
}
tot=360.0/w;
a[0].qt=aa/w;
a[0].dt=MAX; //imp锛�
a[0].id=0;
nn=n+1;
sort(a,a+n+1);
for (int i=0;i<=n;i++)
if (a[i].id==0) {res=i;break;}
memset(vis,0,sizeof(vis));
for (int i=0;i<nn;i++)
for (int j=0;j<2;j++)
{
vis[i][i][j]=1;
dp[i][i][j]=(i==res)?0:MAX;
}
double ans=MAX,tmp;
int an1,an2,an3;
for (int i=0;i<=n;i++)
{tmp=f(i,(i-1+nn)%nn,0);
if (tmp<ans) {ans=tmp;an1=i;an2=(i-1+nn)%nn;an3=0;}
tmp=f(i,(i-1+nn)%nn,1);
if (tmp<ans) {ans=tmp;an1=i;an2=(i-1+nn)%nn;an3=1;}
}
if (ans+eps>=MAX) {printf("Impossiblen"); return;}
printf("%.6lfn",ans*60);
print(an1,an2,an3);
}
int main()
{
while (scanf("%lf%lf%d",&aa,&w,&n)!=EOF) doit();
}
/*
0 0.0278 3
280 46 100
90 16 100
180 61 100
*/

//ural 1739
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
const long long INF=1LL<<50;
using namespace std;
typedef long long LL;
typedef pair <int,double> PII;
#define N 12
int a[N];
int tmp[N+1];
int n,tot;
int dp[1<<N][2][2],dp2[1<<N][2][2];
int from1[1<<N][2][2],from2[1<<N][2][2],from3[1<<N][2][2];
int sum[1<<N];
#define M 4097
int c1[M];
int c2[N][M];
int c3[N][M];
int c4[N][N][M];
int c5[N][M];
//c1
//c2[j] j-25%
//c3[j] j+10%
//c4[j1][j2] j1-25% j2+10%
//c5[j] j-25% +10%
bool minize(int &a,int b,int& a1,int b1)
{
if (b>a) return 0;
if (b==a&&b1>=a1) return 0;
a=b;
a1=b1;
return 1;
}
void print(int left,int x1,int x2)
{
if (left==0) return;
int f1=from1[left][x1][x2],f2=from2[left][x1][x2],f3=from3[left][x1][x2];
int v1=f2/13,v2=f2%13;
//printf("~~~~~%d %d %d %d %d %dn",left,x1,x2,f1,f2,f3);
if (f1>0)
{
tmp[0]=0;
for (int i=0;i<n;i++)
if (f1&(1<<i))
tmp[++tmp[0]]=i;
printf("return %d",tmp[0]);
for (int i=1;i<=tmp[0];i++)
printf(" %d",tmp[i]+1);
printf("n");
}
if (v1) printf("dearomatize %dn",v1);
if (v2) printf("aromatize %dn",v2);
f3=left+f1-f3;
if (f3>0)
{
tmp[0]=0;
for (int i=0;i<n;i++)
if (f3&(1<<i))
tmp[++tmp[0]]=i;
printf("take %d",tmp[0]);
for (int i=1;i<=tmp[0];i++)
printf(" %d",tmp[i]+1);
printf("n");
}
f3=from3[left][x1][x2];
print(f3,x1||v1,x2||v2);
}
void f(int left,int x1,int x2)
{
if (dp[left][x1][x2]!=-1) return;
if (left==0) {dp[left][x1][x2]=0;dp2[left][x1][x2]=0;return;}
int &re=dp[left][x1][x2],&re2=dp2[left][x1][x2];
int &f1=from1[left][x1][x2],&f2=from2[left][x1][x2],&f3=from3[left][x1][x2];
re=inf; re2=inf;
int v,j;
for (int i=1;i<=c1[0];i++)
{j=c1[i];
v=left^(left&j);
f(v,x1,x2);
if (minize(re,1+dp[v][x1][x2],re2,1+dp2[v][x1][x2]+((left&j)!=j)))
{
f1=j-(left&j);
f2=0;
f3=v;
}
}
if (!x1)
for (int k=0;k<n;k++)
for (int i=1;i<=c2[k][0];i++)
{j=c2[k][i];
v=left^(left&j);
f(v,1,x2);
if (minize(re,1+dp[v][1][x2],re2,2+dp2[v][1][x2]+((left&j)!=j)))
{
f1=j-(left&j);
f2=(k+1)*13;
f3=v;
}
}
if (!x2)
for (int k=0;k<n;k++)
for (int i=1;i<=c3[k][0];i++)
{j=c3[k][i];
v=left^(left&j);
f(v,x1,1);
if (minize(re,1+dp[v][x1][1],re2,2+dp2[v][x1][1]+((left&j)!=j)))
{
f1=j-(left&j);
f2=(k+1);
f3=v;
}
}
if (!x1&&!x2)
for (int k1=0;k1<n;k1++)
for (int k2=0;k2<n;k2++)
for (int i=1;i<=c4[k1][k2][0];i++)
{j=c4[k1][k2][i];
v=left^(left&j);
f(v,1,1);
if (minize(re,1+dp[v][1][1],re2,3+dp2[v][1][1]+((left&j)!=j)))
{
f1=j-(left&j);
f2=(k1+1)*13+(k2+1);
f3=v;
}
}
if (!x1&&!x2)
for (int k=0;k<n;k++)
for (int i=1;i<=c5[k][0];i++)
{j=c5[k][i];
v=left^(left&j);
f(v,1,1);
if (minize(re,1+dp[v][1][1],re2,3+dp2[v][1][1]+((left&j)!=j)))
{
f1=j-(left&j);
f2=(k+1)*13+(k+1);
f3=v;
}
}
}
void doit()
{
for (int i=0;i<n;i++) scanf("%d",&a[i]);
tot=1<<n; tot--;
memset(sum,0,sizeof(sum));
for (int i=1;i<=tot;i++)
{
int x=i&(-i);
int t=int(log(double(x))/log(2.0)+eps);
sum[i]=sum[i-x]+a[t];
}
memset(c1,0,sizeof(c1));
memset(c2,0,sizeof(c2));
memset(c3,0,sizeof(c3));
memset(c4,0,sizeof(c4));
memset(c5,0,sizeof(c5));
for (int i=1;i<=tot;i++)
if (sum[i]%256==0) {c1[++c1[0]]=i;}
for (int j=0;j<n;j++)
if (a[j]%4==0)
{
int x;
for (int i=1;i<=tot;i++)
{
if (!(i&(1<<j))) continue;
x=sum[i]-a[j]+a[j]/4*3;
if (x%256==0) {c2[j][++c2[j][0]]=i;}
}
}
for (int j=0;j<n;j++)
if (a[j]%10==0)
{
int x;
for (int i=1;i<=tot;i++)
{
if (!(i&(1<<j))) continue;
x=sum[i]-a[j]+a[j]/10*11;
if (x%256==0) {c3[j][++c3[j][0]]=i;}
}
}
for (int j1=0;j1<n;j1++)
for (int j2=0;j2<n;j2++)
if (j1!=j2)
if ((a[j1]*15+a[j2]*22)%20==0)
{
int x;
for (int i=1;i<=tot;i++)
{
if (!(i&(1<<j1)))continue;
if (!(i&(1<<j2)))continue;
x=sum[i]-a[j1]-a[j2]+(a[j1]*15+a[j2]*22)/20;
if (x%256==0) {c4[j1][j2][++c4[j1][j2][0]]=i;}
}
}
for (int j=0;j<n;j++)
if (a[j]%40==0)
{
int x;
for (int i=1;i<=tot;i++)
{
if (!(i&(1<<j))) continue;
x=sum[i]-a[j]+a[j]/40*33;
if (x%256==0) {c5[j][++c5[j][0]]=i;}
}
}
memset(dp,255,sizeof(dp));
f(tot,0,0);
if (dp[tot][0][0]==inf) {printf("IMPOSSIBLEn"); return;}
if (dp2[tot][0][0]>20000) {printf("IMPOSSIBLEn"); return;}
printf("%dn",dp2[tot][0][0]);
print(tot,0,0);
}
int main()
{
while (scanf("%d",&n)!=EOF) doit();
}
/*
2
40 223
*/

//ural 1895
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
const long long INF=1LL<<50;
using namespace std;
typedef long long LL;
typedef pair <int,double> PII;
#define N 1005
#define M 55
#define M2 110
int n,X,K,o;
int f[M2][M][M],from1[M2][M][M],from2[M2][M][M];;
int t[M],ans1[M],ans2[M];
int num[M2];
int vis[N],have[N];
void print(int an1,int an2,int an3)
{
if (an1==0) return;
int t1=from1[an1][an2][an3],t2=from2[an1][an2][an3];
for (int i=t1+1;i<=an2;i++)
ans1[i]=an1;
for (int i=t2+1;i<=an3;i++)
ans2[i]=an1;
print(an1-1,t1,t2);
}
void doit()
{
scanf("%d",&n);
for (int i=1;i<=n;i++)
scanf("%d",&t[i]); t[n+1]=-1;
memset(vis,0,sizeof(vis));
//
for (int i=0;i<t[n];i++)
//
num[++o]=i;
for (int i=1;i<=n;i++)
{int x=t[i]-1;
while (vis[x]&&x) x--; vis[x]=1;
x--;
if (x<0) continue;
while (vis[x]&&x) x--; vis[x]=1;
}
o=0;
for (int i=0;i<t[n];i++)
if (vis[i]) {num[++o]=i;vis[i]=o;}
int lx=1,x=1;
for (int i=1;i<t[n];i++)
{
while (t[x]==i) x++;
have[i]=have[i-1]+x-lx;
lx=x;
}
//
for (int i=1;i<=o;i++)
//
printf("%d %d %dn",i,num[i],have[num[i]]);
memset(f,0x3f,sizeof(f));
f[0][0][0]=0;
for (int i=1;i<=o;i++)
for (int j=0;j<=n;j++)
for (int k=0;k<=j;k++)
{
if (j<have[num[i]]||k<have[num[i]]) continue;
if (f[i][j][k]>f[i-1][j][k])
{
f[i][j][k]=f[i-1][j][k];
from1[i][j][k]=j;
from2[i][j][k]=k;
}
for (int nk=k;nk<=j;nk++)
{
if (nk-k>K)break;
int left=K-(nk-k);
for (int use=0;use<=left;use++)
{
int nj=j+use;
if (nj>n) nj=n;
if (num[i]+X<t[nj]) break;
if (f[i][nj][nk]>f[i-1][j][k]+1)
{f[i][nj][nk]=f[i-1][j][k]+1;
from1[i][nj][nk]=j;
from2[i][nj][nk]=k;
}
}
}
}
if (f[o][n][n]==inf) {printf("%dn",-1); return;}
printf("%dn",f[o][n][n]);
print(o,n,n);
for (int i=1;i<=n;i++)
printf("%d %dn",num[ans1[i]],num[ans2[i]]);
}
int main()
{
while (scanf("%d%d",&X,&K)!=EOF) doit();
}
/*
10 2
3
2 16 25
*/

//ural 1776
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <iostream>
#include <cstdlib>
#include <cmath>
#include <queue>
#include <vector>
#include <map>
#define pb push_back
#define mp make_pair
#define eps 1e-9
#define zero(x) (fabs(x)<eps)
#define pi acos(-1.0)
#define f1 first
#define f2 second
const int inf=0x3f3f3f3f;
const long long INF=1LL<<50;
using namespace std;
typedef long long LL;
typedef pair <int,double> PII;
#define N 402
double dp[N][N],sum[N][N];
int n;
int main()
{
memset(dp,0,sizeof(dp));
memset(sum,0,sizeof(sum));
dp[0][0]=1;
for (int i=0;i<=400;i++)
sum[0][i]=1;
for (int i=1;i<=400;i++)
{
for (int k=1;k<=i;k++)
{
int t=max(k-1,i-k)+1;
for (int j=1;j<=t;j++)
dp[i][j]+= (dp[k-1][j-1] * sum[i-k][j-1]+
sum[k-1][j-1]*
dp[i-k][j-1]-
dp[k-1][j-1] *
dp[i-k][j-1])/i;
}
for (int j=1;j<=400;j++)
sum[i][j]=sum[i][j-1]+dp[i][j];
}
while (scanf("%d",&n)!=EOF)
{
n-=2;
double ans=0;
for (int i=1;i<=n;i++)
ans+=dp[n][i]*i;
printf("%.6lfn",ans*10);
}
}