PAT甲级 1010

Radix

Given a pair of positive integers, for example, 6 and 110, can this equation 6 = 110 be true? The answer is “yes”, if 6 is a decimal number and 110 is a binary number.

Now for any pair of positive integers N1 and N2, your task is to find the radix of one number while that of the other is given.

Input Specification:

Each input file contains one test case. Each case occupies a line which contains 4 positive integers:
N1 N2 tag radix
Here N1 and N2 each has no more than 10 digits. A digit is less than its radix and is chosen from the set {0-9, a-z} where 0-9 represent the decimal numbers 0-9, and a-z represent the decimal numbers 10-35. The last number “radix” is the radix of N1 if “tag” is 1, or of N2 if “tag” is 2.

Output Specification:

For each test case, print in one line the radix of the other number so that the equation N1 = N2 is true. If the equation is impossible, print “Impossible”. If the solution is not unique, output the smallest possible radix.

Sample Input 1:
6 110 1 10
Sample Output 1:
2
Sample Input 2:
1 ab 1 2
Sample Output 2:
Impossible

---

1. 首先需要判断输出radix的范围：

   • 题意已经给了radix>digit
   • 假设tag=1，那么题意暗含了radix<decimal(N1)。
     
     • 否则此式成立：decimal(N2)>=decimal(N1)
       
       • 如果等于的话，此时radix不是最小的
       • 必定大于的话，也就没有判断的必要了

2. 假设tag=1 题中没有给输入的radix一个范围，不知道decimal(N1)的最大值是多少。下面的代码是按照long long int范围来做的。

#include <iostream>
#include <math.h>
#include <algorithm>
#include <string.h>
#include <vector>
#include <stdio.h>
#include <queue>
using namespace std;
#define MAXSIZE 10+1
#define INF 0x7fffffff
typedef long long ll;

void tran(char n[]){
  int len=0;
  while(n[len]!='\0'){
    if(!(n[len]>='0' && n[len]<='9'))n[len]=n[len]-'a'+'0'+10;
    len++;
  }
}

ll getDecimal(char n[],ll radix,int len){
    ll decimal=0;
    int i;
  for(i=0;i<len;i++){
    decimal=decimal*radix+(n[i]-'0');
    if(decimal<0)return -1;
  }
    return decimal;
}

int getMaxDigit(char n[]){
    int m=0;
    int len=0;
    while(n[len]!='\0')m=max(m,n[len++]-'0');
    return m+1;
}

ll work(char n[],ll value,int len){
    ll low=(ll)getMaxDigit(n),high=max(value,low);//here
  ll mid=(low+high)/2,temp=-1;
  while(low<=high){
    temp=getDecimal(n,mid,len);
    if(temp==value)break;
    else if(temp>value || temp==-1)high=mid-1;
    else low=mid+1;
    mid=(low+high)/2;
  }
  if(temp!=value) mid=-1;
  return mid;
}

int main(){
    //freopen("./in","r",stdin);
  int tag;
  ll radix;
  char n1[MAXSIZE],n2[MAXSIZE];

  scanf("%s %s %d %lld",n1,n2,&tag,&radix);

  tran(n1);
  tran(n2);

  if(tag!=1)swap(n1,n2);

  ll rst=work(n2,getDecimal(n1,radix,strlen(n1)),strlen(n2));

  if(rst==-1)printf("Impossible");
  else printf("%lld",rst);
    return 0;
}