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0w1

Monthly Training Farm - May 2017 D. oeis1 ( Offline )

https://oj.icpc.tw/contest/5/problem/D

Problem Description:
Define f( x ) as sum of digits of x, g( x ) as product of digits greater than 0 of x.
Let D[ 1 ] = 8.
D[ i ] = sum( f( D[ j ] ) + g( D[ j ] ) + 10 for j in range( 1, i ) ).
Print D[ X ].

Constraints:
1 ≤ X ≤ 8.7e8

Solution:
A brute force solution obviously can be done with O( 1 ) in both time and space.
Therefore we precompute all values of possible D[ i ] where i % 1e6 == 1, storing it in a static array.
Now we can start from the term of the nearest smaller stored index of x, so we can calculate no more than 1e6 times to retrieve any D[ x ].

Time Complexity:
O( 1e6 )

#include <bits/stdc++.h>
using namespace std;

#define int unsigned long long

int X;

int F( int x ) {
  int res = 0;
  for( int i = x; i; i /= 10 ) {
    res += i % 10;
  }
  return res;
}

int G( int x ) {
  int res = 1;
  for( int i = x; i; i /= 10 ) {
    res *= max( 1ULL, i % 10 );
  }
  return res;
}

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12430172920077300871ULL, 12510033344018143970ULL, 12560111331807167609ULL, 12620300909062507249ULL, 12712327122365584099ULL, 12813120201213924600ULL, 12941165650752604069ULL, 13001015450916236719ULL, 13002102112140524827ULL, 13009101051211529299ULL, 13011011201840674837ULL, 13012011200100272764ULL, 13017871597384507708ULL, 13022112682259429716ULL, 13035303725806214962ULL, 13061207162873518843ULL, 13101110211883417498ULL, 13102307620707874567ULL, 13110011112018854023ULL, 13111001400320058763ULL, 13111510321056866221ULL, 13115311279386832627ULL, 13121402212711908655ULL, 13132110718629054154ULL, 13156105146839315569ULL, 13200313020278778370ULL, 13210011012121171228ULL, 13213423249361668672ULL, 13240312112360909185ULL, 13304815634984262070ULL, 13328169412105586248ULL, 13411042429970294872ULL, 13502668079517138700ULL, 13610101015273120984ULL, 13742172294958738675ULL, 13955010105287823835ULL, 14001111914465436166ULL, 14005718153017034195ULL, 14011091130510814555ULL, 14015104231941588982ULL, 14024198940867158771ULL, 14057011815230816884ULL, 14100450036349198212ULL, 14102112146150632238ULL, 14110110210162382822ULL, 14111301341313317036ULL, 14120017321500977589ULL, 14131110100944863013ULL, 14170725499139830199ULL, 14206111220745099267ULL, 14221102113069094860ULL, 14306120201523498932ULL, 14401016831681598749ULL, 14504145142170066066ULL, 14702246022571246478ULL, 15000932060667308245ULL, 15006213005415560866ULL, 15011203461312037475ULL, 15021114160849623834ULL, 15064132587707205729ULL, 15101114121387175100ULL, 15110121126567543514ULL, 15113111030225503923ULL, 15131111101328059518ULL, 15200211165114008645ULL, 15215231836896581647ULL, 15311015083995691650ULL, 15420613121581116152ULL, 15701101229511434643ULL, 16001112151204909152ULL, 16010411810223887040ULL, 16020101072976056952ULL, 16060135644080345961ULL, 16101351068167771299ULL, 16111071307968405954ULL, 16121617114078184874ULL, 16183314443164332000ULL, 16221811218331501491ULL, 16330217942544948147ULL, 16570104242510786490ULL, 17001104101283642140ULL, 17011116072075965697ULL, 17033456472603589603ULL, 17101190410914354070ULL, 17111218216083481212ULL, 17133141250056037829ULL, 17211101251174495777ULL, 17321130784350135385ULL, 17641014337217993341ULL };

signed main() {
  ios::sync_with_stdio( 0 );
  cin >> X;
  --X;
  if( X == 0 ) {
    cout << 8 << endl;
    exit( 0 );
  }
  int d = dict[ ( X - 1 ) / 1000000 ];
  int prev = d;
  for( int i = ( X - 1 ) / 1000000 * 1000000; i < X; ++i ) {
    d = prev;
    prev += F( d ) + G( d ) + 10;
  }
  cout << d << endl;
  return 0;
}