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fourcolor.cpp
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fourcolor.cpp
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#include <boost/scoped_array.hpp>
#include <iostream>
#include <list>
#include <vector>
#include <cmath>
#include <map>
#include <time.h>
#define LEN 17
// 2 or 4 supported.
#define COLORS 4
int maxFound = 0;
time_t start;
using namespace std;
using boost::scoped_array;
struct Coloring
{
int r;
int c;
int color;
};
// Honestly, C++ committee, is leaving this out REALLY better?
inline unsigned long long pow(int m, int e)
{
unsigned long long ret=1;
unsigned long long add=m;
while(e){
if(e&1 != 0){
ret*=add;
}
add*=add;
e>>=1;
}
return ret;
}
// Returns the size of a mask based on the number of colors in this scenario
unsigned int MaskSizeFromColors(int colors)
{
switch(colors){
case 0:
case 1:
case 2:
return 1;
case 3:
case 4:
return 2;
case 5:
case 6:
case 7:
case 8:
return 3;
}
return -1;
}
// An integer code is used to iterate through the search space.
// 2 colors: 010b = color 1, color 2, color 1
// 4 colors: 00011011 = color0, color1, color 2, color 3 (goes in pairs)
// If we have a square to 4-color:
//
// 1x
// xx
//
// The number 000110 would fill in:
//
// 13
// 12
//
// I.E. it wraps around counterclockwise
int ColorFromCode(unsigned long long num, int index, int colors)
{
int maskSize = MaskSizeFromColors(colors);
return ((num >> (index * maskSize)) & (pow(2, maskSize)-1)) + 1 ;
}
// Converts an integer code into a vector of Colorings
void NumCodeToColorings(unsigned long long toAdd, int colors, int level,
vector<Coloring> &colorings)
{
int slots = 2*level-1;
colorings.resize(slots);
unsigned maskSizeBits = MaskSizeFromColors(colors);
unsigned mask = (1<<maskSizeBits)-1; // msb=2 => (1<<2)-1 => 0x3
int ind = 0;
// Generate column added on left
for(int i=0; i<level; ++i){
colorings[ind].r = i;
colorings[ind].c = level-1;
colorings[ind++].color = ColorFromCode(toAdd, i, colors);
mask<<=2;
}
// Generate row added on bottom
for(int i=0; i<level-1; ++i){
colorings[ind].r = level-1;
colorings[ind].c = i;
colorings[ind++].color
= ColorFromCode(toAdd, slots-1-i, colors);
mask<<=2;
}
}
// Sets all positions on a board to 0
void ClearBoard(int board[LEN][LEN])
{
for(int r=0; r<LEN; ++r)
for(int c=0; c<LEN; ++c)
board[r][c] = 0;
}
// Prints all positions on the board to STDOUT
void PrintBoard(int board[LEN][LEN])
{
for(int r=0; r<LEN; ++r){
for(int c=0; c<LEN; ++c){
cout<<board[r][c];
}
cout<<endl;
}
}
// Draws an integer code onto the board by level and number of colors.
void DrawOnBoard(int board[LEN][LEN], int level, int colors,
unsigned long long toAdd)
{
static vector<Coloring> coloring;
NumCodeToColorings(toAdd, colors, level, coloring);
for(int i=0; i<coloring.size(); ++i){
board[coloring[i].r][coloring[i].c] = coloring[i].color;
}
}
// When iterating through integer codes at a particular level, we will
// inevitably hit constraints based on numbers that had been placed in
// previous levels. Instead of iterating by 1, we can skip all of the
// numbers that match the lowest bit of the conflict.
unsigned long long CalculateJump(int slot, unsigned long long num, int colors)
{
unsigned long long maskSize = MaskSizeFromColors(colors),
one, slotCalc;
slotCalc = slot-1;
one=1;
unsigned long long mod = one<<(slotCalc * maskSize);
return mod - (num % mod);
}
/*
// Counts the constraints that the most recent addition at 'level'
// added.
int CountAddedConstraints(int board[LEN][LEN], int level)
{
int acc=0;
for(int r=0; r<level; ++r){
int rowCol = board[r][level-1];
for(int c=0; c<level; ++c){
int colCol = board[level-1][c];
if(r < level-1){
if(board[r][c] == rowCol)
++acc;
}
if(c < level-1){
if(board[r][c] == colCol)
++acc;
}
}
}
return acc;
}
*/
// Accumulates a distribution of the number of constraints added by
// the most recent number addition at 'level'. Returns the standard
// deviation of this distribution.
double GenerateConstraintScore(int board[LEN][LEN], int level)
{
int constraints[LEN];
for(int i=0; i<LEN; ++i) constraints[i] = 0;
for(int r=0; r<level; ++r){
int rowCol = board[r][level-1];
for(int c=0; c<level; ++c){
int colCol = board[level-1][c];
if(r < level-1){
if(board[r][c] == rowCol)
++constraints[rowCol-1];
}
if(c < level-1){
if(board[r][c] == colCol)
++constraints[colCol-1];
}
}
}
int acc=0;
for(int i=0; i<LEN; ++i) acc+=constraints[i];
double mean = acc / (1.0 * LEN);
double stdDevAcc = 0.;
for(int i=0; i<LEN; ++i){
stdDevAcc+=(pow(mean - constraints[i], 2));
}
return sqrt(stdDevAcc);
}
// Iterates through the search space, and finds the list of
// candidates. I did some benchmarking, and aborting on the first
// mismatch is much faster than finding the best mismatch.
void FindCandidates(int board[LEN][LEN],
multimap<double, unsigned long long> &candidateMap,
int colors,
int level)
{
unsigned long long max = pow(colors, 2*level-1);
unsigned long long rowEndColor, colEndColor;
unsigned long long candidate = 0;
while(candidate < max){
DrawOnBoard(board, level, colors, candidate);
int bestInvalidSlot = 0;
// Search for 2 columns that are invalidated. If we hit, we
// are invalidated on a very high bit, so this gives us the
// biggest jump in our search.
for(int c0=0; c0<level-2; ++c0){
int c0Col=board[level-1][c0];
for(int c1=c0+1; c1<level-1; ++c1){
int c1Col=board[level-1][c1];
if(c0Col == c1Col){
for(int r=0; r<level-1; ++r){
if(board[r][c0] == board[r][c1]
&& board[r][c0] == c0Col){
// cerr<<"invalid col: "<<c0<<" "<<c1<<" "<<(level + (level-c1-1))<<endl;
int nextSlot = (level + (level-c1-1));
bestInvalidSlot =
bestInvalidSlot > nextSlot
? bestInvalidSlot
: nextSlot ;
// If you think of a better way,
// [email protected] :)
goto abort;
}
}
}
}
}
//if(bestInvalidSlot > 0)
//goto abort;
// Now try to strike out on a low bit
// This still could give us a nice jump, but not good enough
for(int r0=0; r0<level-1; ++r0){
int r0Col = board[r0][level-1];
for(int r1=r0+1; r1<level; ++r1){
int r1Col = board[r1][level-1];
if(r0Col == r1Col){
for(int c=0; c<level-1; ++c){
if(board[r0][c] == board[r1][c]
&& board[r0][c] == r0Col){
// cerr<<"invalid row: "<<r0<<" "<<r1<<" "<<(r0 + 1)<<endl;
int nextSlot = r0 + 1;
bestInvalidSlot = bestInvalidSlot > nextSlot
? bestInvalidSlot
: nextSlot;
goto abort;
}
}
}
}
}
abort:
// If we haven't found an invalid slot.
if(0 == bestInvalidSlot ){
double numAdded = GenerateConstraintScore(board, level);
candidateMap
.insert(pair<double, unsigned long long>(numAdded, candidate));
++candidate;
// Good enough for me
if(candidateMap.size() > 20000){
return;
}
else if(level > 9 && candidateMap.size() > 1000)
return;
else if(level > 11 && candidateMap.size() > 10)
return;
}
else
candidate+=CalculateJump(bestInvalidSlot, candidate, colors);
}
}
bool Recur(int board[LEN][LEN], int colors, int level)
{
typedef multimap<double, unsigned long long> multiMapIntUInt64;
if(level > maxFound){
maxFound = level;
cout<<"Solved level "<<level-1<<" in ";
time_t diff = time(NULL) - start;
cout<<diff<<" seconds"<<endl;
PrintBoard(board);
}
if(level <= LEN){
multiMapIntUInt64 candidateMap;
cerr<<"Finding candidates for level "<<level<<endl;
FindCandidates(board, candidateMap, colors, level);
cerr<<"Done finding candidates for level "<<level<<endl;
multiMapIntUInt64::iterator itr = candidateMap.begin(),
end=candidateMap.end();
for(; itr!=end; ++itr){
DrawOnBoard(board, level, colors, itr->second);
if(Recur(board, colors, level+1)){
return true;
}
//cerr<<"level "<<level<<": "<<itr->second<<" sucks"<<endl;
}
return false;
}
else{
return true;
}
}
bool Solve(int board[LEN][LEN], int colors)
{
return Recur(board, colors, 1);
}
int main()
{
int board[LEN][LEN];
ClearBoard(board);
start = time(NULL);
if(!Solve(board, COLORS)){
// Knowing we failed is probably wishful thinking
cout<<"There are no "<<LEN<<" x "<<LEN<<" solutions for "
<<COLORS<<" colors"<<endl;
}
else{
cout<<"Solution found!"<<endl;
PrintBoard(board);
}
return 0;
}