O(n)线性空间的迷宫生成算法

之前所有的迷宫生成算法，空间都是O(mn)，时间同样是O(mn)，时间上已经不可能更优化，

于是，我就从空间优化上着手，研究一个仅用O(n)空间的生成算法。

我初步的想法是，每次生成一行，生成后立即输出，而其连通性的信息用并查集保存。

然而这时却遇到阻力：不可能简单保存连通性信息就能完成。

因为通过上一行的信息去生成下一行的时候，仅靠连通性信息你无法知道你能不能结束当前路径上的连结，

也就是说你不能判断你能不能生成一个死胡同。

后来想了一下，改进了一下并查集，让它的父结点多保存一个信息：当前行与本结点连通的格子数

看这个例子：

　━┳━━┳━━┳━━━━┳━━┳━━┳┓
┣┓┗━┃┃━━┛━━━┓┃┃┃┗┓┃┃┃
┃┗┳━┫┗━━┓┏━━┫┏┛┣━┃┃┃┃

1 1  1 1 2 2 2 2 3 6 6 6 3 3 3 3 3 4 4 5

相同数字的表示它们连通（连向同一个父结点），比如3那个，当前行与本结点连通的格子数=6

也就是3出现的次数。

在生成下一行的时候，每生成一个格子，就得同时更新连通格子数的值，以保证不会生成环路和死路

参考源代码：

#include <iostream>
#include <ctime>
using namespace std;

#define next(n,s) ((n)=(n)%((s)-1)+1)
int fset_size;
struct data
{
    int parent;
    int level;
    int sum;
}*fset;
int fset_pt = 1, fdeep;

int try_alloc(int level)
{
    if (fset[fset_pt].level < level)
    {
        return fset_pt;
    }
    int lpt = fset_pt;
    for (next(fset_pt, fset_size); fset_pt!=lpt; next(fset_pt, fset_size))
    {
        if (fset[fset_pt].level < level)
        {
            return fset_pt;
        }
    }
    return 0;
}

int reg_fset(int level)
{
    int lpt = fset_pt;
    fset[fset_pt].level = level;
    fset[fset_pt].parent = 0;
    fset[fset_pt].sum = 1;
    next(fset_pt, fset_size);
    return lpt;
}

int get_parent(int id)
{
    int d = id;
    for (fdeep=0; fset[id].parent>0; ++fdeep)
        id = fset[id].parent;
    if (d != id) fset[d].parent = id;
    return id;
}

void insert(int a, int b)
{
    int pa = get_parent(a), da = fdeep;
    int pb = get_parent(b), db = fdeep;
    if (pa==pb)
    {
    }
    else if (da<=db)
    {
        fset[pa].parent = pb;
        fset[pb].level = max(fset[pa].level, fset[pb].level);
        fset[pb].sum += fset[pa].sum-1;
    }
    else
    {
        fset[pb].parent = pa;
        fset[pa].level = max(fset[pa].level, fset[pb].level);
        fset[pa].sum += fset[pb].sum-1;
    }
}

int is_connect(int a, int b)
{
    if (a==b || get_parent(a)==get_parent(b))
        return 1;
    return 0;
}

struct tile
{
    int wall;
    int id;
};

char cw[][4]={"　","━","┃","┛","━","━","┗","┻","┃","┓","┃","┫","┏","┳","┣","╋"};

int Gen(int w, int h)
{
    int x,y,lx,ly,p;
    tile* maze = (tile*)malloc(sizeof(tile)*(w+2));
    fset = (data*)malloc(sizeof(data)*(w*2));
    fset_size = w;
    memset(fset, 0, sizeof(data)*(w*2));
    for (x=0; x<=w+1; ++x) maze[x].wall = 7, maze[x].id=0;
    maze[0].wall = 15; maze[w+1].wall = 15;
    for (y=1; y<=h+1; ++y,maze[0].wall = 11,maze[w+1].wall = 14)
    {
        lx = 0; ly = 1; x = 0;
        fset[0].sum = 0;
        p = 15;
        if (lx) p &= ~8;
        if (ly) p &= ~1;
        if (maze[x+1].wall&8) p &= ~4;
        if (is_connect(maze[x].id, maze[x+1].id)) p &= ~2;
        if (y>h) p &= ~8;
        printf(cw[p]);
        p &= 4;
        printf(cw[p]);
        for (x=1; x<=w; ++x)
        {
            int r, _r = 4;
            ly = (maze[x].wall&8);
            int id, dsum = 0;
            if (lx && ly)
            {
                insert(maze[x-1].id, maze[x].id);
            }
            else if (lx==0 && ly==0)
            {
                maze[x].id = try_alloc(y);
                if (maze[x].id==0)
                {
                    fset_size += w;
                    maze[x].id = try_alloc(y);
                    fset_size -= w;
                }
                reg_fset(y+1);
            }
            else if (lx)
            {
                maze[x].id = maze[x-1].id;
            }
            id = get_parent(maze[x].id);
            if ((maze[x+1].wall&8) && is_connect(maze[x].id, maze[x+1].id)) _r = 2;
            if (y==h && x==w)
            {
                r = 0;
            }
            else do
            {
                r = rand() % _r;
                if ((r&2)==0 && try_alloc(y)==0) r |= 2;
                if (y>h)
                {
                    r = 2;
                }
                else if (x==w) r &= ~2;
                if (y==h) r &= ~1;
                dsum = 0;
                if ((r & 1) == 0 ) dsum -= 1;
                if (r & 2) dsum += 1;
            }
            while (y<=h && ((r==0 && (lx==0 && ly==0) || fset[id].sum+dsum<=0 )) );
            maze[x].wall = 0;
            if (ly) maze[x].wall |= 2;
            if (lx)
            {
                maze[x].wall |= 1;
            }
            lx = (r&2);
            if (lx) maze[x].wall |= 4;
            if (r&1) maze[x].wall |= 8;
            p = 15;
            if (maze[x].wall&4) fset[id].sum += 1;
            if ((maze[x].wall&8)==0) fset[id].sum -= 1, fset[0].sum += 1;
            fset[id].level = y+1;
            if (maze[x].wall&4) p &= ~8;
            if (maze[x].wall&2) p &= ~1;
            if (maze[x+1].wall&8) p &= ~4;
            if (maze[x+1].wall&1) p &= ~2;
            printf(cw[p]);
            p &= 4;
            printf(cw[p]);
        }
        puts("");
        if (y<h)
        {
            for (x=0; x<=w; ++x)
            {
                if (maze[x].wall&4) printf(cw[0]); else printf(cw[10]);
                int id = get_parent(maze[x].id);
                maze[x].id = id;
                printf(cw[0]);
            }
            puts("");
        }
        else if (y==h)
        {
            for (x=0; x<=w; ++x)
            {
                if ((maze[x].wall&4) || x==w) printf(cw[0]); else printf(cw[10]);
                int id = get_parent(maze[x].id);
                maze[x].id = id;
                printf(cw[0]);
            }
            puts("");
        }
    }
    free(maze);
    free(fset);
    return 0;
}
int main()
{
    srand((unsigned)time(NULL));
    Gen(15,10);
    return 0;
}