图的表示

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1 邻接矩阵，对于N个点的图，需要N×N的矩阵表示点与点之间是否有边的存在。这种表示法的缺点是浪费空间，尤其是对于N×N的矩阵是稀疏矩阵，即边的数目远远小于N×N的时候，浪费了巨大的存储空间

2 邻接链表，对于任何一个node A，外挂一个邻接链表，如果存在 A->X这样的边，就将X链入链表。这种表示方法的优点是节省空间，缺点是所有链表都存在的缺点，地址空间的不连续造成缓存命中降低，性能有不如临界矩阵这样的数组。

#ifndef __GRAPH_H__#define __GRAPH_H__ typedef struct graph *Graph; Graph graph_create(int n);void graph_destroy(Graph);void graph_add_edge(Graph, int source, int sink);int graph_vertex_count(Graph);int graph_edge_count(Graph);int graph_out_degree(Graph, int source);int graph_has_edge(Graph, int source, int sink);void graph_foreach(Graph g, int source,        void (*f)(Graph g, int source, int sink, void *data),        void *data); #endif#include <stdlib.h>#include <assert.h> #include "graph.h" /* basic directed graph type *//* the implementation uses adjacency lists * represented as variable-length arrays */ /* these arrays may or may not be sorted: if one gets long enough * and you call graph_has_edge on its source, it will be */ struct graph {    int n;                                             /* number of vertices */    int m;                                             /* number of edges */    struct successors {        int d;                                         /* number of successors */        int len;                                       /* number of slots in array */        char is_sorted;                                /* true if list is already sorted */        int list[1];                                                                                          /* actual list of successors */    } *alist[1];}; /* create a new graph with n vertices labeled 0..n-1 and no edges */Graph graph_create(int n){    Graph g;    int i;     g = malloc(sizeof(struct graph) + sizeof(struct successors *) * (n-1));    assert(g);     g->n = n;    g->m = 0;     for(i = 0; i < n; i++) {        g->alist[i] = malloc(sizeof(struct successors));        assert(g->alist[i]);         g->alist[i]->d = 0;        g->alist[i]->len = 1;        g->alist[i]->is_sorted= 1;    }        return g;} /* free all space used by graph */void graph_destroy(Graph g){    int i;     for(i = 0; i < g->n; i++) free(g->alist[i]);    free(g);} /* add an edge to an existing graph */void graph_add_edge(Graph g, int u, int v){    assert(u >= 0);    assert(u < g->n);    assert(v >= 0);    assert(v < g->n);     /* do we need to grow the list? */    while(g->alist[u]->d >= g->alist[u]->len) {        g->alist[u]->len *= 2;        g->alist[u] =            realloc(g->alist[u],                 sizeof(struct successors) + sizeof(int) * (g->alist[u]->len - 1));    }     /* now add the new sink */    g->alist[u]->list[g->alist[u]->d++] = v;    g->alist[u]->is_sorted = 0;     /* bump edge count */    g->m++;} /* return the number of vertices in the graph */int graph_vertex_count(Graph g){    return g->n;} /* return the number of vertices in the graph */int graph_edge_count(Graph g){    return g->m;} /* return the out-degree of a vertex */int graph_out_degree(Graph g, int source){    assert(source >= 0);    assert(source < g->n);     return g->alist[source]->d;} /* when we are willing to call bsearch */#define BSEARCH_THRESHOLD (10) static intintcmp(const void *a, const void *b){    return *((const int *) a) - *((const int *) b);} /* return 1 if edge (source, sink) exists), 0 otherwise */int graph_has_edge(Graph g, int source, int sink){    int i;     assert(source >= 0);    assert(source < g->n);    assert(sink >= 0);    assert(sink < g->n);     if(graph_out_degree(g, source) >= BSEARCH_THRESHOLD) {        /* make sure it is sorted */        if(! g->alist[source]->is_sorted) {            qsort(g->alist[source]->list,                    g->alist[source]->d,                    sizeof(int),                    intcmp);        }                /* call bsearch to do binary search for us */        return             bsearch(&sink,                    g->alist[source]->list,                    g->alist[source]->d,                    sizeof(int),                    intcmp)            != 0;    } else {        /* just do a simple linear search */        /* we could call lfind for this, but why bother? */        for(i = 0; i < g->alist[source]->d; i++) {            if(g->alist[source]->list[i] == sink) return 1;        }        /* else */        return 0;    }} /* invoke f on all edges (u,v) with source u *//* supplying data as final parameter to f */void graph_foreach(Graph g, int source,    void (*f)(Graph g, int source, int sink, void *data),    void *data){    int i;     assert(source >= 0);    assert(source < g->n);     for(i = 0; i < g->alist[source]->d; i++) {        f(g, source, g->alist[source]->list[i], data);    }}

除此外，graph_foreach这种思想也很不错啊。Lisp这种传递函数作用到每一个元素上的方法，相当于Lisp中的mapcar，让人不仅拍案叫绝，很容易就能扩展出很好的功能。（当然也不是完全没瑕疵，比如realloc没有判断失败的情景，白璧微瑕）

既然是图的表示，我们当然很希望能够看到可视化的图。Land of Lisp一书中，学到了Linux下的neato 命令。 Linux下有工具帮助生成图的图片，可以满足我们可视化的需求。

测试代码

#include <stdio.h>#include <assert.h> #include "graph.h" #define TEST_SIZE (6)  static void match_sink(Graph g, int source, int sink, void *data){    assert(data && sink == *((int *) data));} static int node2dot(Graph g){    assert(g != NULL);    return 0;} static void print_edge2dot(Graph g,int source, int sink, void *data){    fprintf(stdout,"%d->%d;n",source,sink);}static int edge2dot(Graph g){    assert( NULL);    int idx = 0;    int node_cnt = graph_vertex_count(g);    for(idx = 0;idx<node_cnt; idx++)    {        graph_foreach(g,idx,print_edge2dot,NULL);    }    return 0;} int graph2dot(Graph g){    fprintf(stdout,"digraph{");    node2dot(g);    edge2dot(g);    fprintf(stdout,"}n");    return 0;} int main(int argc, char **argv){    Graph g;    int i;    int j;     g = graph_create(TEST_SIZE);     assert(graph_vertex_count(g) == TEST_SIZE);     /* check it's empty */    for(i = 0; i < TEST_SIZE; i++) {        for(j = 0; j < TEST_SIZE; j++) {            assert(graph_has_edge(g, i, j) == 0);        }    }     /* check it's empty again */    for(i = 0; i < TEST_SIZE; i++) {        assert(graph_out_degree(g, i) == 0);        graph_foreach(g, i, match_sink, 0);    }     /* check edge count */    assert(graph_edge_count(g) == 0);     for(i = 0; i < TEST_SIZE; i++) {        for(j = 0; j < TEST_SIZE; j++) {            if(i < j) graph_add_edge(g, i, j);        }    }      for(i = 0; i < TEST_SIZE; i++) {        for(j = 0; j < TEST_SIZE; j++) {            assert(graph_has_edge(g, i, j) == (i < j));        }    }     assert(graph_edge_count(g) == (TEST_SIZE*(TEST_SIZE-1)/2));    graph2dot(g);    /* free it     * */    graph_destroy(g);     return 0;}

我们这个测试程序基本测试了graph的API，同时利用graph_foreach函数的高效扩展性，输出了dot文件格式的文件我们看下执行结果。

root@manu:~/code/c/self/graph_2# gcc -o test generate_graph.c graph.c root@manu:~/code/c/self/graph_2# ./test >test.dot

在我的Ubuntu下面用XDot可以看到test.dot已经是个图形文件了。图形如下：

当然了，也可以用 dot命令绘制出PNG格式的图片来：

我们可以看到在当前目录下产生了一个文件名为graph_test.png的PNG格式的图片。打开看下和上面的图是一致的。我就不贴图了。

对dot感兴趣的筒子，可以继续阅读『dot 学习笔记』二. 记录 Shell 的变迁

参考文献：

1 Land of Lisp

2 PineWiki

3 算法：C语言实现。

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