1126 Eulerian Path (25point(s)) - C語言 PAT 甲級

1126 Eulerian Path (25point(s))

In graph theory, an Eulerian path is a path in a graph which visits every edge exactly once. Similarly, an Eulerian circuit is an Eulerian path which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Konigsberg problem in 1736. It has been proven that connected graphs with all vertices of even degree have an Eulerian circuit, and such graphs are called Eulerian. If there are exactly two vertices of odd degree, all Eulerian paths start at one of them and end at the other. A graph that has an Eulerian path but not an Eulerian circuit is called semi-Eulerian. (Cited from https://en.wikipedia.org/wiki/Eulerian_path)

Given an undirected graph, you are supposed to tell if it is Eulerian, semi-Eulerian, or non-Eulerian.

Input Specification:

Each input file contains one test case. Each case starts with a line containing 2 numbers N (≤ 500), and M, which are the total number of vertices, and the number of edges, respectively. Then M lines follow, each describes an edge by giving the two ends of the edge (the vertices are numbered from 1 to N).

Output Specification:

For each test case, first print in a line the degrees of the vertices in ascending order of their indices. Then in the next line print your conclusion about the graph – either Eulerian, Semi-Eulerian, or Non-Eulerian. Note that all the numbers in the first line must be separated by exactly 1 space, and there must be no extra space at the beginning or the end of the line.

Sample Input:

7 12

5 7

1 2

1 3

2 3

2 4

3 4

5 2

7 6

6 3

4 5

6 4

5 6

Sample Output:

2 4 4 4 4 4 2

Eulerian

Sample Input:

6 10

1 2

1 3

2 3

2 4

3 4

5 2

6 3

4 5

6 4

5 6

Sample Output:

2 4 4 4 3 3

Semi-Eulerian

Sample Input:

5 8

1 2

2 5

5 4

4 1

1 3

3 2

3 4

5 3

Sample Output:

3 3 4 3 3

Non-Eulerian

題目大意：

輸入右 N 個節點，M 條邊的無向圖

輸出每個節點的度，并判斷是否為歐拉圖

設計思路：

關鍵詞：歐拉圖，并查集

先用并查集判斷是否為連通圖，若是連通圖再繼續判斷歐拉圖

1. Eulerian（歐拉圖）：連通圖，每個節點度為偶數
2. Semi-Eulerian（半歐拉圖）：連通圖，隻有兩個節點度為奇數，其他均為偶數
3. Non-Eulerian：既不是歐拉圖也不是半歐拉圖

編譯器：C (gcc)

#include <stdio.h>

int father[510], rank[510];

int makeset()
{
        int i;
        for (i = 0; i < 510; i++) {
                father[i] = i;
                rank[i] = 0;
        }
}

int getfather(int v)
{
        if (father[v] == v)
                return v;
        else {
                father[v] = getfather(father[v]);
                return father[v];
        }
}

void judge(int x, int y)
{
        int fx = getfather(x);
        int fy = getfather(y);

        if (rank[fx] > rank[fy]) {
                father[fy] = fx;
        } else {
                father[fx] = fy;
                if (rank[fx] == rank[fy])
                        rank[fy]++;
        }
}

#include <stdio.h>

int father[510], rank[510];

int makeset()
{
        int i;
        for (i = 0; i < 510; i++) {
                father[i] = i;
                rank[i] = 0;
        }
}

int getfather(int v)
{
        if (father[v] == v)
                return v;
        else {
                father[v] = getfather(father[v]);
                return father[v];
        }
}

void judge(int x, int y)
{
        int fx = getfather(x);
        int fy = getfather(y);

        if (rank[fx] > rank[fy]) {
                father[fy] = fx;
        } else {
                father[fx] = fy;
                if (rank[fx] == rank[fy])
                        rank[fy]++;
        }
}