前言

記錄實(shí)驗(yàn)，同時(shí)也是記錄常見數(shù)據(jù)結(jié)構(gòu)算法的實(shí)現(xiàn)。

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廣州大學(xué)學(xué)生實(shí)驗(yàn)報(bào)告

開課實(shí)驗(yàn)室：計(jì)算機(jī)科學(xué)與工程實(shí)驗(yàn)（電子樓416A）
學(xué)院 計(jì)算機(jī)科學(xué)與網(wǎng)絡(luò)工程學(xué)院
實(shí)驗(yàn)課程 數(shù)據(jù)結(jié)構(gòu)實(shí)驗(yàn)
實(shí)驗(yàn)項(xiàng)目 實(shí)驗(yàn)三 圖的操作與實(shí)現(xiàn)

• 一、實(shí)驗(yàn)?zāi)康?#xff1a;
  1、圖的鄰接表和鄰接矩陣存儲
  2、圖的各種遍歷算法實(shí)現(xiàn)
  3、最小生成樹的算法實(shí)現(xiàn)
  4、最短路徑的算法實(shí)現(xiàn)
• 二、使用儀器、器材
  微機(jī)一臺
  操作系統(tǒng)：Win10
  編程軟件：C++
• 三、實(shí)驗(yàn)內(nèi)容及原理
  1、圖的鄰接表和鄰接矩陣存儲 （返回目錄🖱）
  建立下圖的鄰接表或鄰接矩陣，并輸出之；

#include <iostream> using namespace std;#define MaxInt 32767 //定義極大值，即∞ #define MVNum 100 //最大定點(diǎn)數(shù)typedef struct AMGraph //定義鄰接矩陣結(jié)構(gòu) {char vexs[MVNum];int arcs[MVNum][MVNum];int vexnum, arcnum; //分別為結(jié)點(diǎn)數(shù)和邊數(shù) }AMGraph;//獲取某定點(diǎn)在無向圖中的位置 int LocateVex(AMGraph G, char v) {for (int i = 0; i < G.vexnum; i++){if (G.vexs[i] == v){return i;break;}} }void CreatUDN(AMGraph& G) {cout << "請輸入無向圖的總頂點(diǎn)數(shù)和邊數(shù)" << endl;cout << "頂點(diǎn)數(shù):"; cin >> G.vexnum;//設(shè)置一個(gè)循環(huán)用于檢測輸入邊數(shù)是否超過確定頂點(diǎn)數(shù)的無向圖的最大邊數(shù)while (1){cout << "邊數(shù):"; cin >> G.arcnum;if (G.arcnum > ((G.vexnum) * (G.vexnum - 1) / 2))cout << "邊數(shù)超出限制,請重新輸入邊數(shù)" << endl;else break;}//輸入各頂點(diǎn)的信息for (int i = 0; i < G.vexnum; i++){cout << "第" << i + 1 << "個(gè)頂點(diǎn)的名稱: ";cin >> G.vexs[i];}//初始化各邊的權(quán)重（設(shè)置為無窮大）for(int i=0;i<G.vexnum;i++)for (int j = 0; j < G.vexnum; j++){G.arcs[i][j] = MaxInt;}//構(gòu)建鄰接矩陣cout << "請輸入每兩個(gè)頂點(diǎn)及其邊的權(quán)重" << endl;for (int i = 0; i < G.arcnum; i++){char v1, v2;//任意兩個(gè)頂點(diǎn)int weight; //邊權(quán)重cout << "請輸入第" << i + 1 << "條邊及其兩個(gè)頂點(diǎn)的名稱( 形如（1 0 28））: ";cin >> v1 >> v2 >> weight;int index1, index2;//v1,v2的在無向圖中的序號index1 = LocateVex(G, v1); index2 = LocateVex(G, v2);G.arcs[index1][index2] = weight;G.arcs[index2][index1] = G.arcs[index1][index2];}cout << "無向網(wǎng)構(gòu)建完成......"<<endl;}//輸出鄰接矩陣無向網(wǎng) void AMGraph_show(AMGraph G) {for (int i = 0; i < G.vexnum; i++){for (int j = 0; j < G.vexnum; j++){if (G.arcs[i][j] != MaxInt) cout << G.arcs[i][j] << " ";else cout << "∞ ";}cout << endl;} }int main() {AMGraph G;CreatUDN(G);AMGraph_show(G);}

2、圖的各種遍歷算法實(shí)現(xiàn) （返回目錄🖱）
以0結(jié)點(diǎn)為起點(diǎn)實(shí)現(xiàn)上述圖的深度優(yōu)先和廣度優(yōu)先遍歷算法；

#include <iostream> using namespace std;#define MaxInt 32767 //定義極大值，即∞ #define MVNum 100 //最大頂點(diǎn)數(shù) bool visited[MVNum]; typedef struct AMGraph //定義鄰接矩陣結(jié)構(gòu) {char vexs[MVNum];int arcs[MVNum][MVNum];int vexnum, arcnum; //分別為結(jié)點(diǎn)數(shù)和邊數(shù) }AMGraph;//獲取某定點(diǎn)在無向圖中的位置 int LocateVex(AMGraph G, char v) {for (int i = 0; i < G.vexnum; i++){if (G.vexs[i] == v){return i;break;}} }void CreatUDN(AMGraph& G) {cout << "請輸入無向圖的總頂點(diǎn)數(shù)和邊數(shù)" << endl;cout << "頂點(diǎn)數(shù):"; cin >> G.vexnum;//設(shè)置一個(gè)循環(huán)用于檢測輸入邊數(shù)是否超過確定頂點(diǎn)數(shù)的無向圖的最大邊數(shù)while (1){cout << "邊數(shù):"; cin >> G.arcnum;if (G.arcnum > ((G.vexnum) * (G.vexnum - 1) / 2))cout << "邊數(shù)超出限制,請重新輸入邊數(shù)" << endl;else break;}//輸入各頂點(diǎn)的信息for (int i = 0; i < G.vexnum; i++){cout << "第" << i + 1 << "個(gè)頂點(diǎn)的名稱: ";cin >> G.vexs[i];}//初始化各邊的權(quán)重（設(shè)置為無窮大）for (int i = 0; i < G.vexnum; i++){visited[i] = 0;for (int j = 0; j < G.vexnum; j++)G.arcs[i][j] = MaxInt;}//構(gòu)建鄰接矩陣cout << "請輸入每兩個(gè)頂點(diǎn)及其邊的權(quán)重" << endl;for (int i = 0; i < G.arcnum; i++){char v1, v2;//任意兩個(gè)頂點(diǎn)int weight; //邊權(quán)重cout << "請輸入第" << i + 1 <<"條邊及其兩個(gè)頂點(diǎn)的名稱( 形如（1 0 28））: ";cin >> v1 >> v2 >> weight;int index1, index2;//v1,v2的在無向圖中的序號index1 = LocateVex(G, v1); index2 = LocateVex(G, v2);G.arcs[index1][index2] = weight;G.arcs[index2][index1] = G.arcs[index1][index2];}cout << "無向網(wǎng)構(gòu)建完成......" << endl;}//輸出鄰接矩陣無向網(wǎng) void AMGraph_show(AMGraph G) {for (int i = 0; i < G.vexnum; i++){for (int j = 0; j < G.vexnum; j++){if (G.arcs[i][j] != MaxInt) cout << G.arcs[i][j] << " ";else cout << "∞ ";}cout << endl;} }//---------下面實(shí)現(xiàn)深度優(yōu)先遍歷算法--------- void DFS_AM(AMGraph &G, int v) {//訪問名稱為v的結(jié)點(diǎn),并標(biāo)記為已訪問int i;cout <<G.vexs[v]; visited[v] = true; //標(biāo)記為已訪問//遞歸調(diào)用深度優(yōu)先遍歷算法for ( i = 0; i < G.vexnum; i++){if ((G.arcs[v][i] != MaxInt) && !visited[i])DFS_AM(G,i);} } void DFSTraverse(AMGraph &G) {int i;for (i = 0; i < G.vexnum; ++i)visited[i] = 0;for (i = 0; i < G.vexnum; ++i){if (!visited[i])DFS_AM(G, i);}}int main() {AMGraph G;CreatUDN(G);AMGraph_show(G);char x;cout << "請輸入遍歷開始的結(jié)點(diǎn) ";cin >> x;DFSTraverse(G); }

3、最小生成樹的算法實(shí)現(xiàn)（返回目錄🖱）
利用普里姆（Prim）算法或克魯斯卡爾(Kruskal)算法求上圖的最小生成樹，算法實(shí)現(xiàn)代碼必須有注釋。

#include <iostream> using namespace std;#define MaxInt 32767 //定義極大值，即∞ #define MVNum 100 //最大頂點(diǎn)數(shù) bool visited[MVNum]; typedef struct AMGraph //定義鄰接矩陣結(jié)構(gòu) {char vexs[MVNum];int arcs[MVNum][MVNum];int vexnum, arcnum; //分別為結(jié)點(diǎn)數(shù)和邊數(shù) }AMGraph;//獲取某定點(diǎn)在無向圖中的位置 int LocateVex(AMGraph G, char v) {for (int i = 0; i < G.vexnum; i++){if (G.vexs[i] == v){return i;break;}} }//--------------構(gòu)建無向網(wǎng)------------------- void CreatUDN(AMGraph& G) {cout << "請輸入無向圖的總頂點(diǎn)數(shù)和邊數(shù)" << endl;cout << "頂點(diǎn)數(shù):"; cin >> G.vexnum;//設(shè)置一個(gè)循環(huán)用于檢測輸入邊數(shù)是否超過確定頂點(diǎn)數(shù)的無向圖的最大邊數(shù)while (1){cout << "邊數(shù):"; cin >> G.arcnum;if (G.arcnum > ((G.vexnum) * (G.vexnum - 1) / 2))cout << "邊數(shù)超出限制,請重新輸入邊數(shù)" << endl;else break;}//輸入各頂點(diǎn)的信息for (int i = 0; i < G.vexnum; i++){cout << "第" << i + 1 << "個(gè)頂點(diǎn)的名稱: ";cin >> G.vexs[i];}//初始化各邊的權(quán)重（設(shè)置為無窮大）for (int i = 0; i < G.vexnum; i++){visited[i] = 0;for (int j = 0; j < G.vexnum; j++)G.arcs[i][j] = MaxInt;}//構(gòu)建鄰接矩陣cout << "請輸入每兩個(gè)頂點(diǎn)及其邊的權(quán)重" << endl;for (int i = 0; i < G.arcnum; i++){char v1, v2;//任意兩個(gè)頂點(diǎn)int weight; //邊權(quán)重cout << "請輸入第" << i + 1 << "條邊及其兩個(gè)頂點(diǎn)的名稱( 形如（1 0 28））: ";cin >> v1 >> v2 >> weight;int index1, index2;//v1,v2的在無向圖中的序號index1 = LocateVex(G, v1); index2 = LocateVex(G, v2);G.arcs[index1][index2] = weight;G.arcs[index2][index1] = G.arcs[index1][index2];}cout << "無向網(wǎng)構(gòu)建完成......" << endl; }//輸出鄰接矩陣無向網(wǎng) void AMGraph_show(AMGraph G) {for (int i = 0; i < G.vexnum; i++){for (int j = 0; j < G.vexnum; j++){if (G.arcs[i][j] != MaxInt) cout << G.arcs[i][j] << " ";else cout << "∞ ";}cout << endl;} }//輔助數(shù)組，用于篩選權(quán)值最小的鄰接結(jié)點(diǎn) typedef struct {int adjvex;int lowcost; }closedge[MVNum];closedge theclose; //全局變量，輔助數(shù)組//從輔助數(shù)組中找出權(quán)值最小的邊的數(shù)組下標(biāo) int minimun(AMGraph G, closedge c) {int min = MaxInt; //min初始化為MaxIntint min_index = -1; //max_index表示輔助數(shù)組中最小權(quán)值邊的數(shù)組下標(biāo)for (int i = 0; i < G.vexnum; i++){if (c[i].lowcost > 0 && c[i].lowcost < min){min = c[i].lowcost;min_index = i;}}return min_index; }//-------Prim算法-------- void MSTreePrim(AMGraph G,char v) {int k = LocateVex(G,v); //結(jié)點(diǎn)v在頂點(diǎn)數(shù)組中的下標(biāo)//初始化，將結(jié)點(diǎn)v的鄰接邊的權(quán)重和鄰接頂點(diǎn)存入輔助數(shù)組theclosefor (int i = 0; i < G.vexnum; i++){if (i != k){theclose[i].adjvex = k;theclose[i].lowcost = G.arcs[k][i];}}theclose[k].lowcost = 0; //將結(jié)點(diǎn)v放入輔助數(shù)組//繼續(xù)往輔助數(shù)組里添加結(jié)點(diǎn)for (int i = 1; i < G.vexnum; i++){//找出權(quán)值最小的邊的下標(biāo)k = minimun(G, theclose);//輸出權(quán)值最小邊的頂點(diǎn)cout << G.vexs[theclose[k].adjvex] << G.vexs[k] << endl;//將結(jié)點(diǎn)k歸入theclosetheclose[k].lowcost = 0;/*往theclose中加入新結(jié)點(diǎn)之后，需要重新計(jì)算最小邊是哪條,由此頂點(diǎn)出發(fā)，到達(dá)其它各頂點(diǎn)的權(quán)值是否比之前記錄的權(quán)值還要小，如果還小，則更新*/for (int i = 0; i < G.vexnum; i++){if (G.arcs[k][i] < theclose[i].lowcost){theclose[i].lowcost = G.arcs[k][i];theclose[i].adjvex = k;}}}cout << "Prinm算法最小生成樹構(gòu)建完成......"; }int main() {AMGraph G;CreatUDN(G);AMGraph_show(G); //輸出鄰接矩陣cout << "請輸入開始遍歷的頂點(diǎn): ";char x;cin >> x;MSTreePrim(G, x);//輸出最小生成樹 }

4、最短路徑的算法實(shí)現(xiàn)（返回目錄🖱）
利用狄克斯特拉（Dijkstra）算法求上圖中0結(jié)點(diǎn)到其它結(jié)點(diǎn)的最短路徑，算法實(shí)現(xiàn)代碼必須有注釋

#include <iostream> using namespace std;#define MaxInt 32767 //定義極大值，即∞ #define MVNum 100 //最大頂點(diǎn)數(shù) bool visited[MVNum]; typedef struct AMGraph //定義鄰接矩陣結(jié)構(gòu) {char vexs[MVNum];int arcs[MVNum][MVNum];int vexnum, arcnum; //分別為結(jié)點(diǎn)數(shù)和邊數(shù) }AMGraph;//獲取某定點(diǎn)在無向圖中的位置 int LocateVex(AMGraph G, char v) {for (int i = 0; i < G.vexnum; i++){if (G.vexs[i] == v){return i;break;}} }//--------------構(gòu)建無向網(wǎng)------------------- void CreatUDN(AMGraph& G) {cout << "請輸入無向圖的總頂點(diǎn)數(shù)和邊數(shù)" << endl;cout << "頂點(diǎn)數(shù):"; cin >> G.vexnum;//設(shè)置一個(gè)循環(huán)用于檢測輸入邊數(shù)是否超過確定頂點(diǎn)數(shù)的無向圖的最大邊數(shù)while (1){cout << "邊數(shù):"; cin >> G.arcnum;if (G.arcnum > ((G.vexnum) * (G.vexnum - 1) / 2))cout << "邊數(shù)超出限制,請重新輸入邊數(shù)" << endl;else break;}//輸入各頂點(diǎn)的信息for (int i = 0; i < G.vexnum; i++){cout << "第" << i + 1 << "個(gè)頂點(diǎn)的名稱: ";cin >> G.vexs[i];}//初始化各邊的權(quán)重（設(shè)置為無窮大）for (int i = 0; i < G.vexnum; i++){visited[i] = 0;for (int j = 0; j < G.vexnum; j++)G.arcs[i][j] = MaxInt;}//構(gòu)建鄰接矩陣cout << "請輸入每兩個(gè)頂點(diǎn)及其邊的權(quán)重" << endl;for (int i = 0; i < G.arcnum; i++){char v1, v2; //任意兩個(gè)頂點(diǎn)int weight; //邊權(quán)重cout << "請輸入第" << i + 1 << "條邊及其兩個(gè)頂點(diǎn)的名稱( 形如（1 0 28））: ";cin >> v1 >> v2 >> weight;int index1, index2; //v1,v2的在無向圖中的序號index1 = LocateVex(G, v1); index2 = LocateVex(G, v2);G.arcs[index1][index2] = weight;G.arcs[index2][index1] = G.arcs[index1][index2];}cout << "無向網(wǎng)構(gòu)建完成......" << endl; }//輸出鄰接矩陣無向網(wǎng) void AMGraph_show(AMGraph G) {for (int i = 0; i < G.vexnum; i++){for (int j = 0; j < G.vexnum; j++){if (G.arcs[i][j] != MaxInt) cout << G.arcs[i][j] << " ";else cout << "∞ ";}cout << endl;} }//-----------Dijkstra算法的實(shí)現(xiàn)----------- int dist[MVNum]; //存放v0到vi的最短路徑長度 int sign[MVNum]; //若vi已卻定最短路徑長度，則為1，否則為0 int pre[MVNum]; //記錄vi的前驅(qū)結(jié)點(diǎn)，若無前驅(qū)結(jié)點(diǎn)則為-1void Dijkstra(AMGraph G, int v) {//初始化dist、sign、prefor (int i = 0; i < G.vexnum; i++){dist[i] = G.arcs[v][i];sign[i] = 0; //初始化為0if (dist[i] < MaxInt) pre[i] = v; //若存在邊，則將前驅(qū)初始化為原點(diǎn)else pre[i] = -1; //若不存在邊，則將前驅(qū)初始化為-1}sign[v] = 1;for (int i = 0; i < G.vexnum-1; i++){int min_v; //待選點(diǎn)int min = MaxInt;//尋找dist中的最短路徑for (int i = 0; i < G.vexnum; i++){if (sign[i] == 0 && dist[i] < min){min = dist[i];min_v = i;}}sign[min_v] = 1; //將找到最短路徑的頂點(diǎn)的標(biāo)志置為1//重置每個(gè)頂點(diǎn)的最短路徑值for (int i = 0; i < G.vexnum; i++){if (sign[i] == 0 && dist[i] > dist[min_v] + G.arcs[min_v][i]){dist[i] = dist[min_v] + G.arcs[min_v][i];pre[i] = min_v;}}}//打印結(jié)果cout << "輸出頂點(diǎn)" << v << "到各頂點(diǎn)的最短路徑" << endl << "result:" << endl;for (int i = 0; i < G.vexnum; i++){cout << "<" << v << "," << i << "> : ";int p = pre[i];if (p != -1){cout << " " << dist[i] << "\t";cout << i;while (p != v){cout << "<--" << p;p = pre[p]; //逆向輸出路徑}cout << "<--" << v;}else cout << "∞"; //若不可達(dá)則輸出無窮cout << endl;} }int main() {AMGraph G;CreatUDN(G);AMGraph_show(G); //輸出鄰接矩陣Dijkstra(G, 0); //輸出頂點(diǎn)0到各頂點(diǎn)的最短路徑 }

四、實(shí)驗(yàn)過程原始數(shù)據(jù)記錄
1.、圖的鄰接表和鄰接矩陣存儲
建立下圖的鄰接表或鄰接矩陣，并輸出之；

思路：通過構(gòu)建鄰接矩陣來實(shí)現(xiàn)無向圖。

2、圖的各種遍歷算法實(shí)現(xiàn)
以0結(jié)點(diǎn)為起點(diǎn)實(shí)現(xiàn)上述圖的深度優(yōu)先和廣度優(yōu)先遍歷算法；

3、最小生成樹的算法實(shí)現(xiàn)
利用普里姆（Prim）算法或克魯斯卡爾(Kruskal)算法求上圖的最小生成樹，算法實(shí)現(xiàn)代碼必須有注釋。

4、最短路徑的算法實(shí)現(xiàn)
利用狄克斯特拉（Dijkstra）算法求上圖中0結(jié)點(diǎn)到其它結(jié)點(diǎn)的最短路徑，算法實(shí)現(xiàn)代碼必須有注釋。

五、實(shí)驗(yàn)結(jié)果及分析
通過構(gòu)造鄰接矩陣實(shí)現(xiàn)了圖的存儲、圖的深度優(yōu)先遍歷算法、圖的最小生成樹Prim算法、最短路徑的Dijkstra算法，通過對這些算法的實(shí)現(xiàn)，讓我對圖有了更深的理解。

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