Graph เป็น pictorial representation ของชุด objects ที่บางคู่เชื่อมด้วย links Objects ที่เชื่อมต่อกันแทนด้วย points เรียกว่า vertices และ links ที่เชื่อม vertices เรียกว่า edges
อย่างเป็นทางการ graph เป็นคู่ของ sets (V, E) โดย V คือ set of vertices และ E คือ set of edges เชื่อมคู่ vertices มาดู graph ต่อไปนี้:
ใน graph ข้างต้น:
V = {a, b, c, d, e}
E = {ab, ac, bd, cd, de}
Terminology
Mathematical graphs แทนได้ใน data structure เราสามารถแทน graph ด้วย array of vertices และ two-dimensional array of edges ก่อนไปต่อ มาทำความรู้จัก terms สำคัญ:
- Vertex - แต่ละ node ของ graph แทนเป็น vertex labeled circle แทน vertices A ถึง G เป็น vertices สามารถแทนด้วย array ได้ index 0 = A, index 1 = B ฯลฯ
- Edge - แทน path ระหว่างสอง vertices หรือเส้นเชื่อมสอง vertices ใช้ two-dimensional array แทน AB = 1 at row 0 col 1, BC = 1 at row 1 col 2 ฯลฯ อื่นๆ เป็น 0
- Adjacency - สอง nodes/vertices adjacent ถ้าเชื่อมผ่าน edge B adjacent กับ A, C adjacent กับ B ฯลฯ
- Path - ลำดับของ edges ระหว่างสอง vertices ABCD คือ path จาก A ไป D
นี่คือการนำ Graph Class ไปใช้งานอย่างสมบูรณ์ด้วย JavaScript:
ตัวอย่างโค้ด
const Queue = require("./Queue");
const Stack = require("./Stack");
const PriorityQueue = require("./PriorityQueue");
class Graph {
constructor() {
this.edges = {};
this.nodes = [];
}
addNode(node) {
this.nodes.push(node);
this.edges[node] = [];
}
addEdge(node1, node2, weight = 1) {
this.edges[node1].push({ node: node2, weight: weight });
this.edges[node2].push({ node: node1, weight: weight });
}
addDirectedEdge(node1, node2, weight = 1) {
this.edges[node1].push({ node: node2, weight: weight });
}
display() {
let graph = "";
this.nodes.forEach(node => {
graph += node + "->" + this.edges[node].map(n => n.node).join(", ") + "\n";
});
console.log(graph);
}
BFS(node) {
let q = new Queue(this.nodes.length);
let explored = new Set();
q.enqueue(node);
explored.add(node);
while (!q.isEmpty()) {
let t = q.dequeue();
console.log(t);
this.edges[t].filter(n => !explored.has(n)).forEach(n => {
explored.add(n);
q.enqueue(n);
});
}
}
DFS(node) {
let s = new Stack(this.nodes.length);
let explored = new Set();
s.push(node);
explored.add(node);
while (!s.isEmpty()) {
let t = s.pop();
console.log(t);
this.edges[t].filter(n => !explored.has(n)).forEach(n => {
explored.add(n);
s.push(n);
});
}
}
topologicalSortHelper(node, explored, s) {
explored.add(node);
this.edges[node].forEach(n => {
if (!explored.has(n)) {
this.topologicalSortHelper(n, explored, s);
}
});
s.push(node);
}
topologicalSort() {
let s = new Stack(this.nodes.length);
let explored = new Set();
this.nodes.forEach(node => {
if (!explored.has(node)) {
this.topologicalSortHelper(node, explored, s);
}
});
while (!s.isEmpty()) {
console.log(s.pop());
}
}
BFSShortestPath(n1, n2) {
let q = new Queue(this.nodes.length);
let explored = new Set();
let distances = { n1: 0 };
q.enqueue(n1);
explored.add(n1);
while (!q.isEmpty()) {
let t = q.dequeue();
this.edges[t].filter(n => !explored.has(n)).forEach(n => {
explored.add(n);
distances[n] = distances[t] == undefined ? 1 : distances[t] + 1;
q.enqueue(n);
});
}
return distances[n2];
}
primsMST() {
const MST = new Graph();
if (this.nodes.length === 0) return MST;
let s = this.nodes[0];
let edgeQueue = new PriorityQueue(this.nodes.length * this.nodes.length);
let explored = new Set();
explored.add(s);
MST.addNode(s);
this.edges[s].forEach(edge => {
edgeQueue.enqueue([s, edge.node], edge.weight);
});
let currentMinEdge = edgeQueue.dequeue();
while (!edgeQueue.isEmpty()) {
while (!edgeQueue.isEmpty() && explored.has(currentMinEdge.data[1])) {
currentMinEdge = edgeQueue.dequeue();
}
let nextNode = currentMinEdge.data[1];
if (!explored.has(nextNode)) {
MST.addNode(nextNode);
MST.addEdge(currentMinEdge.data[0], nextNode, currentMinEdge.priority);
this.edges[nextNode].forEach(edge => {
edgeQueue.enqueue([nextNode, edge.node], edge.weight);
});
explored.add(nextNode);
s = nextNode;
}
}
return MST;
}
kruskalsMST() {
const MST = new Graph();
this.nodes.forEach(node => MST.addNode(node));
if (this.nodes.length === 0) return MST;
let edgeQueue = new PriorityQueue(this.nodes.length * this.nodes.length);
for (let node in this.edges) {
this.edges[node].forEach(edge => {
edgeQueue.enqueue([node, edge.node], edge.weight);
});
}
let uf = new UnionFind(this.nodes);
while (!edgeQueue.isEmpty()) {
let nextEdge = edgeQueue.dequeue();
let nodes = nextEdge.data;
let weight = nextEdge.priority;
if (!uf.connected(nodes[0], nodes[1])) {
MST.addEdge(nodes[0], nodes[1], weight);
uf.union(nodes[0], nodes[1]);
}
}
return MST;
}
djikstraAlgorithm(startNode) {
let distances = {};
let prev = {};
let pq = new PriorityQueue(this.nodes.length * this.nodes.length);
distances[startNode] = 0;
pq.enqueue(startNode, 0);
this.nodes.forEach(node => {
if (node !== startNode) distances[node] = Infinity;
prev[node] = null;
});
while (!pq.isEmpty()) {
let minNode = pq.dequeue();
let currNode = minNode.data;
let weight = minNode.priority;
this.edges[currNode].forEach(neighbor => {
let alt = distances[currNode] + neighbor.weight;
if (alt < distances[neighbor.node]) {
distances[neighbor.node] = alt;
prev[neighbor.node] = currNode;
pq.enqueue(neighbor.node, distances[neighbor.node]);
}
});
}
return distances;
}
floydWarshallAlgorithm() {
let dist = {};
for (let i = 0; i < this.nodes.length; i++) {
dist[this.nodes[i]] = {};
this.edges[this.nodes[i]].forEach(e => (dist[this.nodes[i]][e.node] = e.weight));
this.nodes.forEach(n => {
if (dist[this.nodes[i]][n] == undefined) dist[this.nodes[i]][n] = Infinity;
if (this.nodes[i] === n) dist[this.nodes[i]][n] = 0;
});
}
this.nodes.forEach(i => {
this.nodes.forEach(j => {
this.nodes.forEach(k => {
if (dist[i][k] + dist[k][j] < dist[i][j])
dist[i][j] = dist[i][k] + dist[k][j];
});
});
});
return dist;
}
}
class UnionFind {
constructor(elements) {
this.count = elements.length;
this.parent = {};
elements.forEach(e => (this.parent[e] = e));
}
union(a, b) {
let rootA = this.find(a);
let rootB = this.find(b);
if (rootA === rootB) return;
if (rootA < rootB) {
if (this.parent[b] != b) this.union(this.parent[b], a);
this.parent[b] = this.parent[a];
} else {
if (this.parent[a] != a) this.union(this.parent[a], b);
this.parent[a] = this.parent[b];
}
}
find(a) {
while (this.parent[a] !== a) {
a = this.parent[a];
}
return a;
}
connected(a, b) {
return this.find(a) === this.find(b);
}
}
