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graph.ts
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import type {
AdjacencyStructure,
AdjacencyStructureConstructor,
TypedArrayConstructors,
} from "./utility-types.ts";
import { BinaryHeap } from "./binary-heap.ts";
export const Colors = {
WHITE: 0,
GRAY: 1,
BLACK: 2,
} as const;
export type Colors = typeof Colors[keyof typeof Colors];
type Vertex = { e: number; w: number };
class VertexHeap extends BinaryHeap<Vertex> {
static compare(a: Vertex, b: Vertex): boolean {
return a.w < b.w;
}
}
/**
* Creates a Graph class extending a given adjacency structure.
*/
export function GraphMixin<
T extends TypedArrayConstructors,
U extends AdjacencyStructureConstructor<T>,
>(Base: U) {
// deno-lint-ignore no-empty-interface
interface Graph extends AdjacencyStructure {}
/**
* Extends an adjacency list/matrix structure and provides methods for traversal (BFS, DFS),
* pathfinding (Dijkstra, Bellman-Ford), spanning tree construction (BFS, Prim), etc.
*/
class Graph extends Base {
_colors?: Uint8Array;
get colors(): Uint8Array {
return (
this._colors ||
((this._colors = new Uint8Array(this.vertices)), this._colors)
);
}
hasColor(vertex: number, color: Colors): boolean {
return this.colors[vertex] === color;
}
/**
* Checks whether the graph is acyclic.
*/
isAcyclic(): boolean {
for (const vertex of this.traverse(true, 0, false, true)) {
if (!this.hasColor(vertex, Colors.WHITE)) return false;
}
return true;
}
/**
* Returns a list of vertices along the shortest path between two given vertices.
*
* @param start the starting vertex
* @param end the ending vertex
* @param isAcyclic whether the graph is acyclic
* @param isNonNegative whether all edges are non-negative
*/
path(
start: number,
end: number,
isAcyclic = false,
isNonNegative = false,
): Array<number> {
const { weighted } = this
.constructor as AdjacencyStructureConstructor<TypedArrayConstructors>;
const { vertices } = this;
const predecessors = new Array(vertices).fill(-1);
const distances = new Array(vertices).fill(Infinity);
const isFound = !weighted
? this.searchUnweighted(start, end, predecessors)
: isAcyclic
? this.searchTopological(start, end, distances, predecessors)
: isNonNegative
? this.searchDijkstra(start, end, distances, predecessors)
: this.searchBellmanFord(start, end, distances, predecessors);
if (!isFound) return [];
const path = [];
let last = end;
while (~last) {
path.unshift(last);
last = predecessors[last];
}
return path;
}
/**
* Resets all coloring of vertices done during traversals.
*/
resetColors(): void {
this.colors.fill(0);
}
/**
* For all.
*
* @param start
* @param end
* @param distances
* @param predecessors
*/
searchBellmanFord(
start: number,
end: number,
distances: Array<number>,
predecessors: Array<number>,
): boolean {
const { vertices } = this;
distances[start] = 0;
let isFound = false;
for (let i = 0; i < vertices; i++) {
for (const edge of this.outEdges(i)) {
const weight = this.getEdge(i, edge);
const distance = distances[i] + weight;
if (distances[edge] > distance) {
distances[edge] = distance;
predecessors[edge] = i;
if (edge === end) {
isFound = true;
}
}
}
}
return isFound;
}
/**
* For non-negative edges.
*
* @param start
* @param end
* @param distances
* @param predecessors
*/
searchDijkstra(
start: number,
end: number,
distances: Array<number>,
predecessors: Array<number>,
): boolean {
this.resetColors();
const heap = new VertexHeap();
distances[start] = 0;
heap.push({ e: start, w: this[start] });
let isFound = false;
while (heap.length) {
const vertex = heap.shift()!;
if (this.hasColor(vertex.e, Colors.GRAY)) continue;
this.setColor(vertex.e, Colors.GRAY);
for (const edge of this.outEdges(vertex.e)) {
const weight = this.getEdge(vertex.e, edge);
const distance = distances[vertex.e] + weight;
if (distance < distances[edge]) {
distances[edge] = distance;
predecessors[edge] = vertex.e;
heap.push({ e: edge, w: distance });
}
if (edge === end) {
isFound = true;
}
}
}
return isFound;
}
/**
* For DAGs only.
*
* @param start
* @param end
* @param distances
* @param predecessors
*/
searchTopological(
start: number,
end: number,
distances: Array<number>,
predecessors: Array<number>,
): boolean {
distances[start] = 0;
let lastPredecessor = start;
let isFound = false;
for (const vertex of this.traverse(true, start, true, true)) {
if (!this.hasColor(vertex, Colors.GRAY)) {
const weight = this.getEdge(lastPredecessor, vertex);
if (distances[vertex] > distances[lastPredecessor] + weight) {
distances[vertex] = distances[lastPredecessor] + weight;
predecessors[vertex] = lastPredecessor;
}
} else if (!this.hasColor(vertex, Colors.BLACK)) {
lastPredecessor = vertex;
}
if (vertex === end) {
isFound = true;
}
}
return isFound;
}
/**
* For unweighted graphs.
*
* @param start the starting vertex
* @param end the ending vertex
* @param predecessors
*/
searchUnweighted(
start: number,
end: number | undefined,
predecessors: Array<number>,
): boolean {
let lastPredecessor = start;
let isFound = false;
for (const vertex of this.traverse(false, start, true, true)) {
if (this.hasColor(vertex, Colors.BLACK)) continue;
if (this.hasColor(vertex, Colors.WHITE)) {
predecessors[vertex] = lastPredecessor;
} else {
lastPredecessor = vertex;
}
if (vertex === end) {
isFound = true;
break;
}
}
return isFound;
}
setColor(vertex: number, color: Colors): void {
this.colors[vertex] = color;
}
/**
* Returns a list of vertexes sorted topologically.
*/
topologicalSort(): Array<number> {
const result: Array<number> = [];
for (const vertex of this.traverse(true, 0, false, false, true)) {
result.unshift(vertex);
}
return result;
}
/**
* Does a Breadth-First or Depth-First traversal of the graph.
*
* @param isDFS whether to do DFS traversal, does BFS otherwise
* @param start the vertex to start at
* @param gray whether to return vertices upon entering
* @param white whether to return edges upon first encountering
* @param black whether to return vertices after processing
* @return the vertex at each step
*/
*traverse(
isDFS = false,
start = 0,
gray = true,
white = false,
black = false,
) {
this.resetColors();
const processing = [start];
const [push, pull]: Array<keyof Array<number>> = isDFS
? ["push", "pop"]
: ["push", "shift"];
while (processing.length) {
const vertex = processing[pull]()!;
this.setColor(vertex, Colors.GRAY);
if (gray) yield vertex;
for (const edge of this.outEdges(vertex)) {
if (this.hasColor(edge, Colors.WHITE)) {
processing[push](edge);
}
if (white) yield edge;
}
this.setColor(vertex, Colors.BLACK);
if (black) yield vertex;
}
}
/**
* Returns a minimal spanning tree of the graph.
* Uses the Prim's algorithm for weighted graphs and BFS tree for unweighted graphs.
*
* @param start
*/
tree(start = 0) {
const { weighted } = this
.constructor as AdjacencyStructureConstructor<TypedArrayConstructors>;
const { vertices } = this;
const predecessors = new Array(vertices).fill(-1);
if (!weighted) {
this.searchUnweighted(start, undefined, predecessors);
return predecessors;
}
this.resetColors();
const distances = new Array(vertices).fill(Infinity);
const heap = new VertexHeap();
distances[start] = 0;
heap.push({ e: start, w: this[0] });
while (heap.length) {
const vertex = heap.shift()!;
if (this.hasColor(vertex.e, Colors.GRAY)) continue;
this.setColor(vertex.e, Colors.GRAY);
for (const edge of this.outEdges(vertex.e)) {
const weight = this.getEdge(vertex.e, edge);
if (this.hasColor(edge, Colors.GRAY) || weight > distances[edge]) {
continue;
}
distances[edge] = weight;
predecessors[edge] = vertex.e;
heap.push({ e: edge, w: weight });
}
}
return predecessors;
}
}
return Graph;
}