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feat: add Prim's algorithm for Minimum Spanning Tree
fix: do not decrease priority in PriorityQueue increasePriority()
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import { PriorityQueue } from '../data_structures/heap/heap' | ||
/** | ||
* @function prim | ||
* @description Compute a minimum spanning tree(MST) of a fully connected weighted undirected graph. The input graph is in adjacency list form. It is a multidimensional array of edges. graph[i] holds the edges for the i'th node. Each edge is a 2-tuple where the 0'th item is the destination node, and the 1'th item is the edge weight. | ||
* @Complexity_Analysis | ||
* Time complexity: O(Elog(V)) | ||
* Space Complexity: O(V) | ||
* @param {[number, number][][]} graph - The graph in adjacency list form | ||
* @return {Edge[], number} - [The edges of the minimum spanning tree, the sum of the weights of the edges in the tree] | ||
* @see https://en.wikipedia.org/wiki/Prim%27s_algorithm | ||
*/ | ||
export const prim = (graph: [number, number][][]): [Edge[], number] => { | ||
if (graph.length == 0) { | ||
return [[], 0]; | ||
} | ||
let minimum_spanning_tree: Edge[] = []; | ||
let total_weight = 0; | ||
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let priorityQueue = new PriorityQueue((e: Edge) => { return e.b }, graph.length, (a: Edge, b: Edge) => { return a.weight < b.weight }); | ||
let visited = new Set<number>(); | ||
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// Start from the 0'th node. For fully connected graphs, we can start from any node and still produce the MST. | ||
visited.add(0); | ||
add_children(graph, priorityQueue, 0); | ||
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while (!priorityQueue.isEmpty()) { | ||
// We always store the previously visited edge in `edge.a`, and the newly visited node in `edge.b`. | ||
let edge = priorityQueue.extract(); | ||
if (visited.has(edge.b)) { | ||
continue; | ||
} | ||
minimum_spanning_tree.push(edge); | ||
total_weight += edge.weight; | ||
visited.add(edge.b); | ||
add_children(graph, priorityQueue, edge.b); | ||
} | ||
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return [minimum_spanning_tree, total_weight]; | ||
} | ||
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const add_children = (graph: [number, number][][], priorityQueue: PriorityQueue<Edge>, node: number) => { | ||
for (let i = 0; i < graph[node].length; ++i) { | ||
let out_edge = graph[node][i]; | ||
// By increasing the priority, we ensure we only add each vertex to the queue one time, and the queue will be at most size V. | ||
priorityQueue.increasePriority(out_edge[0], new Edge(node, out_edge[0], out_edge[1])); | ||
} | ||
} | ||
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export class Edge { | ||
a: number = 0; | ||
b: number = 0; | ||
weight: number = 0; | ||
constructor(a: number, b: number, weight: number) { | ||
this.a = a; | ||
this.b = b; | ||
this.weight = weight; | ||
} | ||
} | ||
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import { Edge, prim } from "../prim"; | ||
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let edge_equal = (x: Edge, y: Edge): boolean => { | ||
return (x.a == y.a && x.b == y.b) || (x.a == y.b && x.b == y.a) && x.weight == y.weight; | ||
} | ||
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let test_graph = (expected_tree_edges: Edge[], other_edges: Edge[], num_vertices: number, expected_cost: number) => { | ||
// First make sure the graph is undirected | ||
let graph: [number, number][][] = []; | ||
for (let _ = 0; _ < num_vertices; ++_) { | ||
graph.push([]); | ||
} | ||
for (let edge of expected_tree_edges) { | ||
graph[edge.a].push([edge.b, edge.weight]); | ||
graph[edge.b].push([edge.a, edge.weight]); | ||
} | ||
for (let edge of other_edges) { | ||
graph[edge.a].push([edge.b, edge.weight]); | ||
graph[edge.b].push([edge.a, edge.weight]); | ||
} | ||
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let [tree_edges, cost] = prim(graph); | ||
expect(cost).toStrictEqual(expected_cost); | ||
for (let expected_edge of expected_tree_edges) { | ||
expect(tree_edges.find(edge => edge_equal(edge, expected_edge))).toBeTruthy(); | ||
} | ||
for (let unexpected_edge of other_edges) { | ||
expect(tree_edges.find(edge => edge_equal(edge, unexpected_edge))).toBeFalsy(); | ||
} | ||
}; | ||
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describe("prim", () => { | ||
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it("should return empty tree for empty graph", () => { | ||
expect(prim([])).toStrictEqual([[], 0]); | ||
}); | ||
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it("should return empty tree for single element graph", () => { | ||
expect(prim([])).toStrictEqual([[], 0]); | ||
}); | ||
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it("should return correct value for two element graph", () => { | ||
expect(prim([[[1, 5]], []])).toStrictEqual([[new Edge(0, 1, 5)], 5]); | ||
}); | ||
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it("should return the correct value", () => { | ||
let expected_tree_edges = [ | ||
new Edge(0, 1, 1), | ||
new Edge(1, 3, 2), | ||
new Edge(3, 2, 3), | ||
]; | ||
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let other_edges = [ | ||
new Edge(0, 2, 4), | ||
new Edge(0, 3, 5), | ||
new Edge(1, 2, 6), | ||
]; | ||
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test_graph(expected_tree_edges, other_edges, 4, 6); | ||
}); | ||
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it("should return the correct value", () => { | ||
let expected_tree_edges = [ | ||
new Edge(0, 2, 2), | ||
new Edge(1, 3, 9), | ||
new Edge(2, 6, 74), | ||
new Edge(2, 7, 8), | ||
new Edge(3, 4, 3), | ||
new Edge(4, 9, 9), | ||
new Edge(5, 7, 5), | ||
new Edge(7, 9, 4), | ||
new Edge(8, 9, 2), | ||
] | ||
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let other_edges = [ | ||
new Edge(0, 1, 10), | ||
new Edge(2, 4, 47), | ||
new Edge(4, 5, 42), | ||
]; | ||
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test_graph(expected_tree_edges, other_edges, 10, 116); | ||
}); | ||
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}) |