diff --git a/README.md b/README.md index baef23fa..405dc34e 100644 --- a/README.md +++ b/README.md @@ -72,7 +72,8 @@ * [Prim’s Algorithm](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/prim) - finding Minimum Spanning Tree (MST) for weighted undirected graph * [Kruskal’s Algorithm](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/kruskal) - finding Minimum Spanning Tree (MST) for weighted undirected graph * [Topological Sorting](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/topological-sorting) - DFS method - * Eulerian path, Eulerian circuit + * [Articulation Points](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/articulation-points) - Tarjan's algorithm + * [Eulerian Path and Eulerian Circuit](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/eulerian-path) * Strongly Connected Component algorithm * Shortest Path Faster Algorithm (SPFA) * **Uncategorized** diff --git a/src/algorithms/graph/articulation-points/README.md b/src/algorithms/graph/articulation-points/README.md new file mode 100644 index 00000000..d9a7eff0 --- /dev/null +++ b/src/algorithms/graph/articulation-points/README.md @@ -0,0 +1,22 @@ +# Articulation Points (or Cut Vertices) + +A vertex in an undirected connected graph is an articulation point +(or cut vertex) iff removing it (and edges through it) disconnects +the graph. Articulation points represent vulnerabilities in a +connected network – single points whose failure would split the +network into 2 or more disconnected components. They are useful for +designing reliable networks. + +For a disconnected undirected graph, an articulation point is a +vertex removing which increases number of connected components. + +![Articulation Points](https://www.geeksforgeeks.org/wp-content/uploads/ArticulationPoints.png) + +![Articulation Points](https://www.geeksforgeeks.org/wp-content/uploads/ArticulationPoints1.png) + +![Articulation Points](https://www.geeksforgeeks.org/wp-content/uploads/ArticulationPoints21.png) + +## References + +- [GeeksForGeeks](https://www.geeksforgeeks.org/articulation-points-or-cut-vertices-in-a-graph/) +- [YouTube](https://www.youtube.com/watch?v=2kREIkF9UAs) diff --git a/src/algorithms/graph/articulation-points/__test__/articulationPoints.test.js b/src/algorithms/graph/articulation-points/__test__/articulationPoints.test.js new file mode 100644 index 00000000..402dfc73 --- /dev/null +++ b/src/algorithms/graph/articulation-points/__test__/articulationPoints.test.js @@ -0,0 +1,143 @@ +import GraphVertex from '../../../../data-structures/graph/GraphVertex'; +import GraphEdge from '../../../../data-structures/graph/GraphEdge'; +import Graph from '../../../../data-structures/graph/Graph'; +import articulationPoints from '../articulationPoints'; + +describe('articulationPoints', () => { + it('should find articulation points in simple graph', () => { + const vertexA = new GraphVertex('A'); + const vertexB = new GraphVertex('B'); + const vertexC = new GraphVertex('C'); + const vertexD = new GraphVertex('D'); + + const edgeAB = new GraphEdge(vertexA, vertexB); + const edgeBC = new GraphEdge(vertexB, vertexC); + const edgeCD = new GraphEdge(vertexC, vertexD); + + const graph = new Graph(); + + graph + .addEdge(edgeAB) + .addEdge(edgeBC) + .addEdge(edgeCD); + + const articulationPointsSet = articulationPoints(graph); + + expect(articulationPointsSet).toEqual([ + vertexC, + vertexB, + ]); + }); + + it('should find articulation points in simple graph with back edge', () => { + const vertexA = new GraphVertex('A'); + const vertexB = new GraphVertex('B'); + const vertexC = new GraphVertex('C'); + const vertexD = new GraphVertex('D'); + + const edgeAB = new GraphEdge(vertexA, vertexB); + const edgeBC = new GraphEdge(vertexB, vertexC); + const edgeCD = new GraphEdge(vertexC, vertexD); + const edgeAC = new GraphEdge(vertexA, vertexC); + + const graph = new Graph(); + + graph + .addEdge(edgeAB) + .addEdge(edgeAC) + .addEdge(edgeBC) + .addEdge(edgeCD); + + const articulationPointsSet = articulationPoints(graph); + + expect(articulationPointsSet).toEqual([ + vertexC, + ]); + }); + + it('should find articulation points in graph', () => { + const vertexA = new GraphVertex('A'); + const vertexB = new GraphVertex('B'); + const vertexC = new GraphVertex('C'); + const vertexD = new GraphVertex('D'); + const vertexE = new GraphVertex('E'); + const vertexF = new GraphVertex('F'); + const vertexG = new GraphVertex('G'); + const vertexH = new GraphVertex('H'); + + const edgeAB = new GraphEdge(vertexA, vertexB); + const edgeBC = new GraphEdge(vertexB, vertexC); + const edgeAC = new GraphEdge(vertexA, vertexC); + const edgeCD = new GraphEdge(vertexC, vertexD); + const edgeDE = new GraphEdge(vertexD, vertexE); + const edgeEG = new GraphEdge(vertexE, vertexG); + const edgeEF = new GraphEdge(vertexE, vertexF); + const edgeGF = new GraphEdge(vertexG, vertexF); + const edgeFH = new GraphEdge(vertexF, vertexH); + + const graph = new Graph(); + + graph + .addEdge(edgeAB) + .addEdge(edgeBC) + .addEdge(edgeAC) + .addEdge(edgeCD) + .addEdge(edgeDE) + .addEdge(edgeEG) + .addEdge(edgeEF) + .addEdge(edgeGF) + .addEdge(edgeFH); + + const articulationPointsSet = articulationPoints(graph); + + expect(articulationPointsSet).toEqual([ + vertexF, + vertexE, + vertexD, + vertexC, + ]); + }); + + it('should find articulation points in graph starting with articulation root vertex', () => { + const vertexA = new GraphVertex('A'); + const vertexB = new GraphVertex('B'); + const vertexC = new GraphVertex('C'); + const vertexD = new GraphVertex('D'); + const vertexE = new GraphVertex('E'); + const vertexF = new GraphVertex('F'); + const vertexG = new GraphVertex('G'); + const vertexH = new GraphVertex('H'); + + const edgeAB = new GraphEdge(vertexA, vertexB); + const edgeBC = new GraphEdge(vertexB, vertexC); + const edgeAC = new GraphEdge(vertexA, vertexC); + const edgeCD = new GraphEdge(vertexC, vertexD); + const edgeDE = new GraphEdge(vertexD, vertexE); + const edgeEG = new GraphEdge(vertexE, vertexG); + const edgeEF = new GraphEdge(vertexE, vertexF); + const edgeGF = new GraphEdge(vertexG, vertexF); + const edgeFH = new GraphEdge(vertexF, vertexH); + + const graph = new Graph(); + + graph + .addEdge(edgeDE) + .addEdge(edgeAB) + .addEdge(edgeBC) + .addEdge(edgeAC) + .addEdge(edgeCD) + .addEdge(edgeEG) + .addEdge(edgeEF) + .addEdge(edgeGF) + .addEdge(edgeFH); + + const articulationPointsSet = articulationPoints(graph); + + expect(articulationPointsSet).toEqual([ + vertexF, + vertexE, + vertexC, + vertexD, + ]); + }); +}); diff --git a/src/algorithms/graph/articulation-points/articulationPoints.js b/src/algorithms/graph/articulation-points/articulationPoints.js new file mode 100644 index 00000000..f34d3e72 --- /dev/null +++ b/src/algorithms/graph/articulation-points/articulationPoints.js @@ -0,0 +1,116 @@ +import depthFirstSearch from '../depth-first-search/depthFirstSearch'; + +/** + * Helper class for visited vertex metadata. + */ +class VisitMetadata { + constructor({ discoveryTime, lowDiscoveryTime }) { + this.discoveryTime = discoveryTime; + this.lowDiscoveryTime = lowDiscoveryTime; + + // We need this to know to which vertex we need to compare discovery time + // when leaving the vertex. + this.childVertex = null; + + // We need this in order to check graph root node, whether it has two + // disconnected children or not. + this.childrenCount = 0; + } +} + +/** + * Tarjan's algorithm for rinding articulation points in graph. + * + * @param {Graph} graph + * @return {GraphVertex[]} + */ +export default function articulationPoints(graph) { + // Set of vertices we've already visited during DFS. + const visitedSet = {}; + + // Set of articulation points found so far. + const articulationPointsSet = []; + + // Time needed to get to the current vertex. + let discoveryTime = 0; + + // Peek the start vertex for DFS traversal. + const startVertex = graph.getAllVertices()[0]; + + const dfsCallbacks = { + /** + * @param {GraphVertex} currentVertex + * @param {GraphVertex} previousVertex + */ + enterVertex: ({ currentVertex, previousVertex }) => { + // Put current vertex to visited set. + visitedSet[currentVertex.getKey()] = new VisitMetadata({ + discoveryTime, + lowDiscoveryTime: discoveryTime, + }); + + // Tick discovery time. + discoveryTime += 1; + + if (previousVertex) { + // Update child vertex information for previous vertex. + visitedSet[previousVertex.getKey()].childVertex = currentVertex; + + // Update children counter for previous vertex. + visitedSet[previousVertex.getKey()].childrenCount += 1; + } + }, + /** + * @param {GraphVertex} currentVertex + * @param {GraphVertex} previousVertex + */ + leaveVertex: ({ currentVertex }) => { + // Detect whether current vertex is articulation point or not. + // To do so we need to check two (OR) conditions: + // 1. Is it a root vertex with at least two independent children. + // 2. If its visited time is <= low time of adjacent vertex. + if (currentVertex === startVertex) { + // Check that it has at least two independent children. + if (visitedSet[currentVertex.getKey()].childrenCount >= 2) { + articulationPointsSet.push(currentVertex); + } + } else { + // Get child vertex low discovery time. + let childVertexLowDiscoveryTime = null; + if (visitedSet[currentVertex.getKey()].childVertex) { + const childVertexKey = visitedSet[currentVertex.getKey()].childVertex.getKey(); + childVertexLowDiscoveryTime = visitedSet[childVertexKey].lowDiscoveryTime; + } + + // Compare child vertex low discovery time with current discovery time to if there + // are any short path (back edge) exists. If we can get to child vertex faster then + // to current one it means that there is a back edge exists (short path) and current + // vertex isn't articulation point. + const currentDiscoveryTime = visitedSet[currentVertex.getKey()].discoveryTime; + if (currentDiscoveryTime <= childVertexLowDiscoveryTime) { + articulationPointsSet.push(currentVertex); + } + + // Update the low time with the smallest time of adjacent vertices. + + // Get minimum low discovery time from all neighbors. + /** @param {GraphVertex} neighbor */ + visitedSet[currentVertex.getKey()].lowDiscoveryTime = currentVertex.getNeighbors().reduce( + (lowestDiscoveryTime, neighbor) => { + const neighborLowTime = visitedSet[neighbor.getKey()].lowDiscoveryTime; + return neighborLowTime < lowestDiscoveryTime ? neighborLowTime : lowestDiscoveryTime; + }, + visitedSet[currentVertex.getKey()].lowDiscoveryTime, + ); + } + }, + allowTraversal: ({ nextVertex }) => { + return !visitedSet[nextVertex.getKey()]; + }, + }; + + // Do Depth First Search traversal over submitted graph. + depthFirstSearch(graph, startVertex, dfsCallbacks); + + return articulationPointsSet; +} diff --git a/src/algorithms/graph/eulerian-path/README.md b/src/algorithms/graph/eulerian-path/README.md new file mode 100644 index 00000000..9f1f9e26 --- /dev/null +++ b/src/algorithms/graph/eulerian-path/README.md @@ -0,0 +1,34 @@ +# Eulerian Path + +In graph theory, an **Eulerian trail** (or **Eulerian path**) is a +trail in a finite graph which visits every edge exactly once. +Similarly, an **Eulerian circuit** or **Eulerian cycle** is an +Eulerian trail which starts and ends on the same vertex. + +Euler proved that a necessary condition for the existence of Eulerian +circuits is that all vertices in the graph have an even degree, and +stated that connected graphs with all vertices of even degree have +an Eulerian circuit. + +![Eulerian Circuit](https://upload.wikimedia.org/wikipedia/commons/7/72/Labelled_Eulergraph.svg) + +Every vertex of this graph has an even degree. Therefore, this is +an Eulerian graph. Following the edges in alphabetical order gives +an Eulerian circuit/cycle. + +For the existence of Eulerian trails it is necessary that zero or +two vertices have an odd degree; this means the Königsberg graph +is not Eulerian. If there are no vertices of odd degree, +all Eulerian trails are circuits. If there are exactly two vertices +of odd degree, all Eulerian trails start at one of them and end at +the other. A graph that has an Eulerian trail but not an Eulerian +circuit is called semi-Eulerian. + +![Königsberg graph](https://upload.wikimedia.org/wikipedia/commons/9/96/K%C3%B6nigsberg_graph.svg) + +The Königsberg Bridges multigraph. This multigraph is not Eulerian, +therefore, a solution does not exist. + +## References + +- [Wikipedia](https://en.wikipedia.org/wiki/Eulerian_path)