feat: Add A search and heap sort implementations with tests

algorithms/graph/a-star/aStar.test.ts

@@ -0,0 +1,212 @@
+import {aStar, createManhattanHeuristic, createEuclideanHeuristic} from './aStar';
+
+describe('A* Search Algorithm', () => {
+  test('should find the shortest path in a simple graph', () => {
+    // Graph: 0 -> 1 -> 2 -> 3
+    const graph = [[{to: 1, weight: 1}], [{to: 2, weight: 1}], [{to: 3, weight: 1}], []];
+    const heuristic = (v: number): number => 3 - v; // Simple heuristic
+
+    const result = aStar(graph, 0, 3, heuristic);
+    expect(result.path).toEqual([0, 1, 2, 3]);
+    expect(result.distance).toBe(3);
+  });
+
+  test('should find optimal path when multiple paths exist', () => {
+    // Graph with two paths: 0->1->3 (cost 5) and 0->2->3 (cost 3)
+    const graph = [
+      [
+        {to: 1, weight: 2},
+        {to: 2, weight: 1},
+      ],
+      [{to: 3, weight: 3}],
+      [{to: 3, weight: 2}],
+      [],
+    ];
+    const heuristic = (): number => 0; // Zero heuristic (degrades to Dijkstra)
+
+    const result = aStar(graph, 0, 3, heuristic);
+    expect(result.path).toEqual([0, 2, 3]);
+    expect(result.distance).toBe(3);
+  });
+
+  test('should return null path when no path exists', () => {
+    const graph = [[{to: 1, weight: 1}], [], [{to: 3, weight: 1}], []];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, 0, 3, heuristic);
+    expect(result.path).toBeNull();
+    expect(result.distance).toBe(Infinity);
+  });
+
+  test('should handle source equals goal', () => {
+    const graph = [[{to: 1, weight: 1}], []];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, 0, 0, heuristic);
+    expect(result.path).toEqual([0]);
+    expect(result.distance).toBe(0);
+  });
+
+  test('should handle invalid source vertex', () => {
+    const graph = [[{to: 1, weight: 1}], []];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, -1, 1, heuristic);
+    expect(result.path).toBeNull();
+    expect(result.distance).toBe(Infinity);
+  });
+
+  test('should handle invalid goal vertex', () => {
+    const graph = [[{to: 1, weight: 1}], []];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, 0, 5, heuristic);
+    expect(result.path).toBeNull();
+    expect(result.distance).toBe(Infinity);
+  });
+
+  test('should work with a grid-based graph using Manhattan heuristic', () => {
+    // 3x3 grid:
+    // 0 - 1 - 2
+    // |   |   |
+    // 3 - 4 - 5
+    // |   |   |
+    // 6 - 7 - 8
+    const graph = [
+      [
+        {to: 1, weight: 1},
+        {to: 3, weight: 1},
+      ],
+      [
+        {to: 0, weight: 1},
+        {to: 2, weight: 1},
+        {to: 4, weight: 1},
+      ],
+      [
+        {to: 1, weight: 1},
+        {to: 5, weight: 1},
+      ],
+      [
+        {to: 0, weight: 1},
+        {to: 4, weight: 1},
+        {to: 6, weight: 1},
+      ],
+      [
+        {to: 1, weight: 1},
+        {to: 3, weight: 1},
+        {to: 5, weight: 1},
+        {to: 7, weight: 1},
+      ],
+      [
+        {to: 2, weight: 1},
+        {to: 4, weight: 1},
+        {to: 8, weight: 1},
+      ],
+      [
+        {to: 3, weight: 1},
+        {to: 7, weight: 1},
+      ],
+      [
+        {to: 4, weight: 1},
+        {to: 6, weight: 1},
+        {to: 8, weight: 1},
+      ],
+      [
+        {to: 5, weight: 1},
+        {to: 7, weight: 1},
+      ],
+    ];
+
+    const goalX = 2;
+    const goalY = 2;
+    const gridWidth = 3;
+    const heuristic = createManhattanHeuristic(goalX, goalY, gridWidth);
+
+    const result = aStar(graph, 0, 8, heuristic);
+    expect(result.distance).toBe(4);
+    expect(result.path?.length).toBe(5);
+    expect(result.path?.[0]).toBe(0);
+    expect(result.path?.[result.path.length - 1]).toBe(8);
+  });
+
+  test('should work with Euclidean heuristic', () => {
+    // Simple 2x2 grid
+    const graph = [
+      [
+        {to: 1, weight: 1},
+        {to: 2, weight: 1},
+      ],
+      [
+        {to: 0, weight: 1},
+        {to: 3, weight: 1},
+      ],
+      [
+        {to: 0, weight: 1},
+        {to: 3, weight: 1},
+      ],
+      [
+        {to: 1, weight: 1},
+        {to: 2, weight: 1},
+      ],
+    ];
+
+    const heuristic = createEuclideanHeuristic(1, 1, 2);
+    const result = aStar(graph, 0, 3, heuristic);
+
+    expect(result.distance).toBe(2);
+    expect(result.path?.length).toBe(3);
+  });
+
+  test('should handle disconnected components', () => {
+    const graph = [
+      [{to: 1, weight: 1}],
+      [{to: 0, weight: 1}],
+      [{to: 3, weight: 1}],
+      [{to: 2, weight: 1}],
+    ];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, 0, 3, heuristic);
+    expect(result.path).toBeNull();
+    expect(result.distance).toBe(Infinity);
+  });
+
+  test('should handle graph with cycles', () => {
+    // Triangle graph: 0 - 1 - 2 - 0
+    const graph = [
+      [
+        {to: 1, weight: 1},
+        {to: 2, weight: 3},
+      ],
+      [
+        {to: 0, weight: 1},
+        {to: 2, weight: 1},
+      ],
+      [
+        {to: 0, weight: 3},
+        {to: 1, weight: 1},
+      ],
+    ];
+    const heuristic = (): number => 0;
+
+    const result = aStar(graph, 0, 2, heuristic);
+    expect(result.path).toEqual([0, 1, 2]);
+    expect(result.distance).toBe(2);
+  });
+});
+
+describe('Heuristic functions', () => {
+  test('Manhattan heuristic should calculate correct distance', () => {
+    const heuristic = createManhattanHeuristic(2, 2, 3);
+    expect(heuristic(0)).toBe(4); // (0,0) to (2,2)
+    expect(heuristic(4)).toBe(2); // (1,1) to (2,2)
+    expect(heuristic(8)).toBe(0); // (2,2) to (2,2)
+  });
+
+  test('Euclidean heuristic should calculate correct distance', () => {
+    const heuristic = createEuclideanHeuristic(3, 0, 4);
+    expect(heuristic(0)).toBe(3); // (0,0) to (3,0)
+    expect(heuristic(3)).toBe(0); // (3,0) to (3,0)
+    expect(heuristic(4)).toBeCloseTo(Math.sqrt(10)); // (0,1) to (3,0)
+  });
+});

algorithms/graph/a-star/aStar.ts

@@ -0,0 +1,171 @@
+/**
+ * A* Search algorithm implementation for finding the shortest path in a weighted graph.
+ * A* uses a heuristic function to guide the search towards the goal more efficiently than Dijkstra.
+ */
+
+interface Edge {
+  to: number;
+  weight: number;
+}
+
+interface AStarResult {
+  path: number[] | null;
+  distance: number;
+}
+
+/**
+ * Implements the A* search algorithm to find the shortest path from source to goal.
+ *
+ * @param graph - Adjacency list representation of the graph where graph[i] is an array of edges from vertex i.
+ * @param source - The source vertex to start the search from.
+ * @param goal - The goal vertex to reach.
+ * @param heuristic - A function that estimates the distance from a vertex to the goal. Must be admissible (never overestimate).
+ * @returns An object containing the path array and the total distance, or null path if no path exists.
+ */
+export function aStar(
+  graph: Edge[][],
+  source: number,
+  goal: number,
+  heuristic: (vertex: number) => number,
+): AStarResult {
+  const n = graph.length;
+
+  if (source < 0 || source >= n || goal < 0 || goal >= n) {
+    return {path: null, distance: Infinity};
+  }
+
+  // g[n] = actual cost from source to n
+  const gScore: number[] = Array(n).fill(Infinity);
+  gScore[source] = 0;
+
+  // f[n] = g[n] + h[n] (estimated total cost)
+  const fScore: number[] = Array(n).fill(Infinity);
+  fScore[source] = heuristic(source);
+
+  const predecessors: number[] = Array(n).fill(-1);
+  const closedSet: boolean[] = Array(n).fill(false);
+  const openSet: boolean[] = Array(n).fill(false);
+  openSet[source] = true;
+
+  while (hasOpenNodes(openSet)) {
+    // Find node in openSet with lowest fScore
+    const current = getLowestFScore(openSet, fScore);
+
+    if (current === goal) {
+      return {
+        path: reconstructPath(source, goal, predecessors),
+        distance: gScore[goal],
+      };
+    }
+
+    openSet[current] = false;
+    closedSet[current] = true;
+
+    for (const edge of graph[current]) {
+      const neighbor = edge.to;
+
+      if (closedSet[neighbor]) {
+        continue;
+      }
+
+      const tentativeGScore = gScore[current] + edge.weight;
+
+      if (!openSet[neighbor]) {
+        openSet[neighbor] = true;
+      } else if (tentativeGScore >= gScore[neighbor]) {
+        continue;
+      }
+
+      // This path is the best so far
+      predecessors[neighbor] = current;
+      gScore[neighbor] = tentativeGScore;
+      fScore[neighbor] = gScore[neighbor] + heuristic(neighbor);
+    }
+  }
+
+  // No path found
+  return {path: null, distance: Infinity};
+}
+
+/**
+ * Checks if there are any nodes in the open set.
+ */
+function hasOpenNodes(openSet: boolean[]): boolean {
+  return openSet.some((isOpen) => isOpen);
+}
+
+/**
+ * Finds the node in the open set with the lowest f score.
+ */
+function getLowestFScore(openSet: boolean[], fScore: number[]): number {
+  let minScore = Infinity;
+  let minIndex = -1;
+
+  for (let i = 0; i < openSet.length; i++) {
+    if (openSet[i] && fScore[i] < minScore) {
+      minScore = fScore[i];
+      minIndex = i;
+    }
+  }
+
+  return minIndex;
+}
+
+/**
+ * Reconstructs the path from source to goal using the predecessors array.
+ */
+function reconstructPath(source: number, goal: number, predecessors: number[]): number[] {
+  const path: number[] = [];
+  let current = goal;
+
+  while (current !== -1) {
+    path.unshift(current);
+    if (current === source) {
+      break;
+    }
+    current = predecessors[current];
+  }
+
+  return path[0] === source ? path : [];
+}
+
+/**
+ * Creates a Manhattan distance heuristic for grid-based pathfinding.
+ * Assumes vertices are numbered row by row in a grid.
+ *
+ * @param goalX - The x coordinate of the goal.
+ * @param goalY - The y coordinate of the goal.
+ * @param gridWidth - The width of the grid.
+ * @returns A heuristic function that calculates Manhattan distance to the goal.
+ */
+export function createManhattanHeuristic(
+  goalX: number,
+  goalY: number,
+  gridWidth: number,
+): (vertex: number) => number {
+  return (vertex: number): number => {
+    const x = vertex % gridWidth;
+    const y = Math.floor(vertex / gridWidth);
+    return Math.abs(x - goalX) + Math.abs(y - goalY);
+  };
+}
+
+/**
+ * Creates a Euclidean distance heuristic for grid-based pathfinding.
+ *
+ * @param goalX - The x coordinate of the goal.
+ * @param goalY - The y coordinate of the goal.
+ * @param gridWidth - The width of the grid.
+ * @returns A heuristic function that calculates Euclidean distance to the goal.
+ */
+export function createEuclideanHeuristic(
+  goalX: number,
+  goalY: number,
+  gridWidth: number,
+): (vertex: number) => number {
+  return (vertex: number): number => {
+    const x = vertex % gridWidth;
+    const y = Math.floor(vertex / gridWidth);
+    return Math.sqrt((x - goalX) ** 2 + (y - goalY) ** 2);
+  };
+}