Add N-Queens.

Author: trekhleb

README.md

@@ -78,6 +78,7 @@
   * [Strongly Connected Components](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/strongly-connected-components) - Kosaraju's algorithm
 * **Uncategorized**  
   * [Tower of Hanoi](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/uncategorized/hanoi-tower)
+  * [N-Queens Problem](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/uncategorized/n-queens)
   * Union-Find
   * Maze
   * Sudoku
@@ -110,9 +111,15 @@
   * [Maximum Subarray](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/sets/maximum-subarray)
   * [Bellman-Ford Algorithm](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/graph/bellman-ford) - finding shortest path to all graph vertices
 * **Backtracking**
+  * [N-Queens Problem](https://github.com/trekhleb/javascript-algorithms/tree/master/src/algorithms/uncategorized/n-queens)
 * **Branch & Bound**
 
-## Running Tests
+## How to use this repository
+
+**Install all dependencies**
+```
+npm i
+```
 
 **Run all tests**
 ```
@@ -135,11 +142,11 @@ Then just simply run the following command to test if your playground code works
 npm test -- -t 'playground'
 ```
 
-## Useful links
+## Useful Information
 
-[▶ Data Structures and Algorithms on YouTube](https://www.youtube.com/playlist?list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
+### References
 
-## Useful Information
+[▶ Data Structures and Algorithms on YouTube](https://www.youtube.com/playlist?list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
 
 ### Big O Notation
 
@@ -185,4 +192,4 @@ Below is the list of some of the most used Big O notations and their performance
 | **Heap sort**         | n log(n)  | n log(n)  | n log(n)      | 1         | No        |
 | **Merge sort**        | n log(n)  | n log(n)  | n log(n)      | n         | Yes       |
 | **Quick sort**        | n log(n)  | n log(n)  | n^2           | log(n)    | No        |
-| **Shell sort**        | n log(n)  | depends on gap sequence  | n (log(n))^2  | 1         | No        |
+| **Shell sort**        | n log(n)  | depends on gap sequence   | n (log(n))^2  | 1         | No        |

src/algorithms/uncategorized/n-queens/QueenPosition.js

@@ -0,0 +1,39 @@
+/**
+ * Class that represents queen position on the chessboard.
+ */
+export default class QueenPosition {
+  /**
+   * @param {number} rowIndex
+   * @param {number} columnIndex
+   */
+  constructor(rowIndex, columnIndex) {
+    this.rowIndex = rowIndex;
+    this.columnIndex = columnIndex;
+  }
+
+  /**
+   * @return {number}
+   */
+  get leftDiagonal() {
+    // Each position on the same left (\) diagonal has the same difference of
+    // rowIndex and columnIndex. This fact may be used to quickly check if two
+    // positions (queens) are on the same left diagonal.
+    // @see https://youtu.be/xouin83ebxE?t=1m59s
+    return this.rowIndex - this.columnIndex;
+  }
+
+  /**
+   * @return {number}
+   */
+  get rightDiagonal() {
+    // Each position on the same right diagonal (/) has the same
+    // sum of rowIndex and columnIndex. This fact may be used to quickly
+    // check if two positions (queens) are on the same right diagonal.
+    // @see https://youtu.be/xouin83ebxE?t=1m59s
+    return this.rowIndex + this.columnIndex;
+  }
+
+  toString() {
+    return `${this.rowIndex},${this.columnIndex}`;
+  }
+}

src/algorithms/uncategorized/n-queens/README.md

@@ -0,0 +1,73 @@
+# N-Queens Problem
+
+The **eight queens puzzle** is the problem of placing eight chess queens 
+on an `8×8` chessboard so that no two queens threaten each other. 
+Thus, a solution requires that no two queens share the same row, 
+column, or diagonal. The eight queens puzzle is an example of the 
+more general *n queens problem* of placing n non-attacking queens 
+on an `n×n` chessboard, for which solutions exist for all natural 
+numbers `n` with the exception of `n=2` and `n=3`.
+
+For example, following is a solution for 4 Queen problem.
+
+![N Queens](https://cdncontribute.geeksforgeeks.org/wp-content/uploads/N_Queen_Problem.jpg)
+
+The expected output is a binary matrix which has 1s for the blocks 
+where queens are placed. For example following is the output matrix 
+for above 4 queen solution.
+
+```
+{ 0,  1,  0,  0}
+{ 0,  0,  0,  1}
+{ 1,  0,  0,  0}
+{ 0,  0,  1,  0}
+```
+
+## Naive Algorithm
+
+Generate all possible configurations of queens on board and print a 
+configuration that satisfies the given constraints.
+
+```
+while there are untried conflagrations
+{
+   generate the next configuration
+   if queens don't attack in this configuration then
+   {
+      print this configuration;
+   }
+}
+```
+
+## Backtracking Algorithm
+
+The idea is to place queens one by one in different columns, 
+starting from the leftmost column. When we place a queen in a 
+column, we check for clashes with already placed queens. In 
+the current column, if we find a row for which there is no 
+clash, we mark this row and column as part of the solution. 
+If we do not find such a row due to clashes then we backtrack 
+and return false.
+
+```
+1) Start in the leftmost column
+2) If all queens are placed
+    return true
+3) Try all rows in the current column.  Do following for every tried row.
+    a) If the queen can be placed safely in this row then mark this [row, 
+        column] as part of the solution and recursively check if placing  
+        queen here leads to a solution.
+    b) If placing queen in [row, column] leads to a solution then return 
+        true.
+    c) If placing queen doesn't lead to a solution then umark this [row, 
+        column] (Backtrack) and go to step (a) to try other rows.
+3) If all rows have been tried and nothing worked, return false to trigger 
+    backtracking.
+```
+
+## References
+
+- [Wikipedia](https://en.wikipedia.org/wiki/Eight_queens_puzzle)
+- [GeeksForGeeks](https://www.geeksforgeeks.org/backtracking-set-3-n-queen-problem/)
+- [On YouTube by Abdul Bari](https://www.youtube.com/watch?v=xFv_Hl4B83A)
+- [On YouTube by Tushar Roy](https://www.youtube.com/watch?v=xouin83ebxE)

src/algorithms/uncategorized/n-queens/__test__/QueensPosition.test.js

@@ -0,0 +1,16 @@
+import QueenPosition from '../QueenPosition';
+
+describe('QueenPosition', () => {
+  it('should store queen position on chessboard', () => {
+    const position1 = new QueenPosition(0, 0);
+    const position2 = new QueenPosition(2, 1);
+
+    expect(position2.columnIndex).toBe(1);
+    expect(position2.rowIndex).toBe(2);
+    expect(position1.leftDiagonal).toBe(0);
+    expect(position1.rightDiagonal).toBe(0);
+    expect(position2.leftDiagonal).toBe(1);
+    expect(position2.rightDiagonal).toBe(3);
+    expect(position2.toString()).toBe('2,1');
+  });
+});

src/algorithms/uncategorized/n-queens/__test__/nQueens.test.js

@@ -0,0 +1,38 @@
+import nQueens from '../nQueens';
+
+describe('nQueens', () => {
+  it('should not hae solution for 3 queens', () => {
+    const solutions = nQueens(3);
+    expect(solutions.length).toBe(0);
+  });
+
+  it('should solve n-queens problem for 4 queens', () => {
+    const solutions = nQueens(4);
+    expect(solutions.length).toBe(2);
+
+    // First solution.
+    expect(solutions[0][0].toString()).toBe('0,1');
+    expect(solutions[0][1].toString()).toBe('1,3');
+    expect(solutions[0][2].toString()).toBe('2,0');
+    expect(solutions[0][3].toString()).toBe('3,2');
+
+    // Second solution (mirrored).
+    expect(solutions[1][0].toString()).toBe('0,2');
+    expect(solutions[1][1].toString()).toBe('1,0');
+    expect(solutions[1][2].toString()).toBe('2,3');
+    expect(solutions[1][3].toString()).toBe('3,1');
+  });
+
+  it('should solve n-queens problem for 6 queens', () => {
+    const solutions = nQueens(6);
+    expect(solutions.length).toBe(4);
+
+    // First solution.
+    expect(solutions[0][0].toString()).toBe('0,1');
+    expect(solutions[0][1].toString()).toBe('1,3');
+    expect(solutions[0][2].toString()).toBe('2,5');
+    expect(solutions[0][3].toString()).toBe('3,0');
+    expect(solutions[0][4].toString()).toBe('4,2');
+    expect(solutions[0][5].toString()).toBe('5,4');
+  });
+});

src/algorithms/uncategorized/n-queens/nQueens.js

@@ -0,0 +1,103 @@
+import QueenPosition from './QueenPosition';
+
+/**
+ * @param {QueenPosition[]} queensPositions
+ * @param {number} rowIndex
+ * @param {number} columnIndex
+ * @return {boolean}
+ */
+function isSafe(queensPositions, rowIndex, columnIndex) {
+  // New position to which the Queen is going to be placed.
+  const newQueenPosition = new QueenPosition(rowIndex, columnIndex);
+
+  // Check if new queen position conflicts with any other queens.
+  for (let queenIndex = 0; queenIndex < queensPositions.length; queenIndex += 1) {
+    const currentQueenPosition = queensPositions[queenIndex];
+
+    if (
+      // Check if queen has been already placed.
+      currentQueenPosition &&
+      (
+        // Check if there are any queen on the same column.
+        newQueenPosition.columnIndex === currentQueenPosition.columnIndex ||
+        // Check if there are any queen on the same row.
+        newQueenPosition.rowIndex === currentQueenPosition.rowIndex ||
+        // Check if there are any queen on the same left diagonal.
+        newQueenPosition.leftDiagonal === currentQueenPosition.leftDiagonal ||
+        // Check if there are any queen on the same right diagonal.
+        newQueenPosition.rightDiagonal === currentQueenPosition.rightDiagonal
+      )
+    ) {
+      // Can't place queen into current position since there are other queens that
+      // are threatening it.
+      return false;
+    }
+  }
+
+  // Looks like we're safe.
+  return true;
+}
+
+/**
+ * @param {QueenPosition[][]} solutions
+ * @param {QueenPosition[]} previousQueensPositions
+ * @param {number} queensCount
+ * @param {number} rowIndex
+ * @return {boolean}
+ */
+function nQueensRecursive(solutions, previousQueensPositions, queensCount, rowIndex) {
+  // Clone positions array.
+  const queensPositions = [...previousQueensPositions].map((queenPosition) => {
+    return !queenPosition ? queenPosition : new QueenPosition(
+      queenPosition.rowIndex,
+      queenPosition.columnIndex,
+    );
+  });
+
+  if (rowIndex === queensCount) {
+    // We've successfully reached the end of the board.
+    // Store solution to the list of solutions.
+    solutions.push(queensPositions);
+
+    // Solution found.
+    return true;
+  }
+
+  // Let's try to put queen at row rowIndex into its safe column position.
+  for (let columnIndex = 0; columnIndex < queensCount; columnIndex += 1) {
+    if (isSafe(queensPositions, rowIndex, columnIndex)) {
+      // Place current queen to its current position.
+      queensPositions[rowIndex] = new QueenPosition(rowIndex, columnIndex);
+
+      // Try to place all other queens as well.
+      nQueensRecursive(solutions, queensPositions, queensCount, rowIndex + 1);
+
+      // Remove the queen from the row to avoid isSafe() returning false.
+      queensPositions[rowIndex] = null;
+    }
+  }
+
+  return false;
+}
+
+
+/**
+ * @param {number} queensCount
+ * @return {QueenPosition[][]}
+ */
+export default function nQueens(queensCount) {
+  // Init NxN chessboard with zeros.
+  // const chessboard = Array(queensCount).fill(null).map(() => Array(queensCount).fill(0));
+
+  // This array will hold positions or coordinates of each of
+  // N queens in form of [rowIndex, columnIndex].
+  const queensPositions = Array(queensCount).fill(null);
+
+  /** @var {QueenPosition[][]} solutions */
+  const solutions = [];
+
+  // Solve problem recursively.
+  nQueensRecursive(solutions, queensPositions, queensCount, 0);
+
+  return solutions;
+}