Add N-Queens.

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Oleksii Trekhleb 2018-05-16 07:54:06 +03:00
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commit e2ef46016d
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* [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**
```
@ -135,12 +142,12 @@ Then just simply run the following command to test if your playground code works
npm test -- -t 'playground'
```
## Useful links
## Useful Information
### References
[▶ Data Structures and Algorithms on YouTube](https://www.youtube.com/playlist?list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
## Useful Information
### Big O Notation
Order of growth of algorithms specified in 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 |

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/**
* 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}`;
}
}

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# 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)

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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');
});
});

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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');
});
});

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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;
}