Backtracking Algorithm Last Updated : 01 Dec, 2024 Comments Improve Suggest changes Like Article Like Report Backtracking algorithms are like problem-solving strategies that help explore different options to find the best solution. They work by trying out different paths and if one doesn't work, they backtrack and try another until they find the right one. It's like solving a puzzle by testing different pieces until they fit together perfectly.BacktrackingBasic of Backtracking Algorithm:Introduction to BacktrackingBacktracking vs RecursionBacktracking vs Branch and BoundStandard ProblemsPermutations of a StringThe Knight’s tour problemRat in a MazeN Queen Problem | Backtracking-3Subset Sum problemm Coloring ProblemHamiltonian CycleSudoku | Backtracking-7Magnet PuzzleRemove Invalid ParenthesesA backtracking approach to generate n bit Gray CodesEasy Problem:Backtracking to find all subsetsCheck if a given string is sum-stringCount all possible paths between two verticesFind all distinct subsets of a given setFind if there is a path of more than k length from a sourcePrint all paths from a given source to a destinationPrint all possible strings that can be made by placing spacesMedium Problems:Tug of War8 queen problemCombinational SumWarnsdorff’s algorithm for Knight’s tour problemFind paths from corner cell to middle cell in mazeFind Maximum number possible by doing at-most K swapsRat in a Maze with multiple steps or jump allowedN Queen in O(n) spaceHard Problems:Power Set in Lexicographic orderWord Break Problem using BacktrackingPartition of a set into K subsets with equal sumLongest Possible Route in a Matrix with HurdlesFind shortest safe route in a path with landminesPrint all palindromic partitions of a stringPrinting all solutions in N-Queen ProblemPrint all longest common sub-sequences in lexicographical orderQuick Links :Learn Data Structure and Algorithms | DSA TutorialTop 20 Backtracking Algorithm Interview Questions‘Practice Problems’ on Backtracking‘Quiz’ on Backtracking‘Videos’ on BacktrackingWhat is Backtracking Algorithm?Backtracking is a problem-solving algorithmic technique that involves finding a solution incrementally by trying different options and undoing them if they lead to a dead end. It is commonly used in situations where you need to explore multiple possibilities to solve a problem, like searching for a path in a maze or solving puzzles like Sudoku. When a dead end is reached, the algorithm backtracks to the previous decision point and explores a different path until a solution is found or all possibilities have been exhausted.How Does a Backtracking Algorithm Work?A backtracking algorithm works by recursively exploring all possible solutions to a problem. It starts by choosing an initial solution, and then it explores all possible extensions of that solution. If an extension leads to a solution, the algorithm returns that solution. If an extension does not lead to a solution, the algorithm backtracks to the previous solution and tries a different extension.The following is a general outline of how a backtracking algorithm works:Choose an initial solution.Explore all possible extensions of the current solution.If an extension leads to a solution, return that solution.If an extension does not lead to a solution, backtrack to the previous solution and try a different extension.Repeat steps 2-4 until all possible solutions have been explored.Example of Backtracking AlgorithmExample: Finding the shortest path through a mazeInput: A maze represented as a 2D array, where 0 represents an open space and 1 represents a wall.Algorithm:Start at the starting point.For each of the four possible directions (up, down, left, right), try moving in that direction.If moving in that direction leads to the ending point, return the path taken.If moving in that direction does not lead to the ending point, backtrack to the previous position and try a different direction.Repeat steps 2-4 until the ending point is reached or all possible paths have been explored.When to Use a Backtracking Algorithm?Backtracking algorithms are best used to solve problems that have the following characteristics:There are multiple possible solutions to the problem.The problem can be broken down into smaller subproblems.The subproblems can be solved independently.Applications of Backtracking AlgorithmBacktracking algorithms are used in a wide variety of applications, including:Solving puzzles (e.g., Sudoku, crossword puzzles)Finding the shortest path through a mazeScheduling problemsResource allocation problemsNetwork optimization problemsCombinatorial problems, such as generating permutations, combinations, or subsets. Comment More infoAdvertise with us Next Article Introduction to Backtracking H harendrakumar123 Follow Improve Article Tags : Backtracking DSA Algorithms-Backtracking Algorithms-Recursion Practice Tags : Backtracking Similar Reads Backtracking Algorithm Backtracking algorithms are like problem-solving strategies that help explore different options to find the best solution. They work by trying out different paths and if one doesn't work, they backtrack and try another until they find the right one. It's like solving a puzzle by testing different pi 4 min read Introduction to Backtracking Backtracking is like trying different paths, and when you hit a dead end, you backtrack to the last choice and try a different route. In this article, we'll explore the basics of backtracking, how it works, and how it can help solve all sorts of challenging problems. It's like a method for finding t 7 min read Difference between Backtracking and Branch-N-Bound technique Algorithms are the methodical sequence of steps which are defined to solve complex problems. In this article, we will see the difference between two such algorithms which are backtracking and branch and bound technique. Before getting into the differences, lets first understand each of these algorit 4 min read What is the difference between Backtracking and Recursion? What is Recursion?The process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function.Real Life ExampleImagine we have a set of boxes, each containing a smaller box. To find an item, we open the outermost box and conti 4 min read Standard problems on backtrackingThe Knight's tour problemGiven an n × n chessboard with a Knight starting at the top-left corner (position (0, 0)). The task is to determine a valid Knight's Tour where where the Knight visits each cell exactly once following the standard L-shaped moves of a Knight in chess.If a valid tour exists, print an n × n grid where 15+ min read Rat in a MazeGiven an n x n binary matrix representing a maze, where 1 means open and 0 means blocked, a rat starts at (0, 0) and needs to reach (n - 1, n - 1).The rat can move up (U), down (D), left (L), and right (R), but:It cannot visit the same cell more than once.It can only move through cells with value 1. 9 min read N Queen ProblemGiven an integer n, the task is to find the solution to the n-queens problem, where n queens are placed on an n*n chessboard such that no two queens can attack each other. The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other. For example, 15+ min read Subset Sum Problem using BacktrackingGiven a set[] of non-negative integers and a value sum, the task is to print the subset of the given set whose sum is equal to the given sum. Examples: Input: set[] = {1,2,1}, sum = 3Output: [1,2],[2,1]Explanation: There are subsets [1,2],[2,1] with sum 3. Input: set[] = {3, 34, 4, 12, 5, 2}, sum = 8 min read M-Coloring ProblemGiven an edges of graph and a number m, the your task is to find weather is possible to color the given graph with at most m colors such that no two adjacent vertices of the graph are colored with the same color.ExamplesInput: V = 4, edges[][] = [[0, 1], [0, 2], [0,3], [1, 3], [2, 3]], m = 3Output: 13 min read Hamiltonian CycleHamiltonian Cycle or Circuit in a graph G is a cycle that visits every vertex of G exactly once and returns to the starting vertex.If a graph contains a Hamiltonian cycle, it is called Hamiltonian graph otherwise it is non-Hamiltonian.Finding a Hamiltonian Cycle in a graph is a well-known NP-complet 12 min read Algorithm to Solve Sudoku | Sudoku SolverGiven an incomplete Sudoku in the form of matrix mat[][] of order 9*9, the task is to complete the Sudoku.A sudoku solution must satisfy all of the following rules:Each of the digits 1-9 must occur exactly once in each row.Each of the digits 1-9 must occur exactly once in each column.Each of the dig 15+ min read Magnet PuzzleThe puzzle game Magnets involves placing a set of domino-shaped magnets (or electrets or other polarized objects) in a subset of slots on a board so as to satisfy a set of constraints. For example, the puzzle on the left has the solution shown on the right:  Each slot contains either a blank entry 15+ min read Remove Invalid ParenthesesGiven a string str consisting only of lowercase letters and the characters '(' and ')'. Your task is to delete minimum parentheses so that the resulting string is balanced, i.e., every opening parenthesis has a matching closing parenthesis in the correct order, and no extra closing parentheses appea 15+ min read A backtracking approach to generate n bit Gray CodesGiven a number n, the task is to generate n bit Gray codes (generate bit patterns from 0 to 2^n-1 such that successive patterns differ by one bit) Examples: Input : 2 Output : 0 1 3 2Explanation : 00 - 001 - 111 - 310 - 2Input : 3 Output : 0 1 3 2 6 7 5 4 We have discussed an approach in Generate n- 6 min read Permutations of given StringGiven a string s, the task is to return all permutations of a given string in lexicographically sorted order.Note: A permutation is the rearrangement of all the elements of a string. Duplicate arrangement can exist.Examples:Input: s = "ABC"Output: "ABC", "ACB", "BAC", "BCA", "CAB", "CBA"Input: s = " 5 min read Easy Problems on BacktrackingPrint all subsets of a given Set or ArrayGiven an array arr of size n, your task is to print all the subsets of the array in lexicographical order.A subset is any selection of elements from an array, where the order does not matter, and no element appears more than once. It can include any number of elements, from none (the empty subset) t 12 min read Check if a given string is sum-stringGiven a string s, the task is to determine whether it can be classified as a sum-string. A string is a sum-string if it can be split into more than two substring such that, the rightmost substring is equal to the sum of the two substrings immediately before it. This rule must recursively apply to th 14 min read Count all possible Paths between two VerticesGiven a Directed Graph with n vertices represented as a list of directed edges represented by a 2D array edgeList[][], where each edge is defined as (u, v) meaning there is a directed edge from u to v. Additionally, you are given two vertices: source and destination. The task is to determine the tot 15+ min read Find all distinct subsets of a given set using BitMasking ApproachGiven an array arr[] that may contain duplicates. The task is to return all possible subsets. Return only unique subsets and they can be in any order.Note: The individual subsets should be sorted.Examples: Input: arr[] = [1, 2, 2]Output: [], [1], [2], [1, 2], [2, 2], [1, 2, 2]Explanation: The total 7 min read Find if there is a path of more than k weight from a sourceGiven an undirected graph represented by the edgeList[][], where each edgeList[i] contains three integers [u, v, w], representing an undirected edge from u to v, having distance w. You are also given a source vertex, and a positive integer k. Your task is to determine if there exist a simple path (w 10 min read Print all paths from a given source to a destinationGiven a directed acyclic graph, a source vertex src and a destination vertex dest, print all paths from given src to dest. Examples:Input: V = 4, edges[ ][ ] = [[0, 1], [1, 2], [1, 3], [2, 3]], src = 0, dest = 3Output: [[0, 1, 2, 3], [0, 1, 3]]Explanation: There are two ways to reach at 3 from 0. Th 12 min read Print all possible strings that can be made by placing spacesGiven a string you need to print all possible strings that can be made by placing spaces (zero or one) in between them. Input: str[] = "ABC" Output: ABC AB C A BC A B C Source: Amazon Interview Experience | Set 158, Round 1, Q 1. Recommended PracticePermutation with SpacesTry It! The idea is to use 11 min read Medium prblems on BacktrackingTug of WarGiven an array arr[] of size n, the task is to divide it into two subsets such that the absolute difference between the sum of elements in the two subsets.If n is even, both subsets must have exactly n/2 elements.If n is odd, one subset must have (n−1)/2 elements and the other must have (n+1)/2 elem 11 min read 8 queen problemGiven an 8x8 chessboard, the task is to place 8 queens on the board such that no 2 queens threaten each other. Return a matrix of size 8x8, where 1 represents queen and 0 represents an empty position. Approach:The idea is to use backtracking to place the queens one by one on the board. Starting from 9 min read Combination SumGiven an array of distinct integers arr[] and an integer target, the task is to find a list of all unique combinations of array where the sum of chosen element is equal to target.Note: The same number may be chosen from array an unlimited number of times. Two combinations are unique if the frequency 8 min read Warnsdorff's algorithm for Knight’s tour problemProblem : A knight is placed on the first block of an empty board and, moving according to the rules of chess, must visit each square exactly once. Following is an example path followed by Knight to cover all the cells. The below grid represents a chessboard with 8 x 8 cells. Numbers in cells indica 15+ min read Find paths from corner cell to middle cell in mazeGiven a square maze represented by a matrix mat[][] of positive numbers, the task is to find all paths from all four corner cells to the middle cell. From any cell mat[i][j] with value n, you are allowed to move exactly n steps in one of the four cardinal directions—North, South, East, or West. That 14 min read Maximum number possible by doing at-most K swapsGiven a string s and an integer k, the task is to find the maximum possible number by performing swap operations on the digits of s at most k times.Examples: Input: s = "7599", k = 2Output: 9975Explanation: Two Swaps can make input 7599 to 9975. First swap 9 with 5 so number becomes 7995, then swap 15 min read Rat in a Maze with multiple steps or jump allowedYou are given an n × n maze represented as a matrix. The rat starts from the top-left corner (mat[0][0]) and must reach the bottom-right corner (mat[n-1][n-1]).The rat can move forward (right) or downward.A 0 in the matrix represents a dead end, meaning the rat cannot step on that cell.A non-zero va 11 min read N Queen in O(n) spaceGiven an integer n, the task is to find the solution to the n-queens problem, where n queens are placed on an n*n chessboard such that no two queens can attack each other.The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other.For example, th 7 min read Hard problems on BacktrackingPower Set in Lexicographic orderThis article is about generating Power set in lexicographical order. Examples : Input : abcOutput : a ab abc ac b bc cThe idea is to sort array first. After sorting, one by one fix characters and recursively generates all subsets starting from them. After every recursive call, we remove last charact 9 min read Word Break Problem using BacktrackingGiven a non-empty sequence s and a dictionary dict[] containing a list of non-empty words, the task is to return all possible ways to break the sentence in individual dictionary words.Note: The same word in the dictionary may be reused multiple times while breaking.Examples: Input: s = “catsanddog†8 min read Partition of a set into K subsets with equal sumGiven an integer array arr[] and an integer k, the task is to check if it is possible to divide the given array into k non-empty subsets of equal sum such that every array element is part of a single subset.Examples: Input: arr[] = [2, 1, 4, 5, 6], k = 3 Output: trueExplanation: Possible subsets of 9 min read Longest Possible Route in a Matrix with HurdlesGiven an M x N matrix, with a few hurdles arbitrarily placed, calculate the length of the longest possible route possible from source to a destination within the matrix. We are allowed to move to only adjacent cells which are not hurdles. The route cannot contain any diagonal moves and a location on 15+ min read Find shortest safe route in a path with landminesGiven a path in the form of a rectangular matrix having few landmines arbitrarily placed (marked as 0), calculate length of the shortest safe route possible from any cell in the first column to any cell in the last column of the matrix. We have to avoid landmines and their four adjacent cells (left, 15+ min read Printing all solutions in N-Queen ProblemGiven an integer n, the task is to find all distinct solutions to the n-queens problem, where n queens are placed on an n * n chessboard such that no two queens can attack each other. Note: Each solution is a unique configuration of n queens, represented as a permutation of [1,2,3,....,n]. The numbe 15+ min read Print all longest common sub-sequences in lexicographical orderYou are given two strings, the task is to print all the longest common sub-sequences in lexicographical order. Examples: Input : str1 = "abcabcaa", str2 = "acbacba" Output: ababa abaca abcba acaba acaca acbaa acbcaRecommended PracticePrint all LCS sequencesTry It! This problem is an extension of lon 14 min read Top 20 Backtracking Algorithm Interview Questions Backtracking is a powerful algorithmic technique used to solve problems by exploring all possible solutions in a systematic and recursive manner. It is particularly useful for problems that require searching through a vast solution space, such as combinatorial problems, constraint satisfaction probl 1 min read Like