Backtracking Algorithm (original) (raw)

Last Updated : 01 Dec, 2024

**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.

Backtracking

**Basic of Backtracking Algorithm:

1. Introduction to Backtracking
2. Backtracking vs Recursion
3. Backtracking vs Branch and Bound

**Standard Problems

• Permutations of a String
• The Knight’s tour problem
• Rat in a Maze
• N Queen Problem | Backtracking-3
• Subset Sum problem
• m Coloring Problem
• Hamiltonian Cycle
• Sudoku | Backtracking-7
• Magnet Puzzle
• Remove Invalid Parentheses
• A backtracking approach to generate n bit Gray Codes

**Easy Problem:

• Backtracking to find all subsets
• Check if a given string is sum-string
• Count all possible paths between two vertices
• Find all distinct subsets of a given set
• Find if there is a path of more than k length from a source
• Print all paths from a given source to a destination
• Print all possible strings that can be made by placing spaces

**Medium Problems:

• Tug of War
• 8 queen problem
• Combinational Sum
• Warnsdorff’s algorithm for Knight’s tour problem
• Find paths from corner cell to middle cell in maze
• Find Maximum number possible by doing at-most K swaps
• Rat in a Maze with multiple steps or jump allowed
• N Queen in O(n) space

**Hard Problems:

• Power Set in Lexicographic order
• Word Break Problem using Backtracking
• Partition of a set into K subsets with equal sum
• Longest Possible Route in a Matrix with Hurdles
• Find shortest safe route in a path with landmines
• Print all palindromic partitions of a string
• Printing all solutions in N-Queen Problem
• Print all longest common sub-sequences in lexicographical order

**Quick Links :

• Learn Data Structure and Algorithms | DSA Tutorial
• Top 20 Backtracking Algorithm Interview Questions
• ‘Practice Problems’ on Backtracking
• ‘Quiz’ on Backtracking
• ‘Videos’ on Backtracking

What 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:

1. Choose an initial solution.
2. Explore all possible extensions of the current solution.
3. If an extension leads to a solution, return that solution.
4. If an extension does not lead to a solution, backtrack to the previous solution and try a different extension.
5. Repeat steps 2-4 until all possible solutions have been explored.

Example of Backtracking Algorithm

**Example: Finding the shortest path through a maze

**Input: A maze represented as a 2D array, where **0 represents an open space and **1 represents a wall.

**Algorithm:

1. Start at the starting point.
2. For each of the four possible directions (up, down, left, right), try moving in that direction.
3. If moving in that direction leads to the ending point, return the path taken.
4. If moving in that direction does not lead to the ending point, backtrack to the previous position and try a different direction.
5. 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 Algorithm

Backtracking algorithms are used in a wide variety of applications, including:

• Solving puzzles (e.g., Sudoku, crossword puzzles)
• Finding the shortest path through a maze
• Scheduling problems
• Resource allocation problems
• Network optimization problems
• Combinatorial problems, such as generating permutations, combinations, or subsets.

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• 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
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Standard problems on backtracking

• The Knight's tour problem Given 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 Maze Given 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 9 min read

• N Queen Problem Given 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 Backtracking Given 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:Â Input:Â set[] = {1,2,1}, sum = 3Output:Â [1,2],[2,1]Explanation:Â There are subsets [1,2],[2,1] with sum 3. Input:Â set[] = {3, 34, 4, 12, 5, 2}, sum = 8 min read

• M-Coloring Problem Given 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. Examples Input: V = 4, edges[][] = [[0, 1], [0, 2], [0,3], [1, 3], [2, 3]], m = 3Output 13 min read

• Hamiltonian Cycle What is Hamiltonian Cycle?Hamiltonian 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 graph contains a Hamiltonian cycle, it is called Hamiltonian graph otherwise it is non-Hamiltonian.Finding a Hamiltonian Cycle in a graph i 15+ min read

• Algorithm to Solve Sudoku | Sudoku Solver Given 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 di 15+ min read

• Magnet Puzzle The 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: Â Each slot contains either a blank entry 15+ min read

• Remove Invalid Parentheses Given 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 Codes Given 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 String Given 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 Backtracking

• Print all subsets of a given Set or Array Given 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) 12 min read

• Check if a given string is sum-string Given a string of digits, determine whether it is a ‘sum-string’. A string S is called a sum-string if the rightmost substring can be written as the sum of two substrings before it and the same is recursively true for substrings before it. Examples: “12243660” is a sum string. Explanation : 24 + 36 12 min read

• Count all possible Paths between two Vertices Count the total number of ways or paths that exist between two vertices in a directed graph. These paths don't contain a cycle, the simple enough reason is that a cycle contains an infinite number of paths and hence they create a problem Examples: For the following Graph: Input: Count paths between 9 min read

• Find all distinct subsets of a given set using BitMasking Approach Given an array of integers arr[], The task is to find all its subsets. The subset can not contain duplicate elements, so any repeated subset should be considered only once in the output. Examples: Input: S = {1, 2, 2}Output: {}, {1}, {2}, {1, 2}, {2, 2}, {1, 2, 2}Explanation: The total subsets of gi 12 min read

• Find if there is a path of more than k weight from a source Given 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 destination Given 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. T 12 min read

• Print all possible strings that can be made by placing spaces Given 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 Backtracking

• Tug of War Given a set of n integers, divide the set in two subsets of n/2 sizes each such that the absolute difference of the sum of two subsets is as minimum as possible. If n is even, then sizes of two subsets must be strictly n/2 and if n is odd, then size of one subset must be (n-1)/2 and size of other su 15 min read

• 8 queen problem The eight queens problem is the problem of placing eight queens on an 8×8 chessboard such that none of them attack one another (no two are in the same row, column, or diagonal). More generally, the n queens problem places n queens on an n×n chessboard. There are different solutions for the problem. 8 min read

• Combination Sum Given 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 frequenc 8 min read

• Warnsdorff's algorithm for Knight’s tour problem Problem : 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 maze Given 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 swaps Given 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 allowed You 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 v 11 min read

• N Queen in O(n) space Given 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, 7 min read

Hard problems on Backtracking

• Power Set in Lexicographic order This 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 Backtracking Given 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 sum Given 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 Hurdles Given 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 landmines Given 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 Problem Given 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 order You 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