Learn How to Efficiently Solve Coding Interview Backtracking Problems

Backtracking is a powerful algorithmic technique that solves problems by incrementally building candidates to solutions and abandoning a candidate ("backtracking") as soon as it determines that the candidate cannot lead to a valid solution. It is especially prevalent in coding interviews, with many common question types relying on backtracking as the optimal approach.

As a full-stack software engineer, I‘ve used backtracking to solve real-world problems like task scheduling, vehicle routing, and game move evaluation. Backtracking is also extremely common in whiteboard coding interviews. According to an analysis of over 1100 Leetcode questions by Hackernoon, backtracking is the 5th most tested algorithm category, showing up in nearly 10% of problems. Understanding the backtracking pattern is a crucial tool for any software engineer preparing for coding interviews.

In this in-depth guide, I‘ll explain the core concepts of backtracking and walk through a general template that can be adapted to solve any backtracking problem. I‘ll also break down solutions to common backtracking interview questions and share some of the tips I‘ve used to quickly identify and solve these problems under pressure. Let‘s get started!

What is Backtracking?

At its core, backtracking solves computational problems by searching the space of all potential solutions. It incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly lead to a valid solution.

The backtracking algorithm consists of the following steps:

1. Choose: Choose a move to make. This is usually accomplished by iterating through available options at the current state.
2. Constraint: Check if the current move is valid based on given constraints. If not, simply skip this move.
3. Goal: Check if the current move is a solution or completes a solution. If so, you have found a solution, return true.
4. Recurse: Recursively make moves on the updated state after making the current move.
5. Backtrack: Undo the current move to restore the state back to before the move was made.

Let‘s visualize this with a simple example – finding all valid arrangements of N queens on an NxN chessboard such that no two queens threaten each other:

We start with an empty board and place queens one by one. Whenever we place a queen, we check if it conflicts with already placed queens. If so, we backtrack and try a different column in the current row. If we manage to place all N queens without conflicts, we have found a valid solution.

The key here is that we don‘t wait until placing all N queens to check for conflicts. By checking after each placement, we can prune invalid paths early and avoid unnecessary work. This is what makes backtracking more efficient than brute force.

When to Use Backtracking

Backtracking is the ideal technique for solving problems that require searching through a large, possibly exponential, number of potential solutions. It is especially applicable when solutions must satisfy complex constraints that can be verified incrementally.

Some common characteristics of backtracking problems:

1. Find all valid solutions to a problem, not just one
2. Solutions are built incrementally and may be abandoned partway through
3. Constraints determine what makes a valid vs. invalid solution
4. Often result in recursion and passing state down the recursive calls

Classic examples of backtracking problems include:

• Finding all valid permutations, subsets, or combinations
• Solving board/grid layout problems like N-Queens, Knight‘s Tour, Rat in a Maze
• Finding all solutions to a Sudoku or crossword puzzle
• Generating all valid parentheses combinations

If a problem has these traits, reach for backtracking as a powerful algorithmic tool.

General Backtracking Template

While the specific details change problem to problem, most backtracking implementations follow the same general template:

def backtrack(candidate):
    if find_solution(candidate):
        output(candidate)
        return

    # iterate all possible candidates.
    for next_candidate in list_of_candidates:
        if is_valid(next_candidate):
            # try this partial candidate solution
            place(next_candidate) 
            # given the candidate, explore further.
            backtrack(next_candidate)
            # backtrack
            remove(next_candidate)

The key components:

1. is_valid function: Encodes the constraints of the problem. Decides if a candidate is valid to be explored further or should be pruned.

2. place and remove functions: Apply and undo a move, updating state for the next recursive call and backtracking afterwards.

3. Recursive backtrack function: Explores all possibilities by recursing on the updated state after making a move. Base case checks for a complete solution.

Let‘s adapt this template to solve a classic backtracking problem.

Example: Generating Subsets

Problem: Given an integer array nums, return all possible subsets (the power set). Solution must not contain duplicate subsets.

Mapping this to the template:

• The state is the current subset we‘re building
• A candidate solution is any valid subset
• is_valid is always true since any subset is valid
• place adds the current element to the subset, remove backtracks by removing it

Here‘s the complete code:

class Solution:
    def subsets(self, nums: List[int]) -> List[List[int]]:
        def backtrack(start, subset):
            # add every subset
            output.append(subset[:])

            # generate next candidates
            for i in range(start, n):  
                # place candidate
                subset.append(nums[i])
                # recurse on updated state  
                backtrack(i + 1, subset)
                # backtrack  
                subset.pop()

        output = []
        n = len(nums)
        backtrack(0, [])
        return output

The recursive backtrack function drives the solution:

• Base case is when start == n, meaning we have checked all elements. Every subset is a valid result, so we always append a copy to output
• Recursive case generates the next candidates by iterating from start to n. We add the current element, recurse, then pop it to backtrack

Let‘s analyze the time and space complexity. For a set of size n, there are 2^n possible subsets. In the worst case, backtracking will generate all 2^n subsets, giving an exponential time complexity of O(2^n). The recursive call stack will go at most n layers deep, resulting in O(n) space complexity.

This matches our intuition that the problem is exponential – for every additional element, the number of subsets doubles. However, backtracking efficiently explores this exponential state space by abandoning invalid candidates early. A brute force approach would wastefully generate all 2^n subsets upfront before checking them.

Real-World Applications

Beyond coding interviews, backtracking has many practical applications across domains:

• Vehicle routing problems in logistics and ride-sharing
• Drug discovery and molecular structure search in biotech/pharma
• Game move evaluation and pruning search trees in chess AIs
• Constraint satisfaction problems in resource allocation and scheduling

For example, in my work I once used backtracking to optimize delivery truck routes. The problem reduced to finding paths on a graph that satisfied complex constraints like truck capacity, delivery time windows, driver shift limits, etc. By modeling it as a backtracking problem, I was able to efficiently search the space of all possible routes and find cost-optimized paths. Brute forcing all routes and filtering valid ones afterwards would have been infeasible given the combinatorial explosion of possibilities.

Recognizing Backtracking Problems

Backtracking problems can be tricky to spot at first, but there are some common signs:

1. Asks for all solutions or combinations, not just one
2. Solutions must meet complex constraints
3. Can incrementally build and validate solutions
4. Involves making and undoing moves/choices
5. Often solved with recursion

If you see a problem with these traits, consider backtracking. Some specific keywords that hint at backtracking:

• "Generate/find all…"
• "Determine if there exists…"
• "Satisfy constraints…"
• "Combinations/permutations/subsets…"

Some common categories of backtracking questions:

• Combinatorial searches (subsets, permutations, combinations)
• Constraint satisfaction (N-Queens, Sudoku, Knight‘s Tour)
• Recursive string/array manipulation (palindrome partitioning, word breaks)
• Exhaustive search (word search, target sum)

By internalizing the template and practicing these common patterns, you can learn to confidently reach for backtracking whenever you encounter one of these problems.

Tips for Backtracking Questions

Having solved countless backtracking problems in my career, here are my top tips:

1. Always start by clearly defining the state you‘re working with. This is usually what you‘re recursively modifying and passing down recursive calls.

2. If you‘re stuck, try to map the problem to the general template. Identify how to make and unmake moves, check for solutions, and generate next candidates.

3. Don‘t be afraid to use pen and paper to draw out the recursive calls! Backtracking is extremely prone to off-by-one errors, so trace a few examples.

4. When debugging, print the state at the beginning of each recursive call. This makes it easy to trace the algorithm‘s progress and spot issues.

5. Time and space complexity are often exponential, so look for opportunities to optimize by pruning invalid states early. The earlier you can abandon failures, the better!

6. If the naive backtracking exceeds time limits, caching recursive results with memoization can dramatically improve performance.

7. Practice, practice, practice! Work through problems on Leetcode tagged with "backtracking" and internalize the patterns.

With these tips and enough practice, backtracking will become an indispensable tool in your problem solving toolkit.

Conclusion

Backtracking is a powerful algorithmic technique for solving complex combinatorial search problems. By incrementally building and pruning candidates according to constraints, it efficiently searches a large state space without wastefully exploring dead ends.

While backtracking has a reputation for being conceptually difficult, it‘s a critical technique to master for coding interviews and real-world problem solving. Recognizing the common patterns, understanding the general template, and mapping problems to this template are the keys to solving any backtracking question.

I encourage you to practice the problems linked throughout this guide and to approach them with a systematic, template-driven approach. Start by clearly defining state and mappings to the template. Recursively explore possibilities with a base case for valid solutions. Analyze and optimize time and space complexity. With practice, these steps will become second nature.

Backtracking is just one of many important algorithmic techniques. However, it is an essential tool for any software engineer to have in their problem solving toolbox. I believe backtracking questions will only become more prevalent in coding interviews as companies seek developers with strong foundations in algorithms and problem solving. Invest time in mastering this technique – it will pay dividends in your job search and career.

Now go forth and conquer some backtracking problems! As always, feel free to connect with me for more in-depth algorithms guidance and career advice. Happy problem solving!