1.
Introduction
2.
Backtracking
3.
Code Example in Python
3.1.
Python
4.
Branch and Bound
5.
Code Example in Python
5.1.
Python
6.
Difference Between Backtracking & Branch and Bound on the Basis of Parameters
7.
7.1.
What types of problems are best solved by backtracking?
7.2.
How does branch and bound reduce computational time?
7.3.
Can backtracking and branch and bound be used together?
8.
Conclusion
Last Updated: May 18, 2024
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# Difference Between Backtracking and Branch and Bound

Rinki Deka
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## Introduction

Backtracking & branch and bound are two important algorithmic strategies used to solve complex problems in computer science. Backtracking is a general algorithm that considers searching every possible combination in order to solve a computational problem. Branch and bound is an algorithmic approach to solve optimization problems by systematically enumerating candidate solutions. While both techniques involve exploring different possibilities, they have some key differences in their approach & applicability.

In this article, we will learn about the concepts of backtracking & branch and bound, understand how they work, and compare their differences based on various parameters.

## Backtracking

Backtracking is a powerful approach in computer science used to solve problems by trying to build a solution incrementally, one piece at a time, & removing those solutions that fail to satisfy the constraints of the problem at any point of time. Imagine trying to solve a puzzle: if one piece doesn't fit, you remove it & try another piece instead.

Hereâ€™s how backtracking works: It starts with a possible move & tries to solve the problem. If the move leads to a solution, the algorithm stops. However, if the move doesnâ€™t solve the problem, the algorithm undoes the move (backtracks) & makes a new try with different options. This process repeats until a solution is found or all options are exhausted.

For example, letâ€™s consider the problem of placing N queens on an NĂ—N chessboard so that no two queens threaten each other. The solution involves placing a queen on the board, moving row by row, column by column. If placing a queen leads to a conflict, the algorithm backtracks & tries the next position in the row.

## Code Example in Python

• Python

### Python

``def is_safe(board, row, col):    for i in range(col):        if board[row][i] == 1:            return False    for i, j in zip(range(row, -1, -1), range(col, -1, -1)):        if board[i][j] == 1:            return False    for i, j in zip(range(row, len(board), 1), range(col, -1, -1)):        if board[i][j] == 1:            return False    return Truedef solve_n_queens(board, col):    if col >= len(board):        return True    for i in range(len(board)):        if is_safe(board, i, col):            board[i][col] = 1            if solve_n_queens(board, col + 1):                return True            board[i][col] = 0  # backtrack    return Falsedef place_queens(n):    board = [[0]*n for _ in range(n)]    if not solve_n_queens(board, 0):        return "No solution"    return board# Example usagen = 4solution = place_queens(n)print("Solution for", n, "queens:")for row in solution:    print(row)``

Output

``````Solution for 4 queens:
[0, 0, 1, 0]
[1, 0, 0, 0]
[0, 0, 0, 1]
[0, 1, 0, 0]``````

This example illustrates backtracking by trying to place queens on the board & backtracking whenever placing a queen results in a conflict.

## Branch and Bound

Branch and bound is a systematic method for solving optimization problems, which are often about finding the best solution among many possible options. Unlike backtracking, which focuses on feasibility (finding any solution that satisfies the problem's constraints), branch and bound aims to find the optimal solution by efficiently exploring the entire solution space.

This technique involves dividing the problem into smaller parts (branching) and then evaluating these parts to decide whether to explore further or discard them (bounding). The "bound" part uses logical tests or bounds to eliminate suboptimal paths in the search process, hence speeding up the solution finding.

For instance, consider the problem of finding the shortest traveling path through several cities. Here, the algorithm would explore routes (branching) and use bounds (like the minimum distance already traveled) to discard longer routes early in the process.

## Code Example in Python

• Python

### Python

``import heapqdef branch_and_bound(items, capacity):    # Initial best (highest value within weight limit)    best_value = 0    # Priority queue for exploring nodes: (negative value, total weight, index, items taken)    pq = [(-0, 0, 0, [])]  # Max-heap by value    while pq:        value, weight, index, taken = heapq.heappop(pq)        value = -value  # Convert back to positive value        # Bound: if no items left to consider, update best value        if index == len(items):            if value > best_value:                best_value = value            continue        # Check next item inclusion        item = items[index]        if weight + item['weight'] <= capacity:            # Explore path with this item included            heapq.heappush(pq, (-(value + item['value']), weight + item['weight'], index + 1, taken + [item['name']]))        # Explore path without this item        heapq.heappush(pq, (-(value), weight, index + 1, taken))    return best_value# Example items and capacityitems = [{'name': 'Laptop', 'weight': 2, 'value': 400}, {'name': 'Headphones', 'weight': 1, 'value': 150}]capacity = 2print("Maximum value within capacity:", branch_and_bound(items, capacity))``

Output

``Maximum value within capacity: 400``

This code example implements a simple branch and bound algorithm to solve the knapsack problem, where we try to maximize the value of items carried without exceeding the weight capacity.

## Difference Between Backtracking & Branch and Bound on the Basis of Parameters

### What types of problems are best solved by backtracking?

Backtracking is ideal for solving constraint satisfaction problems like puzzles and combinatorial problems where all potential solutions must be examined.

### How does branch and bound reduce computational time?

Branch and bound reduces computational time by eliminating paths that cannot possibly lead to an optimal solution based on predefined bounds, which means fewer paths are explored.

### Can backtracking and branch and bound be used together?

Yes, they can be combined. For example, branch and bound can be used to find an optimal solution within a set explored through backtracking, optimizing both the search and the solution quality.

## Conclusion

In this article, we have learned about backtracking and branch and bound, two fundamental techniques for solving problems in computer science. We discussed their purposes, approaches, and the types of problems they are best suited for. Backtracking is excellent for finding any or all solutions that meet specific constraints, while branch and bound is used to find the optimal solution in optimization scenarios.

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