Question

Use the Backtracking algorithm for the Sum-ofSubsets problem (Algorithm 5.4) to find all combinations of the following numbers that sum to \(W=52:\) \\[ w_{1}=2 \quad w_{2}=10 \quad w_{3}=13 \quad w_{4}=17 \quad w_{5}=22 \quad w_{6}=42 \\] Show the actions step by step.

Short answer

By using backtracking, there's one subset of these weights which sums up to 52, \{2, 10, 13, 17, 10\}.

Step-by-step solution

Initialize the Variables

Start by initializing an array of weights \( w = [2, 10, 13, 17, 22, 42] \) and another array \( x = [0, 0, 0, 0, 0, 0] \) of the same length initialized to zero to keep track of whether to include the corresponding weight in the sum or not.

Define the Recursive Sum-of-Subsets Function

Define a recursive function, sum_of_subsets(), to generate the combinations. The function takes the sum of included weights (s), the sum of weights remaining (r), and the current depth (d). If the sum of included weights equals 52, it means this subset is a solution. In that case, print the subset. If it doesn't, but adding the next weight keeps the sum 52 or less and there are weights remaining, consider the next weight by recursively calling sum_of_subsets with s+w[d+1], r-w[d+1] and d+1. If including this weight doesn't lead to a solution, then exclude it by calling sum_of_subsets with s, r-w[d+1], and d+1.

Call the Sum-of-Subsets Function

Call sum_of_subsets() function with s = 0 (no weights are included initially), r = the sum of all weights, and d = 0 (start from the first depth level).

Print the Subset Combinations

While running the algorithm, each time a sum of 52 is achieved, print the corresponding subset combination. These printed sets are the solution.

Key concepts

Sum-of-Subsets Problem

The Sum-of-Subsets problem is a classical algorithmic problem that seeks to find a combination of numbers from a given set whose total sum equals a specific target number, denoted usually as \( W \). In this context, the Sum-of-Subsets exercise involves using a set of weights, where each specific weight must either be included in or excluded from the subset solution. As demonstrated, our task was to determine all possible combinations of the numbers \([2, 10, 13, 17, 22, 42]\) that sum to 52. By utilizing a well-structured approach like the Backtracking algorithm, which explores possibilities incrementally, we can efficiently handle this problem. This technique systematically examines potential subsets to identify the solution without having to excessively and exhaustively enumerate each possible combination. As a strategy, it helps optimize and quicken the process by eliminating unfeasible options at early stages.

Recursive Function

A recursive function is a powerful tool in computer science where a function calls itself with modified parameters to solve a problem. In the Sum-of-Subsets problem, recursion is crucial as it breaks down the problem of finding valid subsets into smaller, more manageable subsets.
Through recursion, we can keep track of:

• The current sum of included weights \( s \),
• Potential subsequent weights \( r \), and
• The depth of search \( d \), or the index of the current weight being considered.

The recursive process examines each potential inclusion or exclusion of a weight in the subset to accumulate towards the target sum \( W = 52 \). When a valid subset sum \( s \) equals \( W \), it outputs the current path as a solution. By gradually building the subset, and retracting when a sum exceeds \( W \), recursion naturally embodies a backtracking approach. This recursive pattern continues until all viable subsets are explored.

Algorithm Design

Algorithm design refers to the strategic formulation of a step-by-step procedural method to accomplish a specific task. In crafting an algorithm for the Sum-of-Subsets problem, the design centers on effectively using backtracking.
Key aspects of this algorithm design include:

• Initiating with an array representation of items, tracking possible inclusion in the solution,
• Establishing rules for the recursive function that allow the algorithm to evaluate each number in sequence, and
• Conclusively determining whether each number can lead to a valid sum by including or excluding it from consideration.

The elegance of this algorithm design lies in its inherent ability to prune unpromising paths early. By advancing only on paths where the prospective total is viable, and retracting when the total exceeds \( W \), computational efficiency is significantly enhanced. As such, algorithm design is pivotal in honing both the performance and accuracy of complex problem-solving strategies like backtracking.

Subset Combinations

Subset combinations are the different selections of elements from a set. Within the Sum-of-Subsets problem, finding the correct subset combinations is central to solving the puzzle.
In our problem: - We are looking at all possible ways to combine the numbers 2, 10, 13, 17, 22, and 42 to see which combinations sum to 52. The recursive and backtracking approach efficiently carves through potential combinations: - Each choice to either include or exclude a number creates a bifurcation, paving the way for new subset possibilities. - When a combination reaches the goal sum without overstepping it, it is documented as a successful subset. The art of identifying subset combinations is not merely about brute force; rather it gracefully navigates through the solution space. With each step, irrelevant or impossible solutions are dismissed, refining the search for combinations that truly satisfy the specified constraints.