Question

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

Short answer

The possible subsets of the given weights satisfying the total weight \(W=52\) are the paths in the tree that sum up to \(W\).

Step-by-step solution

Initialize

The total weight is \(W=52\) and the weights are \[w = \{2, 10, 13, 17, 22, 42\}\]. To start the algorithm, the initial state is created where no item is included in the subset.

Construction of the Level 1 in the tree

From the initial node, we create six branches corresponding to each value in the set \(w\). For each branch, we check if the current weight added to the total weight results in \(W\) or not.

Continue the process and perform backtracking

For each level, if the addition does not exceed \(W\), then keep that branch and move to the next branch (next item in \(w\)). If it does exceed, then backtrack and move to next branch. Continue this process until all the nodes are explored.

Find all subsets

Any node where the sum of the weights equals to \(W\) is a valid solution. Record these subsets as combinations of values that sum up to \(W\).

Key concepts

Algorithmic Problem Solving

When tackling algorithmic problem solving, one is essentially trying to find a step-by-step method to solve a given problem using algorithms. In the case of the Sum-of-Subsets problem, the desired outcome is to find all possible combinations of a given set of numbers that add up to a specified value, namely, a target sum of 52.

This type of problem is common in computer science and mathematics, where one needs to explore a variety of possibilities and eliminate those that do not fit the desired criteria. Through a systematic approach, solutions are formed by slowly building components that align with the conditions of the exercise. The key is to create a clear path of action, which, in this context, involves initializing the problem, constructing the possibility tree, and then using a combinatorial search method, like backtracking, to explore all potential combinations efficiently.

Recursive Algorithms

Recursive algorithms are a fundamental concept in computer science, where a function calls itself to solve smaller instances of the same problem. These smaller instances continue until reaching a base case, which does not require further recursion.

For the Sum-of-Subsets problem, recursion is used to explore each subset possibility in depth. When a current subset's sum does not exceed the target, the algorithm calls itself with the next increment in the set; otherwise, the recursion unwinds—this unwinding is the 'backtrack' process. By employing recursion, the algorithm elegantly traverses the tree of possibilities without the need for complicated loop structures or auxiliary data storage.

Combinatorial Search

Combinatorial search involves finding an optimal solution from a finite set of possibilities through systematic enumeration. Especially when dealing with problems like Sum-of-Subsets, we're handling a significant number of potential combinations, and the combinatorial search helps in pruning non-viable options.

Through the creation of a 'state space tree', the algorithm represents all potential combinations. Each level of the tree represents the inclusion or exclusion of an element. The search is 'combinatorial' as it combines these elements in various ways to find all possible subsets that add to the target value. However, without a strategy for reducing the number of computations, such as backtracking, the process could be overly time-consuming and computationally expensive.

Backtracking Technique

Backtracking is a refined technique for combinatorial search which reduces unnecessary exploration of the solution space. This technique is characterized by trying to build a solution incrementally, one piece at a time, and removing those pieces as soon as the algorithm determines that the piece will not lead to a viable solution.

In context, with every step of the Sum-of-Subsets problem, the increasing subset is compared to the required sum. If the current combination, plus the current element, remains less than or equal to the target sum, the search proceeds by adding the next possible element. This methodology allows us to discard certain branches early in the search process—those that will definitely not sum up to the required total, thus saving computational effort and time.