Backtracking in Data Structure and Algorithm
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Backtracking is an enhanced brute force approach that explores all possible solutions systematically to find the optimal one. It utilizes a tree structure to represent the solution space, employing explicit and implicit constraints to narrow down potential solutions. The technique is particularly useful for solving problems such as finding subsets that meet a certain criterion or generating permutations.
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5.5 back track
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Backtracking in Data Structure and Algorithm
- 2. Backtracking Backtracking is nothing but the modified process of the Brute Force approach. which will try for every possible solutions, then from which select the optimum solution or select best possible solution. It does so by assuming that the solutions are represented by vectors ( vi ……………in) of values and by traversing through the domains of the vectors until the solutions is found.
- 3. Backtracking- Introduction Suppose we have to make a series of decisions, among various choices, where we don’t have enough information to know what to choose – Each decision leads to a new set of choices – Some sequence of choices (possibly more than one) may be a solution to our problem. Backtracking is a methodical way of trying out various sequences of decisions, until you find one that “works”.
- 4. Backtracking and Recursion Backtracking is easily implemented with recursion because: The run-time stack takes care of keeping track of the choices that got us to a given point. Upon failure we can get to the previous choice simply by returning a failure code from the recursive call.
- 5. Backtracking :General Method Backtracking is used to solve problem in which a sequence of objects is chosen from a specified set so that the sequence satisfies some criterion. The desired solution is expressed as an n-tuple (x1……… xn) where each xi Є S, S being a finite set. The solution is based on finding one or more vectors that maximize, minimize, or satisfy a criterion function P((x1……… xn) . Form a solution and check at every step if this has any chance of success. If the solution at any point seems not promising, ignore it. All solutions requires a set of constraints divided into two categories: Explicit and implicit constraints.
- 6. Explicit & implicit Constraints Explicit constraints are rules that restrict each xi to take on values only from a given set. Explicit constraints depend on the particular instance I of problem being solved. Implicit constraints are rules that determine which of the tuples in the solution space of I satisfy the criterion function. Thus, implicit constraints describe the way in which the xi’s must relate to each other. The implicit constraints narrow the solution space down to an answer space.
- 7. Explicit & implicit Constraints Explicit Constraints are rules which restrict the values of xi . Example xi>=0 or xi =0 or 1 or Li<=xi<=Ui. All tuples that satisfy the explicit constraints define a possible solution space for I. Implicit Constraints describe the way in which the xi must relate to each other. Implicit Constraints allow to find those tuples in the solution space that satisfy the criterion function.
- 8. Backtracking :General Method Backtracking is a modified depth first search of a tree. Backtracking algorithms determine problem solutions by systematically searching the solution space for the given problem instance. This search is facilitated by using a tree organization for the solution space. Backtracking is the procedure whereby, after determining that a node can lead to nothing but dead end, we go back (backtrack) to the nodes parent and proceed with the search on the next child. A backtracking algorithm need not actually create a tree. Rather, it only needs to keep track of the values in the current branch being investigated. This is the way we implement backtracking algorithm. We say that the state space tree exists implicitly in the algorithm because it is not actually constructed.
- 9. State Space Tree Its root represents an initial state before the search for a solution begins. State space is the set of paths from root node to other nodes. State space tree is the tree organization of the solution space. A state-space tree for a backtracking algorithm is constructed in the manner of depth-first search. A node in a state-space tree is said to be promising if it corresponds to a partially constructed solution that may still lead to a complete solution; otherwise, it is called nonpromising. Leaves represent either nonpromising dead ends or complete solutions found by the algorithm.
- 10. Backtracking how we find out all the permutations. If you are given three numbers [1,2,3]
- 11. Backtracking
- 12. Backtracking Problem: Find out all 3-bit binary numbers for which the sum of the 1's is greater than or equal to 2. The only way to solve this problem is to check all the possibilities: (000, 001, 010, ....,111) The 8 possibilities are called the search space of the problem. They can be organized into a tree which is called State Space Tree.
- 13. Backtracking
- 14. Terminology Problem state is each node in the depth first search tree. Solution states are the problem states “S” for which the path from the root node to “S” defines a tuple in the solution space. Answer states are those solution states for which the path from root node to S defines a tuple that is a member of the set of solutions. Live node is a node that has been generated but whose children have not yet been generated. E-node is a live node whose children are currently being explored. In other words, an E-node is a node currently being expanded. Dead node is a generated node that is not to be expanded or explored any further. All children of a dead node have already been expanded.
- 15. Sum of subsets problem Problem definition: Find a subset of a given set A = {a1, . . . , an} of n positive integers whose sum is equal to a given positive integer d. For example, for A = {1, 2, 5, 6, 8} and d = 9, there are two solutions: {1, 2, 6} and {1, 8}. Of course, some instances of this problem may have no solutions. It is convenient to sort the set’s elements in increasing order. So, we will assume that a1< a2< . . . < an.
- 16. Sum of subsets problem
- 17. Problems