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Backtracking

For the line search algorithm used in unconstrained optimization, see Backtracking line search.

Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution.[1]

The classic textbook example of the use of backtracking is the eight queens puzzle, that asks for all arrangements of eight chess queens on a standard chessboard so that no queen attacks any other. In the common backtracking approach, the partial candidates are arrangements of k queens in the first k rows of the board, all in different rows and columns. Any partial solution that contains two mutually attacking queens can be abandoned.

Backtracking can be applied only for problems which admit the concept of a "partial candidate solution" and a relatively quick test of whether it can possibly be completed to a valid solution. It is useless, for example, for locating a given value in an unordered table. When it is applicable, however, backtracking is often much faster than brute-force enumeration of all complete candidates, since it can eliminate many candidates with a single test.

Backtracking is an important tool for solving constraint satisfaction problems,[2] such as crosswords, verbal arithmetic, Sudoku, and many other puzzles. It is often the most convenient technique for parsing,[3] for the knapsack problem and other combinatorial optimization problems. It is also the program execution strategy used in the programming languages Icon, Planner and Prolog.

Backtracking depends on user-given "black box procedures" that define the problem to be solved, the nature of the partial candidates, and how they are extended into complete candidates. It is therefore a metaheuristic rather than a specific algorithm – although, unlike many other meta-heuristics, it is guaranteed to find all solutions to a finite problem in a bounded amount of time.

The term "backtrack" was coined by American mathematician D. H. Lehmer in the 1950s.[4] The pioneer string-processing language SNOBOL (1962) may have been the first to provide a built-in general backtracking facility.

  1. ^ Gurari, Eitan (1999). "CIS 680: DATA STRUCTURES: Chapter 19: Backtracking Algorithms". Archived from the original on 17 March 2007.
  2. ^ Biere, A.; Heule, M.; van Maaren, H. (29 January 2009). Handbook of Satisfiability. IOS Press. ISBN 978-1-60750-376-7.
  3. ^ Watson, Des (22 March 2017). A Practical Approach to Compiler Construction. Springer. ISBN 978-3-319-52789-5.
  4. ^ Rossi, Francesca; van Beek, Peter; Walsh, Toby (August 2006). "Constraint Satisfaction: An Emerging Paradigm". Handbook of Constraint Programming. Amsterdam: Elsevier. p. 14. ISBN 978-0-444-52726-4. Retrieved 30 December 2008.

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