• Backtrack 1 32-bit
• Cached

• Installation
  • Install using pip
    • Requirements
• A Backtracking Example

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You can install pybacktrack either:

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1. using pip, or
2. using Docker.

Docker is typically more straightforward since all the dependencies of pybacktrack have been pre-installed.

Python packages installed using pip will typically also have their dependency packages automatically installed also.However pybacktrack requires manual installation of some of its dependencies.

• Requirements

PyBacktrack depends on:

• Generic Mapping Tools (GMT) (>=5.0.0)

NumPy and SciPy are automatically installed by pip when pybacktrack is installed, however GMT (version 5 or above) and pyGPlates need to be manually installed.

GMT is called via the command-line (shell) and so just needs to be in the PATH in order for pyBacktrack to find it.Also ensure that version 5 or above (supports NetCDF version 4) is installed since the bundled grid files in pyBacktrack are in NetCDF4 format.

PyGPlates is not currently installable as a package and so needs to be in the python path (sys.path or PYTHONPATH).Installation instructions are available here.

PyGPlates now supports Python 3 (in addition to Python 2.7) so you can now use pyBacktrack with either Python 2.7 or 3.The Macports install example below shows one approach to selecting the default Python (2.7 or 3) using sudoportselect.Another approach is using Python virtual environments where each environment has its own python, pip and installed packages.However, currently pyGPlates does not yet work in virtual environments (at least on Mac systems).

This is an example demonstrating how to install GMT and pyGPlates on Ubuntu 18.04 (Bionic).

Note

The main difference for other Ubuntu versions will be the pyGPlates install package(you’ll need to select the package appropriate for your Ubuntu version).

First install GMT 5:

Then install Python 3 (and Pip):

Then download the pyGPlates Python 3 debian package pygplates-py3-ubuntu-bionic_2.2_1_amd64.deb,and install it:

Then add the installed location of pyGPlates to the PYTHONPATH environment variable:

This is an example demonstrating how to install GMT and pyGPlates on a Mac system using Macports.

First install GMT 5:

Note

You will likely need to add /opt/local/lib/gmt5/bin/ to your PATH environment variable,for example in your ~/.bashrc, ~/.bash_profile or ~/.zprofile file so that PATHis set each time you open a new terminal window.After doing this, typing gmt should find GMT and show some help options.

Then install Python 3 (and Pip):

Set your default python to Python 3.8:

Note

If you already have python referencing Python 2 then you can instead use python3 to reference Python 3:

…but this will require using python3 on the command-line to runpybacktrack (instead of just python).

Alternatively you can use Python 2.7 to run pyBacktrack (instead of Python 3) by installing pyGPlatesfor Python 2.7 (instead of for Python 3) - see below.

Then download the pyGPlates Mac zip file pygplates_rev28_python38_MacOS64.zip,and extract it to your home directory.

Then add the unzipped location of pyGPlates to the PYTHONPATH environment variable:

Note

The above line can be added to your ~/.bashrc, ~/.bash_profile or ~/.zprofile fileso that PYTHONPATH is set each time you open a new terminal window.

Note

If you are using Python 2.7 to run pyBacktrack (instead of Python 3) then install the Python 2.7 version of pyGPlates instead by installingpygplates_rev28_python27_MacOS64.zip and adding ~/pygplates_rev28_python27_MacOS64 to your PYTHONPATH.

To install the latest stable version, run:

Warning

Backtrack 1 32-bit

Note

We generally recommend using python-mpipinstallpybacktrack instead of pipinstallpybacktrack to ensure pybacktrack is installedinto the python you are actually using. For example, when using Conda Python it might be that python executes the Conda Python interpreterbut pip installs into the system Python (eg, because the base Conda environment is not activated).

If you already have pybacktrack installed and would like to upgrade to the latest version then use the --upgrade flag:

To install the latest development version (requires Git on local system), run:

https://software-yy.mystrikingly.com/blog/video-hd-move. Note

…or download the pyBacktrack source code, extract to a local directory and run:

Note

Installing pyBacktrack will automatically install the NumPy and SciPyrequirements.However, as mentioned in requirements, GMT and pyGPlates still need to be manually installed.

Before running the example below, or any other examples, you’ll also need to install the example data (from the pybacktrack package itself).This assumes you’ve already installed the pybacktrack package.

How to backup my mac. The following command installs the examples (example data and notebooks) to a new sub-directory of your current working directory called pybacktrack_examples:

Note

The current working directory is whatever directory you are in when you run the above command.

Note

This method of running pybacktrack relies on Docker, so before installingthe pybacktrack docker image, ensure you have installed Docker.

Note

Once Docker is installed, open a Docker terminal (command-line interface).

Note

To install the pybacktrack docker image, type:

To run the docker image:

Note

On Linux systems you may have to use sudo when running docker commands. For example:

From the current working directory you can run the backtracking example below,or any other examples in this documentation. For example, you could run:

If you wish to run the example notebooksthen there is a notebook.sh script to start a Jupyter notebook server in the running docker container:

Then you can start a web browser on your local machine and type the following in the URL field:

Note

Once installed, pybacktrack is available to:

1. run built-in scripts (inside pybacktrack), or
2. importpybacktrack into your own script.

The following example is used to demonstrate both approaches. It backtracks an ocean drill site and saves the output to a text file by:

• reading the ocean drill site file pybacktrack_examples/test_data/ODP-114-699-Lithology.txt,

  Note

  However if you have your own ocean drill site file then you can substitute it in the example below if you want.

• backtracking it using:

  • the M2 dynamic topography model, and
  • the Haq87_SealevelCurve_Longterm sea-level model,

• writing the amended drill site to ODP-114-699_backtrack_amended.txt, and

• writing the following columns to ODP-114-699_backtrack_decompat.txt:

  • age
  • compacted_depth
  • compacted_thickness
  • decompacted_thickness
  • decompacted_density
  • decompacted_sediment_rate
  • decompacted_depth
  • dynamic_topography
  • water_depth
  • tectonic_subsidence
  • lithology

Since there is a backtrack module inside pybacktrack that can be run as a script,we can invoke it on the command-line using python-mpybacktrack.backtrack_cli followed by command line options that are specific to that module.This is the easiest way to run backtracking.

To see its command-line options, run:

The backtracking example can now be demonstrated by running the script as:

An alternative to running a built-in script is to write your own script (using a text editor) that imports pybacktrack andcalls its functions. You might do this if you want to combine pyBacktrack functionality with other research functionality into a single script.

The following Python code does the same as the built-in script by calling thepybacktrack.backtrack_and_write_well() function:

If you save the above code to a file called my_backtrack_script.py then you can run it as:

Backtracking is a general algorithm for finding all (or some) 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][2]

The classic textbook example of the use of backtracking is the eight queens puzzle, that asks for all arrangements of eight chessqueens 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,^[3] such as crosswords, verbal arithmetic, Sudoku, and many other puzzles. It is often the most convenient (if not the most efficient^{[citation needed]}) technique for parsing,^[4] for the knapsack problem and other combinatorial optimization problems. It is also the basis of the so-called logic programming languages such as 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.^[5] The pioneer string-processing language SNOBOL (1962) may have been the first to provide a built-in general backtracking facility.

Description of the method[edit]

The backtracking algorithm enumerates a set of partial candidates that, in principle, could be completed in various ways to give all the possible solutions to the given problem. The completion is done incrementally, by a sequence of candidate extension steps.

Conceptually, the partial candidates are represented as the nodes of a tree structure, the potential search tree. Each partial candidate is the parent of the candidates that differ from it by a single extension step; the leaves of the tree are the partial candidates that cannot be extended any further.

The backtracking algorithm traverses this search tree recursively, from the root down, in depth-first order. At each node c, the algorithm checks whether c can be completed to a valid solution. If it cannot, the whole sub-tree rooted at c is skipped (pruned). Otherwise, the algorithm (1) checks whether c itself is a valid solution, and if so reports it to the user; and (2) recursively enumerates all sub-trees of c. The two tests and the children of each node are defined by user-given procedures.

Therefore, the actual search tree that is traversed by the algorithm is only a part of the potential tree. The total cost of the algorithm is the number of nodes of the actual tree times the cost of obtaining and processing each node. This fact should be considered when choosing the potential search tree and implementing the pruning test.

Pseudocode[edit]

Cached

In order to apply backtracking to a specific class of problems, one must provide the data P for the particular instance of the problem that is to be solved, and six procedural parameters, root, reject, accept, first, next, and output. These procedures should take the instance data P as a parameter and should do the following:

1. root(P): return the partial candidate at the root of the search tree.
2. reject(P,c): return true only if the partial candidate c is not worth completing.
3. accept(P,c): return true if c is a solution of P, and false otherwise.
4. first(P,c): generate the first extension of candidate c.
5. next(P,s): generate the next alternative extension of a candidate, after the extension s.
6. output(P,c): use the solution c of P, as appropriate to the application.

The backtracking algorithm reduces the problem to the call bt(root(P)), where bt is the following recursive procedure:

Usage considerations[edit]

The reject procedure should be a boolean-valued function that returns true only if it is certain that no possible extension of c is a valid solution for P. If the procedure cannot reach a definite conclusion, it should return false. An incorrect true result may cause the bt procedure to miss some valid solutions. The procedure may assume that reject(P,t) returned false for every ancestor t of c in the search tree.

On the other hand, the efficiency of the backtracking algorithm depends on reject returning true for candidates that are as close to the root as possible. If reject always returns false, the algorithm will still find all solutions, but it will be equivalent to a brute-force search.

The accept procedure should return true if c is a complete and valid solution for the problem instance P, and false otherwise. It may assume that the partial candidate c and all its ancestors in the tree have passed the reject test.

The general pseudo-code above does not assume that the valid solutions are always leaves of the potential search tree. In other words, it admits the possibility that a valid solution for P can be further extended to yield other valid solutions.

The first and next procedures are used by the backtracking algorithm to enumerate the children of a node c of the tree, that is, the candidates that differ from c by a single extension step. Balenaetcher 1 5 65 equals. The call first(P,c) should yield the first child of c, in some order; and the call next(P,s) should return the next sibling of node s, in that order. Both functions should return a distinctive 'NULL' candidate, if the requested child does not exist.

Together, the root, first, and next functions define the set of partial candidates and the potential search tree. They should be chosen so that every solution of P occurs somewhere in the tree, and no partial candidate occurs more than once. Moreover, they should admit an efficient and effective reject predicate.

Early stopping variants[edit]

The pseudo-code above will call output for all candidates that are a solution to the given instance P. The algorithm can be modified to stop after finding the first solution, or a specified number of solutions; or after testing a specified number of partial candidates, or after spending a given amount of CPU time.

Examples[edit]

Examples where backtracking can be used to solve puzzles or problems include:

• Puzzles such as eight queens puzzle, crosswords, verbal arithmetic, Sudoku^{[nb 1]}, and Peg Solitaire.
• Combinatorial optimization problems such as parsing and the knapsack problem.
• Logic programming languages such as Icon, Planner and Prolog, which use backtracking internally to generate answers.

The following is an example where backtracking is used for the constraint satisfaction problem:

Constraint satisfaction[edit]

The general constraint satisfaction problem consists in finding a list of integers x = (x[1], x[2], …, x[n]), each in some range {1, 2, …, m}, that satisfies some arbitrary constraint (boolean function) F.

For this class of problems, the instance data P would be the integers m and n, and the predicate F. In a typical backtracking solution to this problem, one could define a partial candidate as a list of integers c = (c[1], c[2], …, c[k]), for any k between 0 and n, that are to be assigned to the first k variables x[1], x[2], …, x[k]. The root candidate would then be the empty list (). The first and next procedures would then be

Here length(c) is the number of elements in the list c.

The call reject(P, c) should return true if the constraint F cannot be satisfied by any list of n integers that begins with the k elements of c. For backtracking to be effective, there must be a way to detect this situation, at least for some candidates c, without enumerating all those m^{n − k}n-tuples.

For example, if F is the conjunction of several boolean predicates, F = F[1] ∧ F[2] ∧ … ∧ F[p], and each F[i] depends only on a small subset of the variables x[1], …, x[n], then the reject procedure could simply check the terms F[i] that depend only on variables x[1], …, x[k], and return true if any of those terms returns false. In fact, reject needs only check those terms that do depend on x[k], since the terms that depend only on x[1], …, x[k − 1] will have been tested further up in the search tree.

Assuming that reject is implemented as above, then accept(P, c) needs only check whether c is complete, that is, whether it has n elements.

It is generally better to order the list of variables so that it begins with the most critical ones (i.e. the ones with fewest value options, or which have a greater impact on subsequent choices).

One could also allow the next function to choose which variable should be assigned when extending a partial candidate, based on the values of the variables already assigned by it. Further improvements can be obtained by the technique of constraint propagation.

In addition to retaining minimal recovery values used in backing up, backtracking implementations commonly keep a variable trail, to record value change history. An efficient implementation will avoid creating a variable trail entry between two successive changes when there is no choice point, as the backtracking will erase all of the changes as a single operation.

An alternative to the variable trail is to keep a timestamp of when the last change was made to the variable. Twixl publisher pro plug in for adobe indesign 10 6. The timestamp is compared to the timestamp of a choice point. If the choice point has an associated time later than that of the variable, it is unnecessary to revert the variable when the choice point is backtracked, as it was changed before the choice point occurred.

See also[edit]

Notes[edit]

References[edit]

1. ^Donald E. Knuth (1968). The Art of Computer Programming. Addison-Wesley.
2. ^Gurari, Eitan (1999). 'CIS 680: DATA STRUCTURES: Chapter 19: Backtracking Algorithms'. Archived from the original on 17 March 2007.
3. ^A. Biere; M. Heule; H. van Maaren (29 January 2009). Handbook of Satisfiability. IOS Press. ISBN978-1-60750-376-7.
4. ^Des Watson (22 March 2017). A Practical Approach to Compiler Construction. Springer. ISBN978-3-319-52789-5.
5. ^Rossi, Francesca; Beek, Peter Van; Walsh, Toby (August 2006). 'Constraint Satisfaction: An Emerging Paradigm'. Handbook of Constraint Programming. Foundations of Artificial Intelligence. Amsterdam: Elsevier. p. 14. ISBN978-0-444-52726-4. Retrieved 2008-12-30. Bitner and Reingold credit Lehmer with first using the term 'backtrack' in the 1950s.

Further reading[edit]

• Gilles Brassard, Paul Bratley (1995). Fundamentals of Algorithmics. Prentice-Hall.

External links[edit]

• HBmeyer.de, Interactive animation of a backtracking algorithm
• Solving Combinatorial Problems with STL and Backtracking, Article and C++ source code for a generic implementation of backtracking