cs353 10 backtracking
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BacktrackingTRANSCRIPT
ICS 353: Design and Analysis of Algorithms
Backtracking
King Fahd University of Petroleum & MineralsKing Fahd University of Petroleum & Minerals
Information & Computer Science DepartmentInformation & Computer Science Department
BacktrackingICS 353: Design and Analysis of Algorithms
Reading Assignment• M. Alsuwaiyel, Introduction to Algorithms:
Design Techniques and Analysis, World Scientific Publishing Co., Inc. 1999.• Chapter 13, Sections 1, 2 and 4.
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BacktrackingICS 353: Design and Analysis of Algorithms
Coping with Hard Problems• There are three useful methodologies that could be used to
cope with problems having no efficient algorithm to solve: • A methodic examination of the implicit state space induced by the
problem instance under study.• suitable for problems that exhibit good average time complexity.
• A probabilistic notion of accuracy where a solution is a simple decision maker or test that can accurately perform one task (either passing or failing the alternative) and not say much about the complementary option. An iteration through this test will enable the construction of the solution or the increase in the confidence level in the solution to the desired degree.
• Obtain an approximate solution• Only some classes of hard problems admit such polynomial time
approximations.
• Backtracking belongs to the first category.
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BacktrackingICS 353: Design and Analysis of Algorithms
Backtracking
• A systematic technique of searching• To reduce the search space• Can be considered as an “organized” exhaustive
search
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BacktrackingICS 353: Design and Analysis of Algorithms
3-Coloring Problem
• Optimization Problem• Input: G = (V, E), an undirected graph with n vertices and
m edges.
• Output: A 3-coloring of G, if possible.
• A coloring can be represented as an n-tuple (c1, c2, ..., cn)
• The number of different possible colorings for a graph is ...........
• These possibilities can be represented as a complete ternary tree
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BacktrackingICS 353: Design and Analysis of Algorithms
3-Coloring: Search Tree
Search tree of all possible colorings of a graph with 4 vertices
(When could this be for a graph with 5 vertices?)
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BacktrackingICS 353: Design and Analysis of Algorithms
Example
a
b c
d e
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BacktrackingICS 353: Design and Analysis of Algorithms
Example
• In the example• Nodes are generated in a depth-first search
manner• No need to store the whole search tree, just the
current active path• What is the time complexity of the algorithm in
the worst case
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BacktrackingICS 353: Design and Analysis of Algorithms
Algorithm 3-ColorRec
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BacktrackingICS 353: Design and Analysis of Algorithms
Algorithm 3-ColorIter
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BacktrackingICS 353: Design and Analysis of Algorithms
The General Backtracking Method• Assume that the solution is of the form (x1, x2,
..., xi) where 0 i n and n is a constant that depends on the problem formulation.• Here, the solution is assumed to satisfy certain
constraints.• i in the case of the 3-coloring problem is fixed.• i may vary from one solution to another.
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BacktrackingICS 353: Design and Analysis of Algorithms
The General Backtracking Method: Example
• Consider the following version of the Partition Problem
Input: X = {x1, x2, ..., xn} a set of n integers, and an integer y.
Output: Find a subset Y X such that the sum of its elements is equal to y.• For example, consider X = {10, 20, 30, 40, 50, 60}
and y = 50. There is more than one solution to this problem: What are they and what is their length?
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BacktrackingICS 353: Design and Analysis of Algorithms
The General Backtracking Algorithm (1)• Each xi in the solution vector belongs to a finite linearly
ordered set Xi.• The backtracking algorithm considers the elements of
the Cartesian product X1 X2 ... Xn in lexicographic order.
• Initially starting with the empty vector.• Suppose that the algorithm has detected the partial
solution (x1, x2, ..., xj). It then considers the vector v = (x1, x2, ..., xj, xj+1). We have the following cases:
1. If v represents a final solution to the problem, the algorithm records it as a solution and either terminates in case only one solution is desired or continues to find other solutions.
2. (The advance Step): If v represents a partial solution, the algorithm advances by choosing the least element in the set Xj+2.
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BacktrackingICS 353: Design and Analysis of Algorithms
The General Backtracking Algorithm (2)
3. If v is neither a final nor a partial solution, we have two sub-cases:
a. If there are still more elements to choose from in the set Xj+1, the algorithm sets xj+1 to the next member of Xj+1.
b. (The Backtrack Step): If there are no more elements to choose from in the set Xj+1, the algorithm backtracks by setting xj to the next member of Xj . If again there are no more elements to choose from in the set Xj , the algorithm backtracks by setting xj1 to the next member of Xj1, and so on.
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BacktrackingICS 353: Design and Analysis of Algorithms
BacktrackRec
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BacktrackingICS 353: Design and Analysis of Algorithms
BacktrackIter
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