Backtracking

1
Backtracking

• Reading Material Chapter 13, Sections 1, 2, 4,
  and 5.

2
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.
• We will study the first one.

3
Backtracking

• A systematic technique of searching
• To reduce the search space
• Can be considered as an organized exhaustive
  search

4
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

5
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?)
6
Example
a
b
c
d
e
7
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

8
Algorithm 3-ColorRec
9
Algorithm 3-ColorIter
3
10
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.

11
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 60. There is more than one solution
  to this problem What are they and what is their
  length?

12
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, xj1).
  We have the following cases
• 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.
• (The advance Step) If v represents a partial
  solution, the algorithm advances by choosing the
  least element in the set Xj2.

13
The General Backtracking Algorithm (2)

• If v is neither a final nor a partial solution,
  we have two sub-cases
• If there are still more elements to choose from
  in the set Xj1, the algorithm sets xj1 to the
  next member of Xj1.
• (The Backtrack Step) If there are no more
  elements to choose from in the set Xj1, 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 xj??1 to the next
  member of Xj?1, and so on.

14
BacktrackRec
15
BacktrackIter