Learning Activity: Backtracking

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Learning Activity Backtracking

• Lecture 27 Learning Activity
• Hand back Midterm
• Sum of Subsets Problem
• Knapsack revisited

• Lecture 25 Backtracking
• n-Queens example
• Formulation on a Tree
• Search
• Node Generation
• Solution Check
• Backtracking
• Node Generation
• Bounding Function
• Sum of Subsets problem

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Appeals Policy

• The purpose of appeals for grades is to ensure
• 1) That our grading template is applied equally
  and fairly from person to person, and
• 2) That feedback about the grading template
  itself is encouraged to make sure the template is
  representative of a students ability to
  internalize and use the material we learned in
  class (not tricks or material outside the
  course mainstream).
• Review your test. Understand correct solutions
  to each problem and compare/contrast them with
  your solution. Understand how points were
  assigned to concepts and ensure any concepts you
  demonstrated earned the points they deserve.
  Highlight discrepancies in a written appeal. You
  may also write an appeal suggesting another
  template assigning points to concepts and arguing
  its superiority over the existing template.
• Return your tests, with or without appeals, on
  Friday so grades can be recorded. No appeals
  will be considered later.

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Sum of Subsets Problem

• Given a set of integers and a value
• Find all subsets of those integers that sum to
  the value.
• Example Given 1,2,4,6 and 7.
• Answer 1,2,4, 1,6
• Since 124 7 and 16 7

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Points to Ponder

• How would you formulate the sum of subsets
  problem?
• What would be a good bounding function?
• Does the order in which one considers elements
  for inclusion matter?
• Come prepared to share and discuss your
  backtracking solution.

• Lets do it!

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Sum of Subsets
One formulation for sets of positive integers
Given a set w1, w2, w3, w4, let x 1 0 1 1
represent w1, w3, w4.
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Sum of Subsets
One formulation for sets of positive integers
Generating the next xi with this representation
is trivial either 1 or 0.
Is there a function of x1,,xk that can reveal
dead ends?
Given a set w1, w2, w3, w4, let x 1 0 1 1
represent w1, w3, w4.
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Sum of Subsets
One formulation for sets of positive integers
Generating the next xi with this representation
is trivial either 1 or 0.
Given a set w1, w2, w3, w4, let x 1 0 1 1
represent w1, w3, w4.
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Sum of Subsets
One formulation for sets of positive integers
If w1ltw2ltltwn, then
Given a set w1, w2, w3, w4, let x 1 0 1 1
represent w1, w3, w4.
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Sum of Subsets

• procedure sumofsubs(s, k, r)
• // s sum(j1 to k-1)w(j)x(j) and r sum(jk
  to n)w(j)
• // w(j)s are in nondecreasing order.
• // It is assumed that w(1) ltM and sum(j1 to
  n)w(j)gtM
• // generate left child. Note that sw(k)ltM
  because Bk-1 true
• X(k)?1
• if sw(k)M
• then print x(j), j?1,...,k
• else if sw(k)w(k1)ltM //sw(k)ltM and
  srgtM?r!w(k), so k1ltn
• //note other part of B is satisfied,
  dont need to check
• then call sumofsubs(sw(k), k1, r-w(k))
• // generate right child and evaluate Bk
• if sr-w(k)gtM and sw(k1)ltM // Bk true
• then x(k)?0
• call sumofsubs(s, k1, r-w(k))

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Sum of Subsets Example
w 5, 10, 12, 13, 15, 18 and M30
s,k,r
Key
0,1,73
5,2,68
0,2,68
15,3,58
5,3,58
10,3,58
0,3,58
27,4,46
15,4,46
17,4,46
5,4,46
10,4,46
12,4,46
0,4,46
10,5,33
12,5,33
13,5,33
0,5,33
15,5,33
5,5,33
20,6,18
12,6,18
13,6,18
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Knapsack Revisited

• Consider 0-1 knapsack problem
• What does the state-space tree look like?
• What would a good bounding function be?
• Book considers a variation for types of
  objectsshould have read through chapter 9 by
  now.
• Come to class prepared to discuss your solution