Intro to Computer Algorithms Lecture 18

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Intro to Computer Algorithms Lecture 18

• Phillip G. Bradford
• Computer Science
• University of Alabama

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Announcements

• Advisory Boards Industrial Talk Series
• http//www.cs.ua.edu/9IndustrialSeries.shtm
• 2-Dec Mike Thomas, CIO, Gulf States Paper
• Next Research Colloquia
• Prof. Nael Abu-Ghazaleh
• 10-Nov _at_ 1100am
• Active Routing in Mobile Ad Hoc Networks

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Computer Security Research Group

• Meets every Friday from 1100 to 1200
• In 112 Houser
• Computer Security, etc.
• Email me to be on the mailing list!

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CS Story Time

• Prof. Jones research group
• See http//cs.ua.edu/StoryHourSlide.pdf

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Next Midterm

• Tuesday before Thanksgiving !
• 25-November

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Outline

• Complete Dynamic Programming
• Principle of Optimality
• Start the Greedy Design Paradigm

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Dynamic Programming

• Principle of optimality
• Consider any well-formed substructure s of an
  optimal structure S
• Then s is optimal itself.
• This allows the combination of well-formed
  substructures into optimal superstructures

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Knapsack Problem

• Section 3.4 from the book

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Knapsack Problem

• Suppose the knapsacks capacity is
• W10
• What subsets can we choose within the knapsacks
  capacity while maximizing the subsets value?
• Note 3,4 is optimal
• Does not include item 1 (of the highest value)

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Apparent worst-case?

• Try all subsets!
• Can we find a better algorithm?
• It is not clear!

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Other than exhaustive search?

• Dynamic programming
• Vi,j is the optimal solution for
• w1,,wi (having values v1,,vi)
• And capacity j W gt j gt 1.
• Want to find Vn,W
• Suppose W is a base-10 integer, then how large is
  this table???

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DP Recurrence for Knapsack

• Can we fit the ith item?
• No, then optimal solution is Vi-1,j
• Yes, then optimal solution
• Includes ith item vi Vi-1,j-wi
• Does not include the ith item Vi-1,j
• If j-wi gt 0
• Vi,j ? max Vi-1,j, vi Vi-1,j-wi

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DP Recurrence for Knapsack

• Base Cases
• V0,j 0
• No items in knapsack (with j capacity) has no
  value
• Vi,0 0
• Zero capacity, putting items 1,,i does no good

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Computing this DP Recurrence
Capacity
0,
1,
j-wi
W
0
0
0
0
0
1
0
Value
n
0
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The Greedy Paradigm

• Greedy Principle
• Start with an (optimal) solution and adding
  locally optimal structure eventually gives a
  globally optimal solution

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The Greedy Paradigm

• Change Making Problem
• Not necessarily optimal
• d1gtd2gt gt dn
• Give back correct change
• Greedy Method
• Start with largest denomination
• Exhaust it,
• On to next smaller denomination

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Prims Algorithm

• Spanning Tree of an undirected graph
• Acyclic subgraph containing all nodes
• Minimum weight spanning tree
• Weighted edges
• Among the least cost spanning trees
• The algorithm
• The inductive Proof of optimality