ECE 665 Spring 2005 Computer Algorithms

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ECE 665Spring 2005Computer Algorithms

• Dynamic Programming
• Knapsack revisited

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Recurrence equation

• Problem statement max ?i pi,
• s.to
• ?i wi ? M knapsack volume
• pi benefit of item i
• wi weight of item i
• Optimum solution at decision variable xn
• Fn(M) maxfn-1(M), fn-1(M-wn)pn)

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Recurrence equation

• For arbitrary fi(X)
• fi(X) max fi-1(X), fi-1(X-wi)pi
• If item i is not added.
• If item i is added (with benefit cost pi,
  contributing weight wi)
• We can represent solution space conveniently in
  terms of pairs (wi, pi)

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Knapsack Solution space
(pi,wi)
M6
1 2
2 3
5 4

• Purge solution (Pj, Wj) dominated by (Pk, Wk) if
• Pj? Pk and Wj? Wk

• Delete solutions with W gt M (knapsack volume
  exceeded