Knapsack III

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Knapsack III

• Lecture 20
• CS 312

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Details

• Project 1 turn in corrected code within 1 week
  of today for 66 of your missed credit
• Project 2 Comments/Questions?
• Project 3 8-12 Queens, limit on nodes visited
  rather than execution time.

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Extra Credit

• Report on what each value in a time command
  measures. gt3 pages.
• upto 50 project points
• Identify and report on an application of a
  greedy, dynamic programming or backtracking
  algorithm that was not covered in class. gt3
  pages.
• upto 50 project points

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Objectives

• Design a DFS backtracking algorithm for a variant
  of the Knapsack problem.
• Compare backtracking and dynamic programming
  solutions to Knapsack
• Compare DFS and BFS backtracking algorithms.

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YAKP

• Same scenario objects with weights and values
  and a capacity
• Maximize the profit of what goes in the knapsack
• Twist now have an unlimited supply of each type
  of object.

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Backtracking algorithm

• First cover how then cover why
• backpack (i,r) returns max profit for a
  combination of items i through numItems with max
  weight r.
• Recursively, top-down using backtracking.

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Backpack (p.308)
function backpack (items,remaining) bestSoFar
0 for choice items1 to numItems if
weightchoice lt remaining then
bestSoFar max (bestSoFar,
valuechoice backpack (choice,
remaining - weightchoice)) return bestSoFar
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Things to Think About

• Why can the algorithm recurse on only items item
  through numItems rather than items 1 through
  numitems?
• Can backtracking be done in a breadth-first
  algorithm?
• How much of the tree is in memory at any given
  time?
• Can you solve this problem using dynamic
  programming?

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function bfs-backpack (items,remaining) Queue
empty bestSoFar 0 veryBest 0
enqueue (items, remaining, bestSoFar) while
(not empty(Queue)) (choice, remaining,
bestSoFar) pop (Queue) if weightchoice
lt remaining then bestSoFar max
(bestSoFar, bestsofar valuechoice
veryBest max (bestSoFar, veryBest)
for nextChoice choice to numItems
enqueue ((nextChoice, remaining -
weightchoice, bestSoFar), Queue)
endfor endwhile return veryBest
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Things to Think About

• When is the BFS algorithm required?
• When do neither BFS nor DFS work?
• Whats the most memory used by the BFS algorithm?
  
• Which finds the solution with the smallest number
  of items?
• what if all 1 through numItems had to be
  searched?

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BFS in Real Life
Every 10,000 nodes explored
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Killing Dead Variables
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Homework

• Problem 9.42 Adapt knapsack to give the
  composition and value of the best load.
• Due November 2
• Helpful to work out some small examples first.
• Trust the recursion

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function bfs-backpack (items,remaining) Queue
empty bestSoFar 0 veryBest 0
enqueue (items, remaining, bestSoFar) while
(not empty(Queue)) (choice, remaining,
bestSoFar) pop (Queue) if weightchoice
lt remaining then bestSoFar max
(bestSoFar, bestsofar valuechoice
veryBest max (bestSoFar, veryBest)
for nextChoice choice to numItems
enqueue ((nextChoice, remaining -
weightchoice, bestSoFar), Queue)
endfor endwhile return veryBest
In Picasso, the BFS algorithm avoids storing so
much state on the stack (ie, in the SV) because
the BFS has to restart for every division at
every state. If the algorithm could save by
starting at n, then the BFS version could avoid
lots of computation by storing the division
indices on the stack. DFS has to store the
division indices on the stack, otherwise the same
sets of indices would be explored at each
backtracking step. This explodes the number of
states badly? Have to think carefully about
that. Might avoid large queues in BFS searches
by going DFS in Picasso. Very interesting.