Algorithm Paradigms

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Algorithm Paradigms
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Algorithm Paradigms

• Search (BFS, DFS)
• often good when guided by a heuristic
• Divide and Conquer
• Good when subproblems are independent
• Randomization
• Good when any choice is equally good

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Algorithm Paradigms

• Greedy Algorithms
• Dijkstras Algorithm for shortest distance
• Dynamic Programming
• Longest common subsequence
• (programming isnt computer programming)

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Greedy Algorithm

• A greedy approach does not always lead to an
  optimal solution
• Appropriate if a good local choice is also a good
  global choice
• Examples
• Huffman coding (computing optimal code)
• Kruskals algorithm

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Dynamic Programming

• Simple subproblems
• Subproblem optimization
• Subproblem overlap
• Optimal solutions to subproblems are computed
  only once, and stored in a table
• consider the space complexity

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Knapsack problem

• n items, each worth vi and weighing wi
• Goal is to fill knapsack of capacity W pounds
  with maximum value of items
• 0 1 knapsack problem include or reject each
  item
• fractional knapsack problem can include
  fractional amounts of each item

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Fractional Knapsack problem

• Greedy algorithm
• Compute each items value per pound vi / wi
• Sort items by value per pound
• Fill knapsack by taking most valuable item first,
  until knapsack is full

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0 1 Knapsack problem

• Greedy approach is not optimal
• Item 1 value 5 weight 3 pounds 1.67
• Item 2 value 3.1 weight 2 pounds 1.55
• Item 3 value 2 weight 2 pounds 1.00
• Knapsack holds 4 pounds

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0 1 Knapsack problem

• Exhaustive enumeration (brute-force approach)
  isnt too speedy
• There are n items, so 2n subsets.
• Dont want to check each one!

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0 1 Knapsack problem

• Dynamic programming approach
• Build a 2-D table Ti,j
• i means using items 1 i ( 1 i n )
• j means max capacity ( 1 j w )
• Ti,j is the maximum value obtainable with items
  1 through i weighing at most j
• Ti,j max (Ti-1,j , vi Ti-1, j-wi )
• max ( dont include item i, or do and optimize
  remaining space in knapsack)

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0 1 Knapsack problem

• Ti,j max (Ti-1,j , vi Ti-1, j-wi )
• Fill in i 1 row with vi if wi gt j
• T , lt 0 is 0

j 1 2 3 4 1 0 0 5 5
i 2 0 3.1 5 5 3 0 3.1 5 5.1