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Greedy Algorithms

Greedy Algorithms. Peter Schröder. Greedy Algorithms. Optimization problems Example: dynamic programming Considers all possible solutions Exploits optimal subproblem property Common features? Choice of steps along the way What about always picking the best one at any given step?

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Greedy Algorithms

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  1. Greedy Algorithms Peter Schröder

  2. Greedy Algorithms • Optimization problems • Example: dynamic programming • Considers all possible solutions • Exploits optimal subproblem property • Common features? • Choice of steps along the way • What about always picking the best one at any given step? • Greedy algorithms

  3. Greedy Algorithms • Choice • Take best choice at a given moment • Locally optimal • Globally optimal? • For some problems: yes • Example problem • Activity selection

  4. Activity Selection • Problem statement • Given a resource and several competing activities drawing on this resource, how should they be scheduled so that the largest number of activities can be performed? • Scheduling a lecture hall for a number of different classes • Get the most classes scheduled • NOT the most hours… just the most classes • Suggestions…

  5. Activity Selection • Formal setup • Set of activities • Each activity has start time and finish time • Activities are compatible if • Find maximal subset of S such that its members are compatible • Assume activities ordered by finish time

  6. Algorithm • Greedy-Activity-Selection(s,f) • n := length[s] • a := {1} • j := 1 • For i:=2 to n • do if si >= fj • then a := a + {i} • j := I • Return a

  7. Example Run

  8. Analysis • Greedy strategy • What is it here? • What does it maximize? • Correctness proof • Some optimal solution contains element 1 • Remaining problem is subset of original problem

  9. Runtime • What is the runtime of Greedy-Activity-Selection? • What about a dynamic programming approach to this problem? • Give an algorithm • What is its running time?

  10. When does it work? • Elements of the greedy strategy • No general strategy, but… • Greedy choice property • Optimal substructure property

  11. Greedy Choice Property • Locally optimal choices • Make best choice available at a given moment • Solve subproblems after choice has been made • Contrast with dynamic programming • Choice at a given step may depend on solutions to subproblems

  12. Optimal Substructure • When does a problem exhibit optimal substructure? • If an optimal solution to the entire problem contains within it optimal solutions to subproblems • But this is also true of dynamic programming… • How to tell the difference?

  13. Greedy vs. Dynamic • Knapsack problem • 0-1 version • Include or don’t include • Fractional version • May take fraction of an item • Problem statement • Fill knapsack with items of weight wi and value vi; maximize value in knapsack

  14. Knapsack Problem • Optimal substructure property • Take out item from optimal solution and prove that remaining problem is still optimal

  15. 20 30 50 10 60$ 120$ 100$ Knapsack Problem • Solution • Greedy strategy works for fractional version • Prove it • Need dynamic programming for o-1 version

  16. Another Example • Huffman codes • Efficient encoding of a finite alphabet and a given distribution of occurances

  17. Prefix Codes • Desirable for coding • Optimal compression can always be achieved by a prefix code • No codeword is a prefix of another codeword • Easy on the decoder • Representation of the code book • Suggestions?

  18. Optimal Encoding • Finding optimal coding • Search over all complete binary trees with |C| leaves • Cost function • Algorithm • Suggestions?

  19. Huffman’s Algorithm Huffman(C) n := |C| Q := C For i := 1 to n-1 do z := AllocNode() x := left[z] := ExtractMIN(Q) y := right[z] := ExtractMIN(Q) f[z] := f[x] + f[y] Insert(Q,z) Return ExtractMIN(Q)

  20. Analysis • Running time • Need Heap data structure • Remember properties • Cost for construction • Cost for extraction • Cost for insertion

  21. Analysis • Correctness • Greedy choice property • Optimal substructure • Lemma • Let X and Y be two lowest frequency characters; then there exists and optimal prefix code for C in which X and Y have the same length and differ only in the last bit

  22. Proof of Lemma • Idea: Modify existing optimal tree • Assume B and C are siblings at lowest depth in tree • Assume wlog that • Exchange X with B and Y with C • Show that cost of new tree must be the same as cost of old tree • Assertion of Lemma follows

  23. Analysis • What is the greedy choice property here? • Cost of tree is cost of all mergers • Greedy choice is merger of least cost, i.e., lowest frequency symbols at the time • Optimal substructure? • Lemma: Let T be a full binary tree representing an optimal prefix code; consider any two leaves X and Y and parent Z; consider Z as a character and replace it accordingly; the new tree is optimal

  24. Proof of Lemma • Consider component costs • hence • the assertion follows by contradiction • Correctness of Huffman • follows from Lemmas

  25. Questions • Describe tree • for encoding a file, must also encode structure of tree; how many bits? • Can you do better? • will Huffman always give the best possible encoding? • how about a file consisting of random characters?

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