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Efficient Algorithms Lecture: Divide and Conquer

Explore greedy algorithms, dynamic programming, problem-solving techniques, and analysis for optimizing runtime. Discover the power of recursion and memory utilization in algorithm design.

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Efficient Algorithms Lecture: Divide and Conquer

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  1. Lecture 34 CSE 331 Nov 18, 2011

  2. HW 9 due today Q1, Q 2 and Q3 in separate piles I will not take any HW after 1:15pm

  3. HW related stuff Solutions to HW 9 at the end of the lecture Graded HW 8 will be available for pickup form Monday No HW handed out today!

  4. Lecture on Monday I’ll be out of town My student Swapnoneel will cover for me

  5. A preview of what I’m up to List Decoding: The master of disguise Practice talk: 3:15pm, Friday, Commons 9

  6. Need volunteers for today/Mon

  7. Dividing up P R Q First n/2 points according to the x-coord

  8. Recursively find closest pairs R Q δ = min (blue, green)

  9. Pruning out some points δ δ >δ R Q > δ > δ δ = min (blue, green)

  10. All we have to do now δ δ R Q S Figure if a pair in S has distance <δ δ = min (blue, green)

  11. Kickass Property Lemma If s ≠ s’ in S have d(s,s’) ≤ δ, then s and s’ are within 15 positions of each other in Sy

  12. Today’s Agenda Prove the kickass property lemma

  13. HW 9 due today Q1, Q 2 and Q3 in separate piles I will not take any HW after 1:15pm

  14. High level view of CSE 331 Problem Statement Problem Definition Three general techniques Algorithm “Implementation” Data Structures Analysis Correctness+Runtime Analysis

  15. Greedy Algorithms Natural algorithms Reduced exponential running time to polynomial

  16. Divide and Conquer Recursive algorithmic paradigm Reduced large polynomial time to smaller polynomial time

  17. A new algorithmic technique Dynamic Programming

  18. Dynamic programming vs. Divide & Conquer

  19. Same same because Both design recursive algorithms

  20. Different because Dynamic programming is smarter about solving recursive sub-problems

  21. Using Memory to be smarter Pow (a,n) Pow (a,n) // n is even and ≥ 2 // n is even and ≥ 2 t= Pow(a,n/2) return Pow(a,n/2) * Pow(a, n/2) return t * t O(n) as we recompute! O(logn) as we compute only once

  22. End of Semester blues Can only do one thing at any day: what is the optimal schedule to obtain maximum value? Write up a term paper (10) Party! (2) Exam study (5) 331 HW (3) Project (30) Monday Tuesday Wednesday Thursday Friday

  23. Previous Greedy algorithm Order by end time and pick jobs greedily Greedy value = 5+2+3= 10 Write up a term paper (10) OPT = 30 Party! (2) Exam study (5) 331 HW (3) Project (30) Monday Tuesday Wednesday Thursday Friday

  24. Today’s agenda Formal definition of the problem Start designing a recursive algorithm for the problem

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