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Introduction to Algorithms: Course Overview and Stable Matching

This lecture introduces the Algorithms course and covers topics such as mechanics, weekly homework, exams, and algorithm design. A specific algorithm, Stable Matching, is discussed as an introductory problem, including its formal notions and an intuitive idea for an algorithm. The lecture concludes with a discussion of the termination and stability of the algorithm.

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Introduction to Algorithms: Course Overview and Stable Matching

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  1. CSE 421Algorithms Richard Anderson Lecture 1

  2. Course Introduction • Instructor • Richard Anderson, anderson@cs.washington.edu • Teaching Assistant • Yiannis Giotas, giotas@cs.washington.edu

  3. All of Computer Science is the Study of Algorithms

  4. Mechanics • It’s on the web • Weekly homework • Midterm • Final exam • Subscribe to the mailing list

  5. Text book • Algorithm Design • Jon Kleinberg, Eva Tardos • Read Chapters 1 & 2

  6. How to study algorithms • Zoology • Mine is faster than yours is • Algorithmic ideas • Where algorithms apply • What makes an algorithm work • Algorithmic thinking

  7. Introductory Problem:Stable Matching • Setting: • Assign TAs to Instructors • Avoid having TAs and Instructors wanting changes • E.g., Prof A. would rather have student X than her current TA, and student X would rather work for Prof A. than his current instructor.

  8. Formal notions • Perfect matching • Ranked preference lists • Stability m1 w1 m2 w2

  9. m1: w1 w2 m2: w2 w1 w1: m1 m2 w2: m2 m1 m1: w1 w2 m2: w1 w2 w1: m1 m2 w2: m1 m2 Examples

  10. m1: w1 w2 m2: w2 w1 w1: m2 m1 w2: m1 m2 Examples

  11. Intuitive Idea for an Algorithm • m proposes to w • If w is unmatched, w accepts • If w is matched to m2 • If w prefers m to m2, w accepts • If w prefers m2 to m, w rejects • Unmatched m proposes to highest w on its preference list

  12. Algorithm Initially all m in M and w in W are free While there is a free m w highest on m’s list that m has not proposed to if w is free, then match (m, w) else suppose (m2, w) is matched if w prefers m to m2 unmatch (m2, w) match (m, w)

  13. Does this work? • Does it terminate? • Is the result a stable matching? • Begin by identifying invariants and measures of progress • m’s proposals get worse • Once w is matched, w stays matched • w’s partners get better

  14. Claim: The algorithms stops in at most n2 steps • Why?

  15. The algorithm terminates with a perfect matching • Why?

  16. The resulting matching is stable • Suppose • m1 prefers w2 to w1 • w2 prefers m1 to m2 • How could this happen? m1 w1 m2 w2

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