Announcements Its on the web. Homework 1, Due October 4 CSE 421 - PDF document

Announcements • It’s on the web. • Homework 1, Due October 4 CSE 421 – It’s on the web Algorithms • Subscribe to the mailing list Richard Anderson • Richard’s office hours: Autumn 2006 – Tuesday, 2:30-3:20 pm, Friday, 2:30-3:20 pm. Lecture 2 • Ning’s office hours: – Monday, 12:30-1:20 pm, Tuesday, 4:30-5:20 pm. A closer look Algorithm under specified • Stable matchings are • Many different ways of picking m’s to propose not necessarily fair • Surprising result – All orderings of picking free m’s give the same result m 1 w 1 m 1 : w 1 w 2 w 3 m 2 : w 2 w 3 w 1 • Proving this type of result m 2 w 2 m 3 : w 3 w 1 w 2 – Reordering argument – Prove algorithm is computing something mores m 3 w 3 specific w 1 : m 2 m 3 m 1 • Show property of the solution – so it computes a specific w 2 : m 3 m 1 m 2 stable matching w 3 : m 1 m 2 m 3 How many stable matchings can you find? Proposal Algorithm finds the best Proof possible solution for M • Formalize the notion of best possible solution • See the text book – pages 9 – 12 • (m, w) is valid if (m, w) is in some stable • Related result: Proposal algorithm is the matching worst case for W • best(m): the highest ranked w for m such that • Algorithm is the M-optimal algorithm (m, w) is valid • Proposal algorithms where w’s propose is • S* = {(m, best(m)} W-Optimal • Every execution of the proposal algorithm computes S* 1

Best choices for one side are bad But there is a stable second choice for the other • Design a configuration for m 1 : • Design a configuration for m 1 : problem of size 4: problem of size 4: m 2 : m 2 : – M proposal algorithm: – M proposal algorithm: m 3 : m 3 : • All m’s get first choice, all w’s • All m’s get first choice, all w’s get last choice get last choice m 4 : m 4 : – W proposal algorithm: – W proposal algorithm: • All w’s get first choice, all m’s • All w’s get first choice, all m’s w 1 : w 1 : get last choice get last choice – There is a stable matching w 2 : w 2 : where everyone gets their second choice w 3 : w 3 : w 4 : w 4 : Key ideas M-rank and W-rank of matching • Formalizing real world problem • m-rank: position of m 1 : w 1 w 2 w 3 m 1 w 1 matching w in – Model: graph and preference lists m 2 : w 1 w 3 w 2 preference list – Mechanism: stability condition m 3 : w 1 w 2 w 3 m 2 w 2 • M-rank: sum of m- • Specification of algorithm with a natural w 1 : m 2 m 3 m 1 ranks operation w 2 : m 3 m 1 m 2 – Proposal • w-rank: position of w 3 : m 3 m 1 m 2 m 3 w 3 matching m in • Establishing termination of process through preference list invariants and progress measure What is the M-rank? • W-rank: sum of w- • Under specification of algorithm ranks • Establishing uniqueness of solution What is the W-rank? Suppose there are n m’s, and n w’s Random Preferences • What is the minimum possible M-rank? Suppose that the preferences are completely random • What is the maximum possible M-rank? m 1 : w 8 w 3 w 1 w 5 w 9 w 2 w 4 w 6 w 7 w 10 m 2 : w 7 w 10 w 1 w 9 w 3 w 4 w 8 w 2 w 5 w 6 … w 1 : m 1 m 4 m 9 m 5 m 10 m 3 m 2 m 6 m 8 m 7 • Suppose each m is matched with a w 2 : m 5 m 8 m 1 m 3 m 2 m 7 m 9 m 10 m 4 m 6 random w, what is the expected M-rank? … If there are n m’s and n w’s, what is the expected value of the M-rank and the W-rank when the proposal algorithm computes a stable matching? 2

Expected Ranks Expected M rank • Expected M rank • Expected M rank is the number of steps until all M’s are matched – (Also is the expected run time of the algorithm) • Expected W rank • Each steps “selects a w at random” – O(n log n) total steps – Average M rank: O(log n) Guess – as a function of n Expected W-rank Probabilistic analysis • If a w receives k random proposals, the • Select items with replacement from a set expected rank for w is n/(k+1). of size n. What is the expected number of items to be selected until every item has been selected at least once. • On the average, a w receives O(log n) proposals • Choose k values at random from the – The average w rank is O(n/log n) interval [0, 1). What is the expected size of the smallest item. What is the run time of the Stable O(1) time per iteration Matching Algorithm? • Find free m Initially all m in M and w in W are free Executed at most n 2 times While there is a free m • Find next available w w highest on m’s list that m has not proposed to if w is free, then match (m, w) • If w is matched, determine m 2 else • Test if w prefer m to m 2 suppose (m 2 , w) is matched if w prefers m to m 2 • Update matching unmatch (m 2 , w) match (m, w) 3

What does it mean for an algorithm to be efficient? 4