algorithm_1introdunction.pdf

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Iris Hui-Ru JiangFall 2014 CHAPTER 1 INTRODUCTION IRIS H.-R. JIANG Outline ¨ Content: ¤ Opening: stable matching ¤ Five representative problems ¨ Reading: ¤ Chapter 1 Opening 2 Opening topic Stable Matching 3 Opening IRIS H.-R. JIANG Opening ¨ We start from an algorithmic problem that illustrates many of the themes we will be emphasizing. 1. It is motivated by some very natural and practical concerns. 2. From these, we formulate a clean and simple statement of a problem. 3. The algorithm to solve the problem is very clean as well. 4. Most of our work will be spent in proving that it is correct and giving an acceptable bound on the amount of time it takes to terminate with an answer. Opening 4

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IRIS H.-R. JIANG IfYou Are a Matchmaker… ¨ Imagine you are a matchmaker ¤ One hundred female clients, and one hundred male clients ¤ Each woman has given you a complete list of the hundred men, ordered by her preference: first choice, second choice, and so on ¤ Each man has given you a list of the women, ranked analogously ¤ Your job is to arrange one hundred happy marriages n Neither singlehood nor polygamy n Happy and stable (You don’t want them to divorce or to develop outside relationships Opening 5 IRIS H.-R. JIANG The Stable Matching Problem ¨ Given n men and n women, find a suitable matching ¤ Participants rank members of opposite gender ¤ Each man lists women in order of preference from best to worst ¤ Each woman lists men in order of preference from best to worst Opening 6 favorite least favorite 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile favorite least favorite 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile IRIS H.-R. JIANG Example (1/2) ¨ Try this way… ¤ Z-A, Y-B, X-C ¨ Q: Is assignment X-C, Y-B, Z-A stable? ¨ A: Opening 7 favorite least favorite 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile favorite least favorite 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile Bertha Amy Clare Xavier Yancey Zeus IRIS H.-R. JIANG Example (2/2) ¨ Q: Is assignment X-C, Y-B, Z-A stable? ¨ A: No. Bertha and Xavier will hook up. ¨ Q: What has gone wrong? ¨ A: The process is not self-enforcing. Opening 8 favorite least favorite 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile favorite least favorite 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile Bertha Xavier

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IRIS H.-R. JIANG TheStable Matching Problem ¨ Perfect matching: Everyone is matched monogamously. ¤ Each man gets exactly one woman ¤ Each woman gets exactly one man ¨ Stability: No incentive for some pair of participants to undermine assignment by joint action. ¤ In matching M, an unmatched pair (m, w) is unstable if man m and woman w prefer each other to current partners. ¤ Unstable pair (m, w) could each improve by eloping. ¨ Stable matching: Perfect matching without unstable pairs. ¨ The stable matching problem: Given the preference lists of n men and n women, find a stable matching if one exists. ¤ The preference lists are complete and have no ties Opening 9 IRIS H.-R. JIANG Simple but Invalid ¨ Random matching and fixing up ¨ Q: Valid? ¨ A: This will NOT work since a loop can occur. Swaps can be made that might continually result in unstable pairs. ¤ Terminate? Opening 10 Simple-But-Invalid 1. Start with some matching 2. while (there is an unstable pair) do 3. Swap mates to make the pair stable IRIS H.-R. JIANG Propose and Reject ¨ Status of each person: free (unmarried) ~ engaged ~ married ¤ Deferred decision making: You may hesitate for a while before you are pretty sure you have got a reasonably good partner ¨ Goal: self-enforcing ¤ [Gale-Shapley 1962] Intuitive method: act in one’s self-interest ¨ Q: Correct? (Stable? or even perfect?) Terminate? Opening 11 Gale-Shapley 1. initialize each person to be free 2. while (some man m is free and hasn't proposed to every woman) do 3. w = highest ranked woman in m's list to whom m has not yet proposed 4. if (w is free) then 5. (m, w) become engaged 6. else if (w prefers m to her fiancé m') then 7. (m, w) become engaged 8. m' become free 9. return the set S of engaged pairs D. Gale and L. S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, vol. 69, pp. 9-14, 1962. 2012 Nobel Prize in Economics Example of Gale-Shapley Algorithm 12 Opening A flash demo can be found at http://mathsite.math.berkeley.edu/smp/smp.html

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Five Representative Problems 13 Opening IRISH.-R. JIANG Interval Scheduling ¨ Given: Set of jobs with start times and finish times ¨ Goal: Find maximum cardinality subset of mutually compatible jobs Opening 14 jobs don’t overlap Time 0 1 2 3 4 5 6 7 8 9 10 11 f g h e a b c d h e b Maximum compatible set {b, e, h} IRIS H.-R. JIANG Weighted Interval Scheduling ¨ Given: Set of jobs with start times, finish times, and weights ¨ Goal: Find maximum weight subset of mutually compatible jobs Opening 15 Time 0 1 2 3 4 5 6 7 8 9 10 11 20 11 16 13 23 12 20 26 16 26 Maximum weighted compatible set {26, 16} IRIS H.-R. JIANG Bipartite Matching ¨ Given: Bipartite graph ¨ Goal: Find maximum cardinality matching Opening 16 1 2 3 4 5 A B C D E Maximum matching {(A, 1), (B, 2), (C, 4), (D, 3), (E, 5)}

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IRIS H.-R. JIANG IndependentSet ¨ Given: Graph ¨ Goal: Find maximum cardinality independent set Opening 17 subset of nodes s.t. no two joined by an edge 1 2 3 4 5 6 7 1 4 5 6 Maximum independent set {1, 4, 5, 6} IRIS H.-R. JIANG Competitive Facility Location ¨ Given: Graph with weight on each node ¨ Game: ¤ Two competing players alternate in selecting nodes ¤ Do not allow to select a node if any of its neighbors have been selected ¨ Goal: Select a maximum weight subset of nodes Opening 18 Second player can guarantee 20, but not 25 10 5 1 5 15 1 5 1 10 15 IRIS H.-R. JIANG Five Representative Problems ¨ Efficiently solvable ¤ Interval scheduling: nlogn greedy algorithm ¤ Weighted interval scheduling: nlogn dynamic programming algorithm ¤ Bipartite matching: nk max-flow based algorithm ¨ Hard ¤ Independent set: NP-complete ¤ Competitive facility location: PSPACE-complete (even harder!) Opening 19 Proofs Back to Stable Matching 20 Opening

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IRIS H.-R. JIANG Formulatingthe Problem ¨ The stable matching problem ¨ Given: ¤ M: n men ¤ W: n women ¤ M x W: Each person has ranked all members of the opposite sex with a unique number between 1 and n in order of preference ¨ Goal: ¤ Marry the men and women off such that ¤ There are no two people of opposite sex who would both rather have each other than their current partners. n If there are no such people, all the marriages are stable. Opening 21 Clean! IRIS H.-R. JIANG Designing the Algorithm ¨ Correctness: ¤ Termination: G-S terminates after at most n2 iterations. ¤ Perfection: Everyone gets married. ¤ Stability: The marriages are stable. ¨ Male-optimal and female-pessimal ¨ All executions yield the same matching Opening 22 Clean! Gale-Shapley 1. initialize each person to be free 2. while (some man m is free and hasn't proposed to every woman) do 3. w = highest ranked woman in m's list to whom m has not yet proposed 4. if (w is free) then 5. (m, w) become engaged 6. else if (w prefers m to her fiancé m') then 7. (m, w) become engaged 8. m' become free 9. return the set S of engaged pairs IRIS H.-R. JIANG Correctness: Termination ¨ G-S terminates after at most n2 iterations of the while loop. ¨ Pf: ¤ At each iteration, a man proposes to a new woman. ¤ There are only n2 possible proposals. ¨ Observation 1: The sequence of women to whom m proposes gets worse and worse. ¨ Observation 2: Once a woman w receives her first proposal, w remains engaged; she only trades up. Opening 23 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile Bertha Amy Clare Xavier Yancey Zeus Amy Bertha Zeus Zeus Don’t count free people IRIS H.-R. JIANG Correctness: Perfection ¨ The set S returned at termination is perfect matching. ¤ i.e., all men get married ¤ Prove by contradiction! ¨ Pf: ¤ Suppose that G-S algorithm terminates with a free man m. ¤ At termination, m had already proposed to every woman. Otherwise, the while loop would not have exited. ¤ Since there is a free man m, there must be at least a free woman, say w. w was never proposed to. ¤ This contradicts that m had already proposed to every woman. Opening 24 Gale-Shapley 1. initialize each person to be free 2. while (some man m is free and hasn't proposed to every woman) do 3. w = highest ranked woman in m's list to whom m has not yet proposed 4. if (w is free) then 5. (m, w) become engaged 6. else if (w prefers m to her fiancé m') then 7. (m, w) become engaged 8. m' become free 9. return the set S of engaged pairs

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IRIS H.-R. JIANG Correctness:Stability ¨ Consider an execution returns S. Set S is a stable matching. ¤ Assume an instability w.r.t. S and obtain a contradiction. ¨ Pf. ¤ This instability involve two pairs, (m, w) and (m', w'), in S with n m prefers w' to w n w' prefers m to m' ¤ In the execution, m’s last proposal was w. ¤ Did m propose to w' at some earlier point? n Case 1: No. n m must prefers w to w'. ®¬ n Case 2: Yes. n m was rejected by w' because of m'', w' prefers m'' to m n m'' = m' (final partner) ®¬ n w' prefers her final partner m' to m'' ®¬ Opening 25 w m m' w' IRIS H.-R. JIANG Uniqueness? ¨ Q: For a given instance, is stable matching is unique? ¨ A: Opening 26 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile Bertha Amy Clare Xavier Yancey Zeus 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Amy Bertha Amy Bertha Clare Clare Clare Men’s Preference Profile 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Xavier Yancey Yancey Zeus Zeus Zeus Women’s Preference Profile Bertha Amy Clare Xavier Yancey Zeus IRIS H.-R. JIANG Male-Optimality ¨ S* = {(m, best(m)): m Î M}: best possible outcome for all men ¤ w Î W is a valid partner of a man m if there is a stable matching that contains the pair (m, w) ¤ w = best(m): best valid partner of m if w is a valid partner of m and no woman whom m ranks higher than w is m’s valid partner ¨ Every execution of G-S algorithm results in the set S* ¤ Male-optimal (and female-pessimal) ¤ The same matching for all executions Opening 27 1st 2nd 3rd Xavier Yancey Zeus Amy Bertha Clare Bertha Amy Bertha Clare Clare Amy Men’s Preference Profile 1st 2nd 3rd Amy Bertha Clare Yancey Xavier Xavier Zeus Zeus Yancey Xavier Yancey Zeus Women’s Preference Profile Bertha Amy Clare Xavier Yancey Zeus IRIS H.-R. JIANG Male-Optimality ¨ G-S algorithm results in the set S* ¤ Again, by contradiction! ¨ Pf. ¤ Suppose some execution E of G-S yields S where some man is paired with someone other than his best valid partner. ¤ Since men propose in decreasing order of preference Þ some man is rejected by a valid partner during E. n Let m be first such man & w be first valid woman who rejects m n w = best(m) n By definition, there is a stable matching S' containing (m, w). ¤ In E, m was rejected by w because of m' Þ w prefers m' to m ¤ Assume (m', w') Î S', w' ¹ w. (w' is a valid partner of m') ¤ In E, (m, w) is the first rejection of a valid pair ¤ m' hadn’t been rejected by any valid partner when m' is engaged to w in E. ¤ m' proposed in decreasing order of preference (those women ranked before w are invalid). ¤ And w' is a valid partner of m' Þ m' must prefer w to w' ¤ Since (m', w) Ï S', it follows that (m', w) is an instability in S'. ®¬ Opening 28 w m m' w' S' S'

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IRIS H.-R. JIANG WhatDid We Learn? ¨ Powerful ideas learned from stable matching ¤ Isolate underlying structure of problem ¤ Create effective and efficient algorithms n Effectiveness: Logical proofs n Useful technique: proof by contradiction n Efficiency will be detailed in the next chapter Opening 29 For me, great algorithms are the poetry of computation. Just like verse, they can be terse, allusive, dense, and even mysterious. But once unlocked, they cast a brilliant new light on some aspect of computing. -- Francis Sullivan An algorithm must be seen to be believed. The best way to learn what an algorithm is all about is to try it. The reader should always take pencil and paper and work through an example... -- Donald Knuth IRIS H.-R. JIANG eBook ¨ Chapters 1, 6, 13 ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch1.pdf ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch6.pdf ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch13.pdf ¨ The whole book ¨ http://www.icsd.aegean.gr/kaporisa/index_files/Algorithm_Design.pdf Opening 30

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IRIS H.-R. JIANG WhatDid We Learn? ¨ Powerful ideas learned from stable matching ¤ Isolate underlying structure of problem ¤ Create effective and efficient algorithms n Effectiveness: Logical proofs n Useful technique: proof by contradiction n Efficiency will be detailed in the next chapter Opening 29 For me, great algorithms are the poetry of computation. Just like verse, they can be terse, allusive, dense, and even mysterious. But once unlocked, they cast a brilliant new light on some aspect of computing. -- Francis Sullivan An algorithm must be seen to be believed. The best way to learn what an algorithm is all about is to try it. The reader should always take pencil and paper and work through an example... -- Donald Knuth IRIS H.-R. JIANG eBook ¨ Chapters 1, 6, 13 ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch1.pdf ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch6.pdf ¨ http://www.aw-bc.com/info/kleinberg/assets/downloads/ch13.pdf ¨ The whole book ¨ http://www.icsd.aegean.gr/kaporisa/index_files/Algorithm_Design.pdf Opening 30

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