Fundamentals of Algorithms - PowerPoint PPT Presentation

�������������������� Fundamentals of Algorithms �������������������������������������������������������������� Tyler Moore ��������������������������������������������������������������� ��������������������������������������������������������������������� CSE 3353, SMU, Dallas, TX Lecture 1 �������������������������������������������������������� ������������������������������������ �������������������� Some slides created by or adapted from Dr. Kevin Wayne. For more information see http://www.cs.princeton.edu/~wayne/kleinberg-tardos 2 ��������������������� ����������������������������������������� Internet. Web search, packet routing, distributed file sharing, ... Goal. Given a set of preferences among hospitals and med-school students, Biology. Human genome project, protein folding, … design a self-reinforcing admissions process. Computers. Circuit layout, databases, caching, networking, compilers, … Computer graphics. Movies, video games, virtual reality, … Unstable pair: student � and hospital � are unstable if: Security. Cell phones, e-commerce, voting machines, … � � prefers � to its assigned hospital. Multimedia. MP3, JPG, DivX, HDTV, face recognition, … � � prefers � to one of its admitted students. Social networks. Recommendations, news feeds, advertisements, … Physics. N-body simulation, particle collision simulation, … Stable assignment. Assignment with no unstable pairs. ⋮ � Natural and desirable condition. � Individual self-interest prevents any hospital–student side deal. We emphasize algorithms and techniques that are useful in practice. 6 3

����������������������� ���������������� Goal. Given a set of � men and a set of � women, find a "suitable" matching. Def. A matching � is a set of ordered pairs � � � with � � ∈ � �� and � � ∈ � � s.t. � Participants rank members of opposite sex. � Each man � � ∈ � � appears in at most one pair of � . � Each man lists women in order of preference from best to worst. � Each woman � � ∈ � � appears in at most one pair of � . � Each woman lists men in order of preference from best to worst. Def. A matching �� is perfect if � � � � ����� � � � ����� � � � ���� � . favorite least favorite favorite least favorite 1 st 2 nd 3 rd 1 st 2 nd 3 rd 1 st 2 nd 3 rd 1 st 2 nd 3 rd Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus ��������������������� ����������������������� ���������������������������������������� 4 5 ������������� ����������������������� Def. Given a perfect matching � , man � and woman � are unstable if: Def. A stable matching is a perfect matching with no unstable pairs. � � prefers � to his current partner. � � prefers � to her current partner. Stable matching problem. Given the preference lists of � men and � women, find a stable matching (if one exists). Key point. An unstable pair ��� could each improve partner by joint action. � Natural, desirable, and self-reinforcing condition. � Individual self-interest prevents any man–woman pair from eloping. 1 st 2 nd 3 rd 1 st 2 nd 3 rd 1 st 2 nd 3 rd 1 st 2 nd 3 rd Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus �������������������������������������� ���������������������������������������� 6 7