Lower bound for the Stable Marriage Problem הפקולטה למדעי ההנדסה Faculty of Engineering Sciences אוניברסיטת בן-גוריון בנגב Ben-Gurion University of the Negev Lower bound for the Stable Marriage Problem Nir Amira Dr. Zvi Lotker

The Stable Marriage problem A Matching criteria N Man, N women, each with their own preference list. A Matching is called Stable if no unstable pairs exist in it blocking pair is a pair not matched together, that ranked each other higher then their current mate Basic Model: Complete bipartite directed graph, each edge hold preference. Introduction Lets see an example… Ben-Gurion University - Department of communication systems engineering - Nir Amira

The Stable Marriage problem אני מעדיפה את גדי על-פני בעלי אבי Introduction העדפות: העדפות: Second try: First try: אבי רותי ר, ש, ת ב, ג, א blocking pair בני שרה ש, ת, ר א, ב, ג גדי תמר ר, ת, ש א, ג, ב Unstable Matching Stable Matching אני מעדיפה את אבי כבעלי אבל הוא לא רוצה אותי אני מעדיף את רותי על-פני אשתי תמר Ben-Gurion University - Department of communication systems engineering - Nir Amira

The Gale-Shapley Algorithm Rejected men propose to their next choice Each woman now chooses her new favorite to be engaged to from the group of the new proposers+ her fiancé, and reject the rest Return to step 3 until no man is rejected Each woman chooses her favorite to be engaged to from among those who have proposed to her, and rejects all the rest Each man propose to his favorite woman Rejected men propose to their next choice Each woman now chooses her new favorite to be engaged to from the group of the new proposers+ her fiancé, and reject the rest We gat Man optimal Stable Marriage Gale Shapley Algorithm העדפות: העדפות: אבי רותי ר, ש, ת ב, ג, א בני שרה ש, ת, ר א, ב, ג גדי תמר ר, ת, ש א, ג, ב Ben-Gurion University - Department of communication systems engineering - Nir Amira

Worst case time complexity of: Distributed Approach Each man or woman is an independent unit Parallelism is the word ! The Gale-Shapley Algorithm is distributed compatible Distributed Approach Worst case time complexity of: O(n) Ben-Gurion University - Department of communication systems engineering - Nir Amira

Switch motivation OQ is an optimal Throughput model but non realistic – needs speedup N Using Stable matching it is proved that CIOQ with speedup 2 is operating like an OQ Output Queuing Combined Input Output Queuing buf buf buf Switch Model buf buf buf buf buf buf 100% Throughput Ben-Gurion University - Department of communication systems engineering - Nir Amira

The switch model Model of VOQ switch Simple centralized algorithm: Each IN has buffers to each OUT port Men = IN Ports , women = Out Ports. One sided preferences - undirected edges Can be easily described as a matrix Simple centralized algorithm: Match Max[Matrix] Delete irrelevant Time complexity: O(n) OUT Switch Model 30 25 71 19 3 33 11 8 60 45 53 24 15 7 28 44 38 68 14 5 6 12 49 57 21 IN Ben-Gurion University - Department of communication systems engineering - Nir Amira

Our Algorithm Time Complexity: 3√n = O(√n) Works when: All preferences of the IN on OUT are monotone, means all prefer OUT-1 most and OUT-n least Each node can send different messages on each edge Buffer Buffer 71 71 60 53 30 33 24 44 25 11 15 38 19 8 7 14 8 15 44 Let’s see it in Action: Phase II Phase I R1 R2 Step 0 – Init: we set √n leaders from the OUT Step √n+2: Matched IN tell their match and all leaders that they are matched Step √n+1: Leader-1 calculate first √n matches and send them to all IN In general: Steps √n+i &√n+i+1 : Leader-i calculate next √n matches and send them to all IN, and the IN notify all leaders and their match In general: Step k (from 1 to √n): All IN send their kth pref. to R1, √n+k pref. to R2, i*√n+k pref. to Ri and so on. 71 30 25 19 60 33 11 8 53 24 15 7 44 38 14 Time Complexity: 3√n = O(√n) Our Algorithm Step 1: All IN send their 1st pref. to R1, √n+1 pref. to R2 and so on. Ben-Gurion University - Department of communication systems engineering - Nir Amira

Future work Find the Lower bound of the problem Incomplete preferences lists with ties Compare Stable matching with normal switch scheduling (iSLIP) Future work Ben-Gurion University - Department of communication systems engineering - Nir Amira

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