1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Questions ?