An Extension of Menger's Theorem -...
Transcript of An Extension of Menger's Theorem -...
![Page 1: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/1.jpg)
An Extension of Menger’s Theorem
Li Xu, Weiping Shang, Guangyue Han
The University of Hong Kong
April 2015
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 2: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/2.jpg)
Merging
mergingmerging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 3: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/3.jpg)
Merging
mergingmerging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 4: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/4.jpg)
Merging
merging
merging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 5: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/5.jpg)
Merging
mergingmerging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 6: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/6.jpg)
Merging
merging
merging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 7: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/7.jpg)
Merging
merging
merging
merging
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 8: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/8.jpg)
A “Theorem’’
Mergings ⇒ Congestions a.s.!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 9: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/9.jpg)
A “Proof”
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 10: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/10.jpg)
Motivation in Transportation Networks
Minimizing Mergings ⇒ Maximizing Throughput!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 11: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/11.jpg)
Motivation in Transportation Networks
Minimizing Mergings ⇒ Maximizing Throughput!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 12: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/12.jpg)
Motivation in Transportation Networks
Minimizing Mergings ⇒ Maximizing Throughput!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 13: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/13.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 14: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/14.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 15: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/15.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 16: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/16.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 17: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/17.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 18: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/18.jpg)
Motivation in Computer NetworksS
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 19: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/19.jpg)
Motivation in Computer NetworksS
R1 R2
a b
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a+b
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network coding
Minimizing Mergings ⇒ Minimizing Encoding Complexity!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 20: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/20.jpg)
Karl Menger
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 21: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/21.jpg)
Menger’s Theorem
Theorem (Menger, 1927)
Consider a directed graph G (V ,E ). For any two vertices A,B, themaximum number of pairwise edge-disjoint directed paths from Ato B in G equals the min-cut between A and B, namely theminimum number of edges in E whose deletion destroys alldirected paths from A to B.
Menger’s paths
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 22: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/22.jpg)
Menger’s Theorem
Theorem (Menger, 1927)
Consider a directed graph G (V ,E ). For any two vertices A,B, themaximum number of pairwise edge-disjoint directed paths from Ato B in G equals the min-cut between A and B, namely theminimum number of edges in E whose deletion destroys alldirected paths from A to B.
Menger’s paths
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 23: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/23.jpg)
A Quick Example
A B
E
F
C
DG
H
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 24: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/24.jpg)
A Quick Example
A B
E
F
C
DG
H
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 25: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/25.jpg)
A Quick Example
A B
E
F
C
DG
H
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 26: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/26.jpg)
We are interested in ...
Mergingsamong different groups of Menger’s paths
in networks with multiple sources and sinks.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 27: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/27.jpg)
Rerouting
S1 R1
S2 R2
reroutable
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 28: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/28.jpg)
Rerouting
S1 R1
S2 R2
reroutable
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 29: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/29.jpg)
Rerouting
S1 R1
S2 R2
reroutableLi Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 30: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/30.jpg)
Rerouting
S1 R1
S2 R2
reroutable
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 31: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/31.jpg)
Rerouting
S1 R1
S2 R2
reroutable
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 32: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/32.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 33: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/33.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 34: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/34.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 35: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/35.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 36: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/36.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 37: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/37.jpg)
Rerouting (cont’d)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 38: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/38.jpg)
Network Model and Notations
I G : an acyclic directed graph.
I S1,S2, · · · ,Sn: sources.
I R1,R2, · · · ,Rn: distinct sinks.
I ci : the min-cut between Si and Ri .
I αi = {αi ,1, αi ,2, · · · , αi ,ci}: a set of Menger’s paths from Si toRi .
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 39: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/39.jpg)
Network Model and Notations
I G : an acyclic directed graph.
I S1,S2, · · · ,Sn: sources.
I R1,R2, · · · ,Rn: distinct sinks.
I ci : the min-cut between Si and Ri .
I αi = {αi ,1, αi ,2, · · · , αi ,ci}: a set of Menger’s paths from Si toRi .
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 40: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/40.jpg)
Network Model and Notations
I G : an acyclic directed graph.
I S1,S2, · · · ,Sn: sources.
I R1,R2, · · · ,Rn: distinct sinks.
I ci : the min-cut between Si and Ri .
I αi = {αi ,1, αi ,2, · · · , αi ,ci}: a set of Menger’s paths from Si toRi .
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 41: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/41.jpg)
Network Model and Notations
I G : an acyclic directed graph.
I S1,S2, · · · ,Sn: sources.
I R1,R2, · · · ,Rn: distinct sinks.
I ci : the min-cut between Si and Ri .
I αi = {αi ,1, αi ,2, · · · , αi ,ci}: a set of Menger’s paths from Si toRi .
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 42: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/42.jpg)
Network Model and Notations
I G : an acyclic directed graph.
I S1,S2, · · · ,Sn: sources.
I R1,R2, · · · ,Rn: distinct sinks.
I ci : the min-cut between Si and Ri .
I αi = {αi ,1, αi ,2, · · · , αi ,ci}: a set of Menger’s paths from Si toRi .
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 43: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/43.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 44: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/44.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 45: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/45.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 46: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/46.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 47: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/47.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 48: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/48.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 49: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/49.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 50: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/50.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decision
worst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 51: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/51.jpg)
M∗(c1, c2, · · · , cn)Assume that all sources are identical.
I M∗(G ):the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M∗(c1, c2, · · · , cn): the supremum of M∗(G ) over all possiblechoices of such G .
S
R1
R2
best decisionworst performance
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 52: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/52.jpg)
M(c1, c2, · · · , cn)
Assume that all sources are distinct.
I M(G ): the minimum number of mergings over all possibleMenger’s path sets αi ’s, i = 1, 2, · · · , n.
I M(c1, c2, · · · , cn): the supremum of M(G ) over all possiblechoices of such G .
S1
S2
R1
R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 53: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/53.jpg)
Previous Work
It was first conjectured by Tavory, Feder and Ron that M(c1, c2, · · · , cn)is finite. More specifically the authors proved that if M(c1, c2) is finitefor all c1, c2, then M(c1, c2, · · · , cn) is finite as well.
As for M∗, Fragouli and Soljanin use the idea of “subtreedecomposition” to first prove that
M∗(2, 2, · · · , 2︸ ︷︷ ︸n
) = n − 1.
It was first shown Langberg, Sprintson and Bruck that M∗(c1, c2) isfinite for all c1, c2, and subsequently M∗(c1, c2, · · · , cn) is finite allc1, c2, · · · , cn.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 54: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/54.jpg)
Main Results
Main Results
I For fixed c1, c2, · · · , cn, M(c1, c2, · · · , cn) andM∗(c1, c2, · · · , cn) are always finite.
I And as functions of c1, c2, · · · , cn, they have interestingproperties.
I We give exact values of and tighter bounds on M and M∗
with certain parameters.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 55: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/55.jpg)
Some Remarks
When n = 1, Ford-Fulkerson algorithm can find the min-cut and aset of Menger’s path between S1 and R1 in polynomial time.
The LDP (Link Disjoint Problem) asks if there are twoedge-disjoint paths from S1, S2 to R1, R2, respectively. The factthat the LDP problem is NP-complete suggests the intricacy of theproblem when n ≥ 2.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 56: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/56.jpg)
Some Remarks
When n = 1, Ford-Fulkerson algorithm can find the min-cut and aset of Menger’s path between S1 and R1 in polynomial time.
The LDP (Link Disjoint Problem) asks if there are twoedge-disjoint paths from S1, S2 to R1, R2, respectively. The factthat the LDP problem is NP-complete suggests the intricacy of theproblem when n ≥ 2.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 57: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/57.jpg)
Outline of the Proof
LemmaFor any c1, c2,
M(c1, c2) ≤ c1c2(c1 + c2)/2.
TheoremFor any c1, c2, · · · , cn, we have
M(c1, c2, · · · , cn) ≤∑i<j
M(ci , cj).
ObservationNote that for any c1, c2, · · · , cn, we always have
M∗(c1, c2, · · · , cn) ≤M(c1, c2, · · · , cn)
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 58: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/58.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 59: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/59.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 60: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/60.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 61: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/61.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 62: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/62.jpg)
A Key Idea
If too many mergings:
· · ·
· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 63: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/63.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 64: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/64.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 65: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/65.jpg)
A Key Idea
If too many mergings:
· · ·· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 66: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/66.jpg)
A Key Idea
If too many mergings:
· · ·
· · ·
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 67: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/67.jpg)
Properties of M∗
Proposition
For c1 ≤ c2 ≤ · · · ≤ cn, if c1 + c2 + · · ·+ cn−1 ≤ cn, then
M∗(c1, c2, · · · , cn) =M∗(c1, c2, · · · , cn−1, c1 + c2 + · · ·+ cn−1).
Proposition
For c1 = 1 ≤ c2 ≤ · · · ≤ cn, we have
M∗(c1, c2, · · · , cn) =M∗(c2, · · · , cn−1, cn).
Proposition
For n1 ≤ n2 ≤ · · · ≤ nk ,
M∗(n1, n2, . . . , nk) ≥k−1∑i=1
M∗(ni , ni ).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 68: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/68.jpg)
Properties of M
Proposition
For any c1,0, c1,1, c2, we have
M(c1,0 + c1,1, c2) ≥M(c1,0, c2) +M(c1,1, c2).
Proposition
For any c1, c2, · · · , cn and any fixed k with 1 ≤ k ≤ n, we have
M(c1, c2, · · · , cn) ≥∑
i≤k,j≥k+1
M(ci , cj).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 69: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/69.jpg)
Properties of M
Proposition
For any c1,0, c1,1, c2, we have
M(c1,0 + c1,1, c2) ≥M(c1,0, c2) +M(c1,1, c2).
Proposition
For any c1, c2, · · · , cn and any fixed k with 1 ≤ k ≤ n, we have∑i<j
M(ci , cj) ≥M(c1, c2, · · · , cn) ≥∑
i≤k,j≥k+1
M(ci , cj).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 70: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/70.jpg)
Properties of M (cont’d)
Proposition
For any m ≤ n, we have
M(m, n) ≤ U(m, n) + V (m, n) + m − 2,
where
U(m, n) =m−1∑j=1
(M(j ,m − 1) + 1 +M(m − j , n))+M(m,m−1)+1,
and
V (m, n) =M(m, n−1)+m−1∑j=1
(M(j , n) + 1 +M(m − j , n))−M(1, n).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 71: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/71.jpg)
Properties of M (cont’d)
Proposition
For any fixed k , there exists a positive constant Ck such that forall n,
M(k , n) ≤ Ckn.
Proof.
Ck mergings Ck mergings Ck mergings
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 72: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/72.jpg)
Properties of M (cont’d)
Proposition
For any fixed k , there exists a positive constant Ck such that forall n,
M(k , n) ≤ Ckn.
Proof.
Ck mergings
Ck mergings Ck mergings
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 73: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/73.jpg)
Properties of M (cont’d)
Proposition
For any fixed k , there exists a positive constant Ck such that forall n,
M(k , n) ≤ Ckn.
Proof.
Ck mergings Ck mergings
Ck mergings
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 74: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/74.jpg)
Properties of M (cont’d)
Proposition
For any fixed k , there exists a positive constant Ck such that forall n,
M(k , n) ≤ Ckn.
Proof.
Ck mergings Ck mergings Ck mergings
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 75: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/75.jpg)
Tighter Bounds
It has been established [Langberg et al.] that
n(n − 1)/2 ≤M∗(n, n) ≤ n3.
Next, we give tighter bounds on M∗(n, n) and M(m, n).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 76: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/76.jpg)
Bounds on M∗
Proposition
(n − 1)2 ≤M∗(n, n) ≤⌈n
2
⌉(n2 − 4n + 5).
Proof.
A graph showing M∗(3, 3) ≥ 4.
S
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 77: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/77.jpg)
Bounds on MProposition
2mn−m−n+1 ≤M(m, n) ≤ (m+n−1)+(mn−2)
⌊m + n − 2
2
⌋.
Proof.
A graph showing M(2, 2) ≥ 5.
S1 S2
R1 R2Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 78: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/78.jpg)
Exact Values
I
M∗(1, 1) = 0.
I
M∗(2, 2) = 1.
I
M∗(3, 3) = 4.
I
M∗(4, 4) = 9.
I
M∗(5, 5) = 16.
M∗(m,m) = (m − 1)2?
M∗(6, 6) = 27!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 79: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/79.jpg)
Exact Values
I
M∗(1, 1) = 0.
I
M∗(2, 2) = 1.
I
M∗(3, 3) = 4.
I
M∗(4, 4) = 9.
I
M∗(5, 5) = 16.
M∗(m,m) = (m − 1)2?
M∗(6, 6) = 27!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 80: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/80.jpg)
Exact Values
I
M∗(1, 1) = 0.
I
M∗(2, 2) = 1.
I
M∗(3, 3) = 4.
I
M∗(4, 4) = 9.
I
M∗(5, 5) = 16.
M∗(m,m) = (m − 1)2?
M∗(6, 6) = 27!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 81: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/81.jpg)
Exact ValuesI
M(1, n) = n.
I
M(2, n) = 3n − 1.
I
M(3, 3) = 13.
I
M(3, 4) = 18.
I
M(3, 5) = 23.
I
M(3, 6) = 28.
I
M(4, 4) = 27.Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 82: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/82.jpg)
Exact ValuesI
M∗(1, 1, · · · , 1︸ ︷︷ ︸n
) = 0.
I
M∗(2, 2, · · · , 2︸ ︷︷ ︸n
) = n − 1.
I
M∗(2,m,m) = 1 + (m − 1)2,m = 2, 3, 4, 5.
I
M∗(m,m,m) = 2(m − 1)2,m = 1, 2, 3, 4.
I
M∗(3, 4, 4) = 13.
I For m ≤ n ≤ p and (m, n) ≤ (3, 4) or (2, 5)
M∗(m, n, p) =M∗(m, n, n).
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 83: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/83.jpg)
Exact Values
I
M(1, 1, · · · , 1︸ ︷︷ ︸k
) =
⌊k2
4
⌋.
I
M(1, . . . , 1︸ ︷︷ ︸k
, 2) =
{3k − 1 if k ≤ 6,
bk2
4 c+ k + 2 if k > 6.
I
M(1, 1, n) = 2n + 1.
I
M(1, 2, n) =
{4n if n = 2, 3,4n + 1 if n = 1 or n ≥ 4.
I
M(2, 2, 2) = 11,M(1, 3, 3) = 17,M(2, 2, 3) = 18.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 84: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/84.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 85: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/85.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 86: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/86.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 87: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/87.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 88: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/88.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 89: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/89.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 90: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/90.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 91: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/91.jpg)
The “Finiteness” Result Does Not Hold for Cyclic GraphsS
R1 R2Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 92: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/92.jpg)
Open Problems
Tighter Upper Bounds
There exists C > 0 such that for all m, n,
M(m, n) ≤ Cmn.
Mergings On Other Classes of Graphs
undirected graphs, planar graphs, and so on.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 93: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/93.jpg)
Open Problems
Tighter Upper Bounds
There exists C > 0 such that for all m, n,
M(m, n) ≤ Cmn.
Mergings On Other Classes of Graphs
undirected graphs, planar graphs, and so on.
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 94: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/94.jpg)
ReferencesL. R. Ford and D. R. FulkersonMaximal flow through a network.Canadian Journal of Mathematics 8: 399–404, 1956.
C. Fragouli and E. Soljanin.Information Flow Decomposition for Network Coding.IEEE Transactions on Information Theory, Volume 52, Issue 3, March2006, Page(s):829 - 848.
M. Langberg, A. Sprintson, A. and J. Bruck.The encoding complexity of network codingIEEE Transactions on Information Theory, Volume 52, Issue 6, June 2006,Page(s):2386 - 2397.
K. Menger.Zur allgemeinen Kurventhoerie.Fund. Math., 10:96-115.
A. Tavory, M. Feder and D. Ron.Bounds on Linear Codes for Network Multicast.
Electronic Colloquium on Computational Complexity (ECCC), 10(033),
2003.Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong
![Page 95: An Extension of Menger's Theorem - 上海交通大学数学系math.sjtu.edu.cn/Conference/2015WCA/talk/HanGuangyue/slides.pdf · An Extension of Menger’s Theorem Li Xu, Weiping](https://reader033.fdocuments.net/reader033/viewer/2022052708/5a7402777f8b9aa3688b84b6/html5/thumbnails/95.jpg)
Thank You!
Li Xu, Weiping Shang, Guangyue Han The University of Hong Kong