Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
[ACM-ICPC] Minimum Cut
Search
KuoE0
January 21, 2013
Programming
2
100
[ACM-ICPC] Minimum Cut
KuoE0
January 21, 2013
Tweet
Share
More Decks by KuoE0
See All by KuoE0
Protocol handler in Gecko
kuoe0
0
100
面試面試面試,因為很重要所以要說三次!
kuoe0
2
250
應徵軟體工程師
kuoe0
0
170
面試心得分享
kuoe0
0
420
Windows 真的不好用...
kuoe0
0
290
Python @Wheel Lab
kuoe0
0
210
Introduction to VP8
kuoe0
0
250
Python @NCKU_CSIE
kuoe0
0
120
[ACM-ICPC] Tree Isomorphism
kuoe0
1
250
Other Decks in Programming
See All in Programming
育てるアーキテクチャ:戦い抜くPythonマイクロサービスの設計と進化戦略
fujidomoe
1
150
Web技術を最大限活用してRAW画像を現像する / Developing RAW Images on the Web
ssssota
2
1.2k
(Extension DC 2025) Actor境界を越える技術
teamhimeh
1
220
Playwrightはどのようにクロスブラウザをサポートしているのか
yotahada3
7
2.3k
ててべんす独演会〜Flowの全てを語ります〜
tbsten
1
220
Let's Write a Train Tracking Algorithm
twocentstudios
0
220
開発生産性を上げるための生成AI活用術
starfish719
1
170
どの様にAIエージェントと 協業すべきだったのか?
takefumiyoshii
2
600
After go func(): Goroutines Through a Beginner’s Eye
97vaibhav
0
230
CSC509 Lecture 02
javiergs
PRO
0
410
ポスターセッション: 「まっすぐ行って、右!」って言ってラズパイカーを動かしたい 〜生成AI × Raspberry Pi Pico × Gradioの試作メモ〜
komofr
0
950
複雑化したリポジトリをなんとかした話 pipenvからuvによるモノレポ構成への移行
satoshi256kbyte
1
770
Featured
See All Featured
Bootstrapping a Software Product
garrettdimon
PRO
307
110k
No one is an island. Learnings from fostering a developers community.
thoeni
21
3.5k
The Cost Of JavaScript in 2023
addyosmani
53
9k
The MySQL Ecosystem @ GitHub 2015
samlambert
251
13k
Building a Scalable Design System with Sketch
lauravandoore
462
33k
Principles of Awesome APIs and How to Build Them.
keavy
127
17k
A better future with KSS
kneath
239
17k
How GitHub (no longer) Works
holman
315
140k
Design and Strategy: How to Deal with People Who Don’t "Get" Design
morganepeng
132
19k
Why Our Code Smells
bkeepers
PRO
339
57k
Rails Girls Zürich Keynote
gr2m
95
14k
Become a Pro
speakerdeck
PRO
29
5.5k
Transcript
Minimum Cut ֲࢸݢʢKuoE0ʣ
[email protected]
KuoE0.ch
Cut
cut (undirected)
1 3 5 6 7 8 9 2 4 undirected
graph A partition of the vertices of a graph into two disjoint subsets
1 3 5 6 7 8 9 2 4 undirected
graph A partition of the vertices of a graph into two disjoint subsets
1 3 5 6 7 8 9 2 4 undirected
graph A partition of the vertices of a graph into two disjoint subsets
1 2 8 5 4 7 9 3 6 A
partition of the vertices of a graph into two disjoint subsets undirected graph
1 2 8 5 4 7 9 3 6 Cut-set
of the cut is the set of edges whose end points are in different subsets. undirected graph
1 2 8 5 4 7 9 3 6 Cut-set
of the cut is the set of edges whose end points are in different subsets. Cut-set undirected graph
1 2 8 5 4 7 9 3 6 weight
= number of edges or sum of weight on edges weight is 7 undirected graph
cut (directed)
1 3 5 6 7 8 9 2 4 directed
graph A partition of the vertices of a graph into two disjoint subsets
1 3 5 6 7 8 9 2 4 directed
graph A partition of the vertices of a graph into two disjoint subsets
1 3 5 6 7 8 9 2 4 directed
graph A partition of the vertices of a graph into two disjoint subsets
1 2 8 5 4 7 9 3 6 directed
graph A partition of the vertices of a graph into two disjoint subsets
1 2 8 5 4 7 9 3 6 directed
graph Cut-set of the cut is the set of edges whose end points are in different subsets.
1 2 8 5 4 7 9 3 6 directed
graph Cut-set of the cut is the set of edges whose end points are in different subsets.
1 2 8 5 4 7 9 3 6 Cut-set
directed graph Cut-set of the cut is the set of edges whose end points are in different subsets.
1 2 8 5 4 7 9 3 6 weight
is 5⇢ or 2⇠ directed graph weight = number of edges or sum of weight on edges
s-t cut 1. one side is source 2. another side
is sink 3. cut-set only consists of edges going from source’s side to sink’s side
1 3 5 6 7 8 9 2 4 flow
network Source Sink Other
1 3 5 6 7 8 9 2 4 flow
network Source Sink
1 3 5 6 7 8 9 2 4 flow
network Source Sink
1 2 8 5 4 7 9 3 6 flow
network cut-set only consists of edges going from source’s side to sink’s side
1 2 8 5 4 7 9 3 6 weight
is 6 flow network cut-set only consists of edges going from source’s side to sink’s side
Max-Flow Min-Cut Theorem
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 3 = 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 3 = 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 4 - 1 = 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 3 + 4 - 1 = 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
Observation 1 The network flow sent across any cut is
equal to the amount reaching sink. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 total flow = 6, flow on cut = 4 + 2= 6
Observation 2 Then the value of the flow is at
most the capacity of any cut. 1 3 2 4 5 6 3 8 4 2 4 4 3 It’s trivial!
Observation 2 Then the value of the flow is at
most the capacity of any cut. 1 3 2 4 5 6 3 8 4 2 4 4 3 It’s trivial!
Observation 3 Let f be a flow, and let (S,T)
be an s-t cut whose capacity equals the value of f. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 f is the maximum flow (S,T) is the minimum cut
Observation 3 Let f be a flow, and let (S,T)
be an s-t cut whose capacity equals the value of f. 1 3 2 4 5 6 3/3 3/8 4/4 2/2 1/4 4/4 2/3 f is the maximum flow (S,T) is the minimum cut
Max-Flow EQUAL Min-Cut
Example
1 3 2 4 5 6 3 8 4 2
4 4 3
1 3 2 4 5 6 3/3 3/8 4/4 2/2
1/4 4/4 2/3 Maximum Flow = 6
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Residual Network
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Minimum Cut = 6
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Minimum Cut = 6
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Minimum Cut = 6
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Minimum Cut = 6
The minimum capacity limit the maximum flow!
find a s-t cut
1 3 2 4 5 6 3/3 3/8 4/4 2/2
1/4 4/4 2/3 Maximum Flow = 6
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 Travel on Residual Network
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 start from source
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 don’t travel through full edge
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 don’t travel through full edge
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 no residual edge
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 no residual edge
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 s-t cut
1 3 2 4 5 6 0/3 5/8 0/4 0/2
3/4 0/4 1/3 s-t cut
result of starting from sink 1 3 2 4 5
6 0/3 5/8 0/4 0/2 3/4 0/4 1/3
result of starting from sink 1 3 2 4 5
6 0/3 5/8 0/4 0/2 3/4 0/4 1/3
Minimum cut is non-unique!
time complexity: based on max-flow algorithm Ford-Fulkerson algorithm O(EF) Edmonds-Karp
algorithm O(VE2) Dinic algorithm O(V2E)
Stoer Wagner only for undirected graph time complexity: O(N3) or
O(N2log2N)
UVa 10480 - Sabotage Practice Now
Problem List UVa 10480 UVa 10989 POJ 1815 POJ 2914
POJ 3084 POJ 3308 POJ 3469
Reference • http://www.flickr.com/photos/dgjones/335788038/ • http://www.flickr.com/photos/njsouthall/3181945005/ • http://www.csie.ntnu.edu.tw/~u91029/Cut.html • http://en.wikipedia.org/wiki/Cut_(graph_theory) •
http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem • http://www.cs.princeton.edu/courses/archive/spr04/cos226/lectures/ maxflow.4up.pdf • http://www.cnblogs.com/scau20110726/archive/ 2012/11/27/2791523.html
Thank You for Your Listening.