Max-flow min-cut
Overview of the Max-flow problem
with sample code and example problem.
Georgi Stoyanov
Student at
Sofia University
http://backtrack-it.blogspot.com
Table of Contents
1.
Definition of the problem
2.
Where does it occur?
3.
Max-flow min-cut theorem
4.
Example
5.
Max-flow algorithm
6.
Run-time estimation
7.
Questions
2
Definition of the problem
Definition of the problem
Maximum flow problems
Finding feasible flow
Through a single -source, -sink flow network
Flow is maximum
Many problems solved by Max-flow
The problem is often present at
algorithmic competitions
The Max-flow algorithm
Additional
definitions
Edge capacity – maximum flow that can go
through the edge
Residual edge capacity – maximum flow that
can pass after a certain amount has passed
residualCapacity = edgeCapacity – alreadyPassedFlow
Augmented path – path starting from source to
sink
Only edges with residual capacity above zero
5
Where does it occur?
Where does it occur?
In any kind of network with certain capacity
Network of pipes – how much water can pass
through the pipe network per unit of time?
7
Where does it occur?
Electricity network – how much electricity can
go through the grid?
8
Where does it occur?
The internet network – how much traffic can go
through a local network or the internet?
9
Where does it occur?
In other problems
Matching problem
Group of N guys and M girls
Every girl/guy likes a certain amount of people
from the other group
What is the maximum
number of couples, with
people who like each other?
10
Where does it occur?
Converting the matching problem to a max-flow
problem:
We add an edge with capacity one for every couple
that is acceptable
We add two bonus nodes – source and sink
We connect the source with the first group and the
second group with the sink
11
Max-flow min-cut theorem
Max-flow min-cut theorem
The max-flow min-cut theorem states that in a flow
network, the maximum amount of flow passing from
the source to the sink is equal to the minimum capacity
that when removed in a specific way from the network
causes the situation that no flow can pass from the
source to the sink.
13
Example
Example
Example
min( cf(A,D), cf(D,E), cf(E,G)) =
min( 3 – 0, 2 – 0, 1 – 0) =
min( 3, 2, 1) = 1
maxFlow = maxFlow + 1 = 1
15
Example
Example
min( cf(A,D), cf(D,F), cf(F,G)) =
min( 3 – 1, 6 – 0, 9 – 0) =
min( 2, 6, 9) = 2
maxFlow = maxFlow + 2 = 3
16
Example
Example
min( cf(A,B), cf(B,C), cf(C,D), cf(D,F), cf(F,G)) =
min( 3 – 0, 4 – 0, 1 – 0, 6 – 2, 9 - 2) =
min( 3, 4, 1, 4, 7) = 1
maxFlow = maxFlow + 1 = 4
17
Example
The flow in the previous slide is not optimal!
Reverting some of the flow through a different
path will achieve the optimal answer
To do that for each directed edge (u, v) we will
add an imaginary reverse edge (v, u)
The new edge shall
be used only if a certain
amount of flow has already passed through
the original edge!
18
Example
Example
min( cf(A,B), cf(B,C), cf(C,E), cf(E,D), cf(D,F), cf(F,g) ) =
min( 3 – 1, 4 – 1, 2 – 0, 0 – -1, 6 – 3, 9 - 3) =
min( 2, 3, 2, 1, 3, 6 ) = 1
maxFlow = maxFlow + 1 = 5 (which is the final answer)
19
The Max-flow algorithm
The Max-flow algorithm
The Edmonds-Karp algorithm
Uses a graph structure
Uses matrix of the capacities
Uses matrix for the passed flow
21
The Max-flow algorithm
The Edmonds-Karp algorithm
Uses breadth-first search on each iteration to
find a path from the source to the sink
Uses parent table to store the path
Uses path capacity table to store the value of
the maximum flow to a node in the path
22
The Max-flow algorithm initialization
#include<cstdio>
#include<queue>
#include<cstring>
#include<vector>
#include<iostream>
#define MAX_NODES 100
// the maximum number of nodes in the graph
#define INF 2147483646 // represents infity
#define UNINITIALIZED -1 // value for node with no parent
using namespace std;
// represents the capacities of the edges
int capacities[MAX_NODES][MAX_NODES];
// shows how much flow has passed through an edge
int flowPassed[MAX_NODES][MAX_NODES];
// represents the graph. The graph must contain the negative edges too!
vector<int> graph[MAX_NODES];
//shows the parents of the nodes of the path built by the BFS
int parentsList[MAX_NODES];
//shows the maximum flow to a node in the path built by the BFS
int currentPathCapacity[MAX_NODES];
23
The Max-flow algorithm - core
The “heart” of the algorithm:
int edmondsKarp(int startNode, int endNode) {
int maxFlow=0;
while(true) {
int flow=bfs(startNode, endNode);
if(flow==0) break;
maxFlow +=flow;
int currentNode=endNode;
while(currentNode != startNode) {
int previousNode = parentsList[currentNode];
flowPassed[previousNode][currentNode] += flow;
flowPassed[currentNode][previousNode] -= flow;
currentNode=previousNode;
}
}
return maxFlow;
}
24
The Max-flow algorithm –
Breadth-first search
Breadth-first
search
int bfs(int startNode, int endNode)
{
memset(parentsList, UNINITIALIZED, sizeof(parentsList));
memset(currentPathCapacity, 0, sizeof(currentPathCapacity));
queue<int> q;
q.push(startNode);
parentsList[startNode]=-2;
currentPathCapacity[startNode]=INF;
. . .
25
The Max-flow algorithm –
Breadth-first search
...
while(!q.empty()) {
int currentNode = q.front(); q.pop();
for(int i=0; i<graph[currentNode].size(); i++) {
int to = graph[currentNode][i];
if(parentsList[to] == UNINITIALIZED
&& capacities[currentNode][to] - flowPassed[currentNode][to] > 0) {
parentsList[to] = currentNode;
currentPathCapacity[to] = min(currentPathCapacity[currentNode],
capacities[currentNode][to] - flowPassed[currentNode][to]);
if(to == endNode) return currentPathCapacity[endNode];
q.push(to);
}
}
}
return 0;
}
26
Run-time estimation
Breaking
down the algorithm:
The BFS will cost O(E) operations to find a path
on each iteration
We will have total O(VE) path augmentations
(proved with Theorem and Lemmas)
This gives us total run-time of O(VE*E)
27
Run-time estimation
There are other algorithms that can run in O(V³)
time but are far more complicated to implement
! Note - this algorithm can also run in O(V³) time
for sparse graphs
28
The Max-flow algorithm
Perks of using the Edmonds-Karp algorithm
Runs relatively fast in sparse graphs
Represents a refined version of the FordFulkerson algorithm
Unlike the Ford-Fulkerson algorithm, this will
always terminate
It is relatively simple to implement
29
Summary
Many problems can be transformed to a max-flow
problem. So keep your eyes open!
The Edmonds-Karp algorithm is:
fairly fast for sparse graphs – O(V³)
easy to implement
runs in O(VE²) time
30
Summary
Don’t forget to add the reverse edges to your
graph!
The algorithm
Looks for augmenting path
from source to sink on each
iteration
Maximum flow == smallest residual capacity of an
edge in that path
31
Resources
Video lectures (in kind of English)
http://www.youtube.com/watch?v=J0wzih3_5Wo
http://en.wikipedia.org/wiki/Maximum_flow_problem
http://en.wikipedia.org/wiki/Edmonds%E2%80%93Kar
p_algorithm
http://en.wikipedia.org/wiki/Matching_(graph_theory)
Nakov’s book: Programming = ++Algorithms;
32
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