max-flow min-cut
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Max-flow min-cut. Overview of the Max-flow problem with sample code and example problem.. Georgi Stoyanov. Sofia University. http:// backtrack-it.blogspot.com. Student at. Table of Contents. Definition of the problem Where does it occur? Max-flow min-cut theorem Example - PowerPoint PPT PresentationTRANSCRIPT
Max-flow min-cutOverview of the Max-flow problem
with sample code and example problem.
Georgi Stoyanov
Sofia University
http://backtrack-it.blogspot.com
Student at
Table of Contents1. Definition of the problem2. Where does it occur?3. Max-flow min-cut theorem4. Example5. Max-flow algorithm6. Run-time estimation7. Questions
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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 zero5
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?
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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
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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.
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Example
Example Example
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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
Example Example
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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
Example Example
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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
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!
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Example Example
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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)
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
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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
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The Max-flow algorithm - initialization
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#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 edgesint capacities[MAX_NODES][MAX_NODES];// shows how much flow has passed through an edgeint 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 BFSint parentsList[MAX_NODES];//shows the maximum flow to a node in the path built by the BFSint currentPathCapacity[MAX_NODES];
The Max-flow algorithm - core
The “heart” of the algorithm:
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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;}
The Max-flow algorithm – Breadth-first search
Breadth-first search
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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;
. . .
The Max-flow algorithm – Breadth-first search
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... 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;}
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)
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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
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The Max-flow algorithm Perks of using the Edmonds-Karp algorithm Runs relatively fast in sparse
graphs Represents a refined version of the
Ford-Fulkerson algorithm Unlike the Ford-Fulkerson
algorithm, this will always terminate
It is relatively simple to implement29
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
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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%93Karp_algorithm
http://en.wikipedia.org/wiki/Matching_(graph_theory)
Nakov’s book: Programming = ++Algorithms;
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