Overview of the Max-flow problem with sample code and example problem. Georgi Stoyanov Sofia University Student at.

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 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

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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?

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

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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

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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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