15.082J, 6.855J, and ESD.78J
September 21, 2010
Eulerian Walks
Flow Decomposition and
Transformations
Eulerian Walks in Directed Graphs in O(m) time.
Step 1. Create a breadth first search tree into node
1. For j not equal to 1, put the arc out of j in T
l
15.082J/6.855J/ESD.78J
September 14, 2010
Data Structures
Overview of this Lecture
A very fast overview of some data structures that
we will be using this semester
lists, sets, stacks, queues, networks, trees
a variation on the well known heap data
stru
15.082J & 6.855J & ESD.78J
October 14, 2010
Maximum Flows 2
Review of the FordFulkerson Algorithm
x := 0;
create the residual network G(x);
while there is some directed path from s to t in G(x) do
let P be a path from s to t in G(x);
* := (P);
send * uni
15.082J & 6.855J & ESD.78J
September 23, 2010
Dijkstras Algorithm for the Shortest
Path Problem
Single source shortest path problem
2
4
4
2
2
1
1
2
3
4
6
2
3
3
5
Find the shortest path from a source node to
each other node.
Assume: (1) all arc lengths are
15.082J & 6.855J & ESD.78J
September 30, 2010
The Label Correcting Algorithm
Overview of the Lecture
A generic algorithm for solving shortest path
problems
negative costs permitted
but no negative cost cycle (at least for now)
The use of reduced costs
A
15.082J,6.855J,andESD.78J
Sept16,2010
Lecture3.GraphSearch
BreadthFirstSearch
DepthFirstSearch
Introtoprogramverification
TopologicalSort
Overview
Today: Different ways of searching a graph
a generic approach
breadth first search
depth first search
pr
15.082and6.855JFall2010
Network Optimization
J.B. Orlin
WELCOME!
Welcome to 15.082/6.855J
Introduction to Network Optimization
Instructor: James B. Orlin
TA: David Goldberg
Textbook: Network Flows: Theory, Algorithms,
and Applications by Ahuja, Magnanti,
15.082J & 6.855J & ESD.78J
Shortest Paths 2:
Bucket implementations of Dijkstras Algorithm
RHeaps
A Simple Bucketbased Scheme
Let C = 1 + max(cij : (i,j) A); then nC is an upper
bound on the minimum length path from 1 to n.
RECALL: When we select nodes
15.082J & 6.855J & ESD.78J
October 7, 2010
Introduction to Maximum Flows
The Max Flow Problem
G
xij =
=
(N,A)
flow on arc (i,j)
uij
=
capacity of flow in arc (i,j)
s
t
=
=
source node
sink node
Maximize
v
Subject to
j xij
j xsj

k xki=
0 for each i s,