Presentation is loading. Please wait.

Presentation is loading. Please wait.

Correctness of Edmonds-Karp

Published byAmanda Bradley Modified over 5 years ago

Similar presentations

Presentation on theme: "Correctness of Edmonds-Karp"— Presentation transcript:

1 Correctness of Edmonds-Karp
An edge serves as bottleneck at most n/2 times

2 An edge serves as bottleneck
Step k: Step k’’:

3 Bottleneck edge Step k: Step k+1 Step k’’:

4 Bottleneck edge Step k: Step k+1 Step k’ Step k’’:

5 The level number never decreases

6 The k-th iteration Residual graph BFS tree: the edges in red Level 3
2 4 4 4 4 6 8 6 2 6 s 1 3 9 5 1 t 9 9 BFS tree: the edges in red Level 3 2 4 4 4 4 6 8 6 2 6 s 1 3 9 5 1 t 9 9 Level 0 Level 1 Level 2

7 The (k+1)-th iteration Residual graph BFS tree: the edges in red
2 4 4 5 4 1 5 6 2 6 7 s 1 3 9 5 t 9 10 BFS tree: the edges in red Level 3 2 4 4 5 4 1 5 6 2 6 7 s 1 3 9 5 t 9 9 Level 0 Level 1 Level 2 Level 4

Download ppt "Correctness of Edmonds-Karp"

Similar presentations

The Build-up of the Red Sequence at z

The Build-up of the Red Sequence at z
Simple Graph Warmup. Cycles in Simple Graphs A cycle in a simple graph is a sequence of vertices v 0, …, v n for some n>0, where v 0, ….v n-1 are distinct,

Simple Graph Warmup. Cycles in Simple Graphs A cycle in a simple graph is a sequence of vertices v 0, …, v n for some n>0, where v 0, ….v n-1 are distinct,
Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.
* Bellman-Ford: single-source shortest distance * O(VE) for graphs with negative edges * Detects negative weight cycles * Floyd-Warshall: All pairs shortest.

* Bellman-Ford: single-source shortest distance * O(VE) for graphs with negative edges * Detects negative weight cycles * Floyd-Warshall: All pairs shortest.
Max Flow Problem Given network N=(V,A), two nodes s,t of V, and capacities on the arcs: uij is the capacity on arc (i,j). Find non-negative flow fij for.

Max Flow Problem Given network N=(V,A), two nodes s,t of V, and capacities on the arcs: uij is the capacity on arc (i,j). Find non-negative flow fij for.
Network Optimization Problems: Models and Algorithms This handout: Minimum Spanning Tree Problem.

Network Optimization Problems: Models and Algorithms This handout: Minimum Spanning Tree Problem.
What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.
Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.
Determine whether each curve below is the graph of a function of x. Select all answers that are graphs of functions of x: 1234567891011121314151617181920.

Determine whether each curve below is the graph of a function of x. Select all answers that are graphs of functions of x:
Prim’s Algorithm and an MST Speed-Up

Prim’s Algorithm and an MST Speed-Up
The Mathematics of Networks Chapter 7. Trees A tree is a graph that –Is connected –Has no circuits Tree.

The Mathematics of Networks Chapter 7. Trees A tree is a graph that –Is connected –Has no circuits Tree.
MST: Red Rule, Blue Rule. 2 MST Instance 1 3 8 2 6 7 4 5.

MST: Red Rule, Blue Rule. 2 MST Instance
$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.

$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.
Exercise Find a schedule that minimises makespan..

Exercise Find a schedule that minimises makespan..
Spanning Trees. A spanning tree for a connected, undirected graph G is a graph S consisting of the nodes of G together with enough edges of G such that:

Spanning Trees. A spanning tree for a connected, undirected graph G is a graph S consisting of the nodes of G together with enough edges of G such that:
Runtime O(VE), for +/- edges, Detects existence of neg. loops

Runtime O(VE), for +/- edges, Detects existence of neg. loops
3 t 4 1 2 s2s2 s1s1 2 1 4 23 3 1 3 3 3 1 the black number next to an arc is its capacity.

3 t s2s2 s1s the black number next to an arc is its capacity.
Tree Diagrams A tree is a connected graph in which every edge is a bridge. There can NEVER be a circuit in a tree diagram!

Tree Diagrams A tree is a connected graph in which every edge is a bridge. There can NEVER be a circuit in a tree diagram!
Dijkstra animation. Dijksta’s Algorithm (Shortest Path Between 2 Nodes) 2 Phases:initialization;iteration Initialization: 1. Included:(Boolean) 2. Distance:(Weight)

Dijkstra animation. Dijksta’s Algorithm (Shortest Path Between 2 Nodes) 2 Phases:initialization;iteration Initialization: 1. Included:(Boolean) 2. Distance:(Weight)
Parallel Lines.   Two lines that have the same slope.   If you graph them they will never intersect.   We can find the equation of a line parallel.

Parallel Lines.   Two lines that have the same slope.   If you graph them they will never intersect.   We can find the equation of a line parallel.

Similar presentations

Ads by Google