cse 421 algorithms
Download
Skip this Video
Download Presentation
CSE 421 Algorithms

Loading in 2 Seconds...

play fullscreen
1 / 17

CSE 421 Algorithms - PowerPoint PPT Presentation


  • 53 Views
  • Uploaded on

CSE 421 Algorithms. Richard Anderson Lecture 21 Shortest Path Network Flow Introduction. Announcements. Friday, 11/18, Class will meet in CSE 305 Reading 7.1-7.3, 7.5-7.6 Section 7.4 will not be covered. Find the shortest paths from v with exactly k edges. 7. y. v. 5. 1. -2. -2.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' CSE 421 Algorithms' - kelsie-wilkinson


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
cse 421 algorithms

CSE 421Algorithms

Richard Anderson

Lecture 21

Shortest Path

Network Flow Introduction

announcements
Announcements
  • Friday, 11/18, Class will meet in CSE 305
  • Reading 7.1-7.3, 7.5-7.6
    • Section 7.4 will not be covered
express as a recurrence
Express as a recurrence
  • Optk(w) = minx [Optk-1(x) + cxw]
  • Opt0(w) = 0 if v=w and infinity otherwise
algorithm version 1
Algorithm, Version 1

foreach w

M[0, w] = infinity;

M[0, v] = 0;

for i = 1 to n-1

foreach w

M[i, w] = minx(M[i-1,x] + cost[x,w]);

algorithm version 2
Algorithm, Version 2

foreach w

M[0, w] = infinity;

M[0, v] = 0;

for i = 1 to n-1

foreach w

M[i, w] = min(M[i-1, w], minx(M[i-1,x] + cost[x,w]))

algorithm version 3
Algorithm, Version 3

foreach w

M[w] = infinity;

M[v] = 0;

for i = 1 to n-1

foreach w

M[w] = min(M[w], minx(M[x] + cost[x,w]))

correctness proof for algorithm 3
Correctness Proof for Algorithm 3
  • Key lemma – at the end of iteration i, for all w, M[w] <= M[i, w];
  • Reconstructing the path:
    • Set P[w] = x, whenever M[w] is updated from vertex x

7

y

v

5

1

-2

-2

3

1

x

3

z

if the pointer graph has a cycle then the graph has a negative cost cycle
If the pointer graph has a cycle, then the graph has a negative cost cycle
  • If P[w] = x then M[w] >= M[x] + cost(x,w)
    • Equal after update, then M[x] could be reduced
  • Let v1, v2,…vk be a cycle in the pointer graph with (vk,v1) the last edge added
    • Just before the update
      • M[vj] >= M[vj+1] + cost(vj+1, vj) for j < k
      • M[vk] > M[v1] + cost(v1, vk)
    • Adding everything up
      • 0 > cost(v1,v2) + cost(v2,v3) + … + cost(vk, v1)

v1

v4

v2

v3

negative cycles
Negative Cycles
  • If the pointer graph has a cycle, then the graph has a negative cycle
  • Therefore: if the graph has no negative cycles, then the pointer graph has no negative cycles
finding negative cost cycles
Finding negative cost cycles
  • What if you want to find negative cost cycles?
network flow definitions
Network Flow Definitions
  • Capacity
  • Source, Sink
  • Capacity Condition
  • Conservation Condition
  • Value of a flow
flow example
Flow Example

u

20

10

30

s

t

10

20

v

residual graph
Residual Graph

u

u

15/20

0/10

5

10

15

15/30

s

t

15

15

s

t

5

5/10

20/20

5

20

v

v

ad