1 / 15

Max-Flow Min-Cut Applications:

Max-Flow Min-Cut Applications:. Playoff Elimination. Quick Review of MLB Playoffs. Two leagues, American and National: independent Three divisions per league Who makes playoffs? Top team in each division Team with best record among all second-place teams in league (“wildcard”).

wesley-mccullough
Télécharger la présentation

Max-Flow Min-Cut Applications:

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Max-Flow Min-Cut Applications: Playoff Elimination

  2. Quick Review of MLB Playoffs • Two leagues, American and National: independent • Three divisions per league • Who makes playoffs? • Top team in each division • Team with best record among all second-place teams in league (“wildcard”)

  3. End-of-Season Scenario American League • New York Wins East • Cleveland Wins Central • Texas Wins West • Chicago Wins Wildcard East Won Lost Pct. GB New York 77 61 0.558 - Baltimore 73 65 0.529 4 Boston 71 68 0.511 6.5 Toronto 63 76 0.453 14.5 Detroit 50 89 0.360 27.5 Central Won Lost Pct. GB Cleveland 81 57 0.587 - Chicago 75 65 0.536 7 Minnesota 69 70 0.496 12.5 Milwaukee 68 72 0.486 14 Kansas City 65 75 0.464 17 West Won Lost Pct. GB Texas 78 60 0.565 - Seattle 72 66 0.522 6 Oakland 67 74 0.475 12.5 California 64 75 0.460 14.5

  4. Can Detroit Win This Division? 8/30/96 W L GB GR New York 75 59 - 28 Baltimore 71 63 4 28 Boston 69 66 6.5 27 Toronto 63 72 12.5 27 Detroit 49 86 26.5 27

  5. Remaining Series in the AL East 8/30/96 Teams Games Baltimore vs. Boston 2 Baltimore vs. New York 3 Baltimore vs. Toronto 7 Boston vs. New York 8 New York vs. Toronto 7

  6. Proof of Detroit’s Elimination • Try to create a scenario where Detroit wins the division. • Detroit can finish with at most 76 wins. • New York already has 75 wins. • If New York wins 2 or more games, Detroit is eliminated. Team W L New York 77 85 Detroit 76 86

  7. Scenarios Where Detroit Can Win the Division. • Case 1: New York loses all it’s remaining games. • Case 2: New York wins 1 more game.

  8. Team W L Boston 77 85 Detroit 76 86 New York 75 87 Case 1: New York Loses All It’s Remaining Games. • New York plays Boston 8 more times. • Boston already has 69 wins. • Implies Boston finishes ahead of Detroit. • Detroit is eliminated in Case 1.

  9. Team W L Boston 76 86 Detroit 76 86 New York 76 86 Case 2: New York Wins Only 1 More Game • New York must win 1 game against Boston. 75 + 1= 76. • Boston cannot beat any teams other than New York. 69 + (8-1) = 76. • Implies that Detroit, Boston and New York each finish with 76 wins. What about Baltimore ?

  10. Team W L Baltimore 76 86 Boston 76 86 Detroit 76 86 New York 76 86 Baltimore • Already has 71 wins. • Plays New York 3 more times. • Plays Boston 2 more times. • Baltimore will finish with at least 76 wins. • A 4-way tie for first is possible, but only if Baltimore only beats New York and Boston. • What about Toronto ?

  11. Team W L Toronto 77 85 Baltimore 76 86 Boston 76 86 Detroit 76 86 New York 76 86 Toronto • Already has 63 wins. • Plays New York 7 more times. • Plays Baltimore 7 more times. • Previous assumptions imply Toronto finishes with at least 77 wins. • Detroit cannot win the division in Case 2 either.

  12. Literature Review • B.Schwartz, 1966 • A. Hoffman and J. Rivlin, 1967 • Solve First Place Elimination Problem via Network Flows • L. Robinson, 1991 • Extensions to Other Sports • D. Gusfield and C. Martel, 1992 • Minimum Number of Games to Avoid Elimination From First Place • K. Wayne, 2001 • First Place Elimination Number via network flows

  13. Bal vs. Bos. Bal Bos Bal. vs. N.Y. Bal. vs. Tor. Bos. vs. N.Y. N.Y. N.Y. vs. Tor. Tor Detroit’s Elimination Network Game Nodes Team Nodes 76-71=5 2 3 76-69=7 7 t s 76-75=1 8 7 76-63=13 Arcs with  capacity.

  14. Bal vs. Bos. Bal Bos Bal. vs. N.Y. Bal. vs. Tor. Bos. vs. N.Y. N.Y. N.Y. vs. Tor. Tor Detroit’s Elimination Network Game Nodes Team Nodes 5 2 3 7 7 t s 1 8 7 13 u[S,T] = 26 < # wins remaining = 27

  15. More Interesting Questions! • I. Adler, A. Erera, D. Hochbaum and E. Olinick, 2002 • http://riot.ieor.berkeley.edu/~baseball • “Clinching” Questions • Is my team guaranteed to win division? Make playoffs? • “Magic Number” Questions • How many games must my team to win to clinch? • How many games must they win to avoid elimination?

More Related