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Richard Hoshino Quest University Canada

Richard Hoshino Quest University Canada. Applying Combinatorics to Inspire Change. Game Of Fifteen. There are nine integers on the whiteboard: 1 2 3 4 5 6 7 8 9 You and I take turns selectin g one of these numbers, and then crossing it off the board.

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Richard Hoshino Quest University Canada

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  1. Richard HoshinoQuest University Canada • Applying Combinatorics to Inspire Change

  2. Game Of Fifteen There are nine integers on the whiteboard: 1 2 3 4 5 6 7 8 9 You and I take turns selecting one of these numbers, and then crossing it off the board. The winner is the first person to select three numbers adding up to 15. Can you beat me?

  3. Playing Tic-Tac-Toe! The Game of Fifteen is identical (isomorphic) to Tic-Tac-Toe!

  4. Eureka Moment I am really good at recognizing isomorphisms, i.e., situations when hard real-life societal problems can be converted into simpler equivalent math problems. This is because of my training in Discrete Mathematics, especially in graph theory and combinatorics.

  5. PART ONE HIGH SCHOOL OUTREACH

  6. CMS National Math Camp

  7. Nova Scotia Math League

  8. Nova Scotia Math Circles

  9. Writing a Novel

  10. Papers from my Ph.D. Thesis

  11. PART TWO MATH IN GOVERNMENT

  12. Canada Border Services Agency

  13. Marine Container Shipping

  14. Improving Risk-Assessment

  15. Reducing Wait Times

  16. Iris Biometrics Source: http://www.cbsr.ia.ac.cn/users/zfhe/research_IR.html

  17. Hamming Distance Comparison • Say a passenger has the following 20-digit iris code: 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 • Compare it to each of the images/codes in the gallery:

  18. Hamming Distance Comparison • Say a passenger has the following 20-digit iris code: 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 • Compare it to each of the images/codes in the gallery:

  19. Hamming Distance Comparison • Say a passenger has the following 20-digit iris code: 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 • Compare it to each of the images/codes in the gallery:

  20. Hamming Distance Comparison • Say a passenger has the following 20-digit iris code: 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 • Compare it to each of the images/codes in the gallery:

  21. Hamming Distance Comparison • Say a passenger has the following iris code: 0 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 • Compare it to each of the images/codes in the gallery: Passenger is Carol

  22. Genuine and Impostor Matches Genuine Distributionu* = 0.09, m* = 49

  23. Calibrated Confidence Scoring

  24. PART THREE JAPANESE BASEBALL LEAGUE

  25. Our Life in Chiba, Japan Our Apartment Kanda University Chiba Marine Stadium Train Station

  26. Chiba Lotte Marines

  27. Victory Parade in Chiba

  28. Life in Toronto…

  29. Inspiration

  30. Unexpected Inspiration

  31. Key Insight

  32. Chiba Marines Schedule (2010) (HOME sets are marked in red.)

  33. Nippon Pro Baseball Schedules Five Conditions: At-Most-Three No-Repeat Home-Away Each-Round Diff-Two|H−R| ≤ 2

  34. Traveling Tournament Problem Given an n × n distance matrix, determine the double round-robin tournament schedule that • satisfies At-Most-Three, No-Repeat, and Home-Away. • minimizes the total distance traveled by the n teams.

  35. An Example A-B-C-B-A – C-D-E-D-E is a valid team schedule under the Traveling Tournament Problem (TTP) but not for the Japanese Pro Baseball (violates Each-Round and Diff-Two).

  36. History of the TTP • TTP-solving algorithms are a complex hybrid of integer programming and constraint programming. • The TTP is NP-complete. Best solved instance is 10 teams.

  37. Multi-Round Balanced TTP Given an n × n distance matrix, find the distance-optimal tournament schedule that lasts 2k rounds (k blocks) and satisfies all five conditions: At-Most-Three, No-Repeat, Home-Away, Each-Round, Diff-Two.

  38. Graph-Theoretic Reformulation The length of the tournament is 2k rounds. We create a graph on 2km+2 vertices.

  39. Explanation of the variable m There are ways to select the home teams of any column. For n = 6, m = 120 × 20 = 2400. In general, So there are ways to select the three matches of any column.

  40. Graph-Theoretic Reformulation

  41. Graph-Theoretic Reformulation • Each team starts and ends the season at home (vstart, vend)

  42. Graph-Theoretic Reformulation Each vertex xt,u with 1 ≤ u ≤ m, represents the first two columns of the tth block (matches then home teams)

  43. Graph-Theoretic Reformulation Each vertex yt,u with 1 ≤ u ≤ m, represents the last two columns of the tth block (home teams then matches)

  44. The Edges xt,u→yt,v

  45. Construction of Edge xt,u→ yt,v • xt,u→ yt,v is an edge iff there exists a (feasible) block satisfying the five conditions. • The weight of edge xt,u→ yt,v is the minimum possible total distance traveled by the n teams within that block. x1,u y1,v

  46. The Edges yt,v→ xt+1,u

  47. Construction of Edge yt,v→ xt+1,u • yt,v→ xt+1,u is an edge iff the n × 4 concatenation matrix does not violate the at-most-three or no-repeat conditions. • The weight of edge yt,v→ xt+1,u is the distance traveled by the n teams moving from set 2t(n-1) to 2t(n-1)+1. y1,v x2,u

  48. Dijkstra’s Algorithm • The directed graph has 2mk+2 vertices and at most 2m+(2k-1)m2edges. Each edge has a weight. • Now apply Dijkstra’s Algorithm to find the shortest path vstart → x1,u1→ y1,v1 → … → xk,uk → yk,vk→ vend which produces the optimal solution of the mb-TTP.

  49. Optimal NPB Schedule • In the NPB, each team plays 120 intra-league games (40 sets of 3 games), with eight sets (24 games) against each of the other 5 teams. Thus, there are 8 rounds.

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