Richard Hoshino Post-Doctoral Fellow National Institute of Informatics

Applying Combinatorics to Inspire Change

Richard HoshinoQuest University CanadaGame Of FifteenThere 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?

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

Eureka MomentI 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.

PART ONE

HIGH SCHOOL OUTREACH

CMS National Math Camp6

Nova Scotia Math League

7Nova Scotia Math Circles

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Writing a Novel

Papers from my Ph.D. Thesis

PART TWO

MATH IN GOVERNMENT

Canada Border Services Agency

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Marine Container Shipping13

Improving Risk-Assessment14

Reducing Wait Times15

Iris Biometrics

Source: http://www.cbsr.ia.ac.cn/users/zfhe/research_IR.htmlHamming Distance ComparisonSay 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:

Image01011010010100010101HDAlice01100101010011010011Bob10101010010010101001Carol01010010010101010101Hamming Distance ComparisonSay 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:

Image01011010010100010101HDAlice011001010100110100110.55BobCarolHamming Distance ComparisonSay 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:

Image01011010010100010101HDAlice0.55Bob101010100100101010010.50CarolHamming Distance ComparisonSay 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:

Image01011010010100010101HDAlice0.55Bob0.50Carol010100100101010101010.10Hamming Distance ComparisonSay 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:

HDAlice0.55Bob0.50Carol0.10Passenger is CarolGenuine and Impostor MatchesGenuine Distributionu* = 0.09, m* = 49

Calibrated Confidence Scoring

PART THREE

JAPANESE BASEBALL LEAGUE

Our Life in Chiba, Japan

Our ApartmentKanda UniversityChiba Marine StadiumTrain Station

Chiba Lotte Marines

Victory Parade in Chiba

Life in Toronto

Inspiration

Unexpected Inspiration

Key InsightR2TFOSHCOSHFOSFTCHTCFSSCHOTFHTCOR3TFOSHCHSFOSTFOCFSCHTOCHTSHOTCFR1SHTFOHFCOSTCOSFFSHTCOTSCHCOFHTR4OSHTFSFOCHFOCSTCHTFSHTSOCTCFHO

Chiba Marines Schedule (2010)12345SaitamaHokkaidoTohokuOrixFukuoka678910SaitamaHokkaidoOrixTohokuFukuoka1112131415SaitamaFukuokaHokkaidoOrixTohoku1617181920OrixHokkaidoFukuokaSaitamaTohoku(HOME sets are marked in red.)

Nippon Pro Baseball Schedules12345SaitamaHokkaidoTohokuOrixFukuoka678910SaitamaHokkaidoOrixTohokuFukuokaFive Conditions:

At-Most-Three

No-Repeat

Home-Away

Each-Round

Diff-Two |HR| 2Traveling Tournament ProblemGiven an n n distance matrix, determine the double round-robin tournament schedule thatsatisfies At-Most-Three, No-Repeat, and Home-Away.minimizes the total distance traveled by the n teams.An ExampleA-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).History of the TTPTTP-solving algorithms are a complex hybrid of integer programming and constraint programming.

The TTP is NP-complete. Best solved instance is 10 teams.Multi-Round Balanced TTPGiven 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.

Graph-Theoretic Reformulation

The length of the tournament is 2k rounds.We create a graph on 2km+2 vertices.Explanation of the variable m12D1E1F1A0B0C0TeamABCDEFFor n = 6, m = 120 20 = 2400. In general,

There are

ways to select the home teams of any column.So there are

ways to select the three matches of any column.

Graph-Theoretic Reformulation

Graph-Theoretic Reformulation

Each team starts and ends the season at home (vstart, vend)

Graph-Theoretic Reformulation

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

Each vertex yt,u with 1 u m, represents the last two columns of the tth block (home teams then matches)The Edges xt,u yt,v

Construction of Edge xt,u yt,v12B0A0F0E1D1C1TeamABCDEF9101F0E1D1C0B0Ax1,uy1,vxt,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.345678????????????????????????????????????The Edges yt,v xt+1,u

Construction of Edge yt,v xt+1,uTeamABCDEF9101F0E1D1C0B0A1112B0A0F0E1D1C1y1,vx2,uyt,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.Dijkstras Algorithm

The directed graph has 2mk+2 vertices and at most 2m+(2k-1)m2 edges. Each edge has a weight.Now apply Dijkstras 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.Optimal NPB ScheduleIn 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.TeamR1R2R3R4R5R6R7R8ChibaSHTFOTFOSHTFOSHOSHTFHOFSTFSTHOTSHOFHOFTSTohokuHFCOSCOSHFCHSFOSFOCHSHOFCOFCSHCHOFSOFSCHHokkaidoTCOSFOSFTCSTFOCFOCSTCTSOFSOFCTFTCSOCSOFTOrixFSHTCHTCFSFSCHTCHTFSFCTHSTHSFCSFTCHTCHSFFukuokaOTSCHSCHOTOCHTSHTSOCOSCTHCTHOSHOSTCSTCHOSaitamaCOFHTFHTCOHOTCFTCFHOTFHCOHCOTFOCFHTFHTOC

Results for NPB Pacific LeagueFor the 6-team NPB Central League, we achieve a 26.8% reduction in total travel distance 14.6% reduction in total trips taken.Team NameDistance(2010)Distance(New)Reduction in TravelTrips(2010)Trips (New)Reduction in TripsChiba 23,26616,60628.6%362919.4%Tohoku23,71017,97524.2%372921.6%Hokkaido28,59920,23429.2%322715.6%Orix24,12818,71322.4%342914.7%Fukuoka33,35221,14336.6%352722.9%Saitama20,88519,4986.6%342817.6%TOTAL153,940114,16925.8%20816918.8%Nippon Pro Baseball League

53NPB Requirements1) Weekday and Weekend balancing20 weekday sets (half home, half away)20 weekend sets (half home, half away)

2) If possible, enforce at-most-two (no 3-set home stands, 3-set road trips)Results with New RequirementsScheduleDistanceTripsCentral League (2012)86,384206mb-TTP (Five Conditions)57,836170All Seven Conditions66,1221957 Conditions + Constraints76,598194Final Central League Schedule (2013)80,006194Inspiring Change with Research

PART FOUR

QUEST UNIVERSITY CANADA

Inspiring Change at Quest1) A new Roommate Matching algorithmMore efficient, increased compatibility

2) A new Course Registration SystemMore equitable, increased effectiveness

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ConclusionWe make a living by what we get.

We make a life by what we give.

Winston Churchill

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