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Last updated: 2009 February 18 Wednesday 17:43:10.513 Central Standard Time = GMT/UTC-06:00
2009 February 18 Wednesday 23:43:10.513 Greenwich Mean Time = GMT/UTC
JULIAN DATE= 2454881
This is a very difficult math problem to get the ultimate "perfect" answer - that is, the absolute
minimum distance sum. The technique used is "simulated annealing". I'll be continually running it and trying to improve the answer.
So check here every once in a while to see if it's been updated.
College Basketball Sectionals
The 1-64 listing of the sectionals is in a rough east-west order. The combining of the sectionals
into regionals will be done in the next stage. So don't read anything into the 1-64 ordering
now.
The goal is to minimize the total mileage - that is, have the teams as close together as possible.
By mileage, I'm referring to the sum of the mileages from the 64 sectionals. The mileage for a
given sectional is the sum of the distances between all possible pairs of teams in a sectional.
In a 5-team sectional (which 37 are),there are 5*4 ÷ 2 = 10 possible pairs of teams.
In a 6-team sectional (which 27 are) there are 6*5 ÷ 2 = 15 possible pairs of teams.
37 sectionals with 5 teams = 185 teams 37 * 10 = 370 pairs
27 sectionals with 6 teams = 162 teams 27 * 15 = 405 pairs
------------- -----------
347 teams 775 pairs
And the numbers to the left of each team's name are simply the code numbers used by the program
for each team. They have no other significance.
The suggestion here is that the host role of the sectionals would rotate among the teams
so that each team gets to host its sectional every 5(6) years.
total miles=67603.1579 total pairs=775
average miles between pairs of teams=87.23
Further Explanation
________________________________________________________________________________
SECTIONAL# 1 total miles=461 average distance between teams=31 pairs=15 teams=6
6 Boston College $
4 Boston U. $
5 Harvard $
11 Holy Cross $
2 New Hampshire $
3 Northeastern $
________________________________________________________________________________
SECTIONAL# 2 total miles=587 average distance between teams=39 pairs=15 teams=6
7 Brown $
9 Bryant College $
12 Connecticut $
14 Massachusetts $
8 Providence $
10 Rhode Island $
________________________________________________________________________________
SECTIONAL# 3 total miles=1548 average distance between teams=155 pairs=10 teams=5
25 Albany-NY $
13 Dartmouth $
1 Maine $
24 Siena $
20 Vermont $
________________________________________________________________________________
SECTIONAL# 4 total miles=401 average distance between teams=27 pairs=15 teams=6
16 Cent. Connecticut State $
22 Fairfield $
15 Hartford $
17 Quinnipiac $
21 Sacred Heart $
18 Yale $
________________________________________________________________________________
SECTIONAL# 5 total miles=568 average distance between teams=38 pairs=15 teams=6
32 Army $
36 Fairleigh Dickinson $
26 Iona $
29 Manhattan $
30 Marist $
19 Stony Brook-NY $
________________________________________________________________________________
SECTIONAL# 6 total miles=179 average distance between teams=12 pairs=15 teams=6
31 Columbia $
28 Fordham $
23 Hofstra $
33 Long Island U. $
34 St. Francis-NY $
27 St. John's $
________________________________________________________________________________
SECTIONAL# 7 total miles=285 average distance between teams=19 pairs=15 teams=6
35 Monmouth-NJ $
39 NJIT(New Jersey Tech) $
41 Rutgers $
37 Saint Peter's $
40 Seton Hall $
38 Wagner $
________________________________________________________________________________
SECTIONAL# 8 total miles=497 average distance between teams=33 pairs=15 teams=6
45 La Salle $
48 Lafayette $
51 Lehigh $
42 Princeton $
43 Rider $
50 Villanova $
________________________________________________________________________________
SECTIONAL# 9 total miles=427 average distance between teams=28 pairs=15 teams=6
55 Delaware $
53 Delaware State $
46 Drexel $
47 Pennsylvania $
49 Saint Joseph's $
44 Temple $
________________________________________________________________________________
SECTIONAL#10 total miles=782 average distance between teams=78 pairs=10 teams=5
56 Binghamton-NY $
69 Bucknell $
52 Colgate $
61 Cornell $
57 Syracuse $
________________________________________________________________________________
SECTIONAL#11 total miles=496 average distance between teams=50 pairs=10 teams=5
60 Hampton $
54 Md.-Eastern Shore $
58 Norfolk State $
59 Old Dominion $
67 William & Mary $
________________________________________________________________________________
SECTIONAL#12 total miles=284 average distance between teams=19 pairs=15 teams=6
66 Coppin State $
65 Loyola-Maryland $
63 Morgan State $
76 Mount St. Mary's $
64 Towson $
68 UMBC(Md-Balt) $
________________________________________________________________________________
SECTIONAL#13 total miles=198 average distance between teams=13 pairs=15 teams=6
74 American U. $
72 George Washington $
73 Georgetown $
71 Howard $
70 Maryland $
62 Navy $
________________________________________________________________________________
SECTIONAL#14 total miles=672 average distance between teams=67 pairs=10 teams=5
75 George Mason $
82 Longwood $
79 Richmond $
78 Va. Commonwealth $
84 Virginia $
________________________________________________________________________________
SECTIONAL#15 total miles=817 average distance between teams=82 pairs=10 teams=5
88 Buffalo $
89 Canisius $
94 Niagara $
80 Penn State $
83 St. Bonaventure $
________________________________________________________________________________
SECTIONAL#16 total miles=662 average distance between teams=44 pairs=15 teams=6
87 Campbell $
92 Duke $
77 East Carolina $
91 NC Central $
86 NC State $
95 North Carolina $
________________________________________________________________________________
SECTIONAL#17 total miles=825 average distance between teams=82 pairs=10 teams=5
108 Charleston Southern $
104 Citadel $
93 Coastal Carolina $
101 College of Charleston $
81 NC Wilmington $
________________________________________________________________________________
SECTIONAL#18 total miles=476 average distance between teams=48 pairs=10 teams=5
105 Duquesne $
103 Pittsburgh $
107 Robert Morris $
85 St. Francis-Pa. $
102 West Virginia $
________________________________________________________________________________
SECTIONAL#19 total miles=709 average distance between teams=71 pairs=10 teams=5
90 James Madison $
96 Liberty $
115 Radford $
97 VMI $
114 Virginia Tech $
________________________________________________________________________________
SECTIONAL#20 total miles=315 average distance between teams=21 pairs=15 teams=6
98 Elon $
106 High Point $
99 N.C. A&T $
100 NC Greensboro $
111 Wake Forest $
110 Winston-Salem State $
________________________________________________________________________________
SECTIONAL#21 total miles=1118 average distance between teams=112 pairs=10 teams=5
109 Florida Atlantic $
133 Florida Gulf Coast $
113 Florida International $
112 Miami-Florida $
142 South Florida $
________________________________________________________________________________
SECTIONAL#22 total miles=996 average distance between teams=100 pairs=10 teams=5
134 Georgia Southern $
135 Presbyterian College $
123 Savannah State $
120 South Carolina $
119 South Carolina State $
________________________________________________________________________________
SECTIONAL#23 total miles=1001 average distance between teams=67 pairs=15 teams=6
132 Appalachian State $
117 Charlotte $
118 Davidson $
140 East Tennessee State $
130 Gardner-Webb $
121 Winthrop $
________________________________________________________________________________
SECTIONAL#24 total miles=1077 average distance between teams=72 pairs=15 teams=6
122 Bethune-Cookman $
124 Central Florida(UCF) $
139 Florida $
129 Jacksonville $
127 North Florida(UNF) $
125 Stetson $
________________________________________________________________________________
SECTIONAL#25 total miles=766 average distance between teams=77 pairs=10 teams=5
128 Akron $
131 Cleveland State $
126 Kent State $
138 Ohio U. $
116 Youngstown State $
________________________________________________________________________________
SECTIONAL#26 total miles=717 average distance between teams=48 pairs=15 teams=6
145 Clemson $
141 Furman $
144 NC Asheville $
137 USC Upstate $
148 Western Carolina $
136 Wofford $
________________________________________________________________________________
SECTIONAL#27 total miles=691 average distance between teams=46 pairs=15 teams=6
154 Bowling Green $
147 Detroit $
153 Eastern Michigan $
156 Michigan $
149 Oakland-Mich. $
152 Toledo $
________________________________________________________________________________
SECTIONAL#28 total miles=567 average distance between teams=57 pairs=10 teams=5
150 Georgia $
163 Georgia State $
164 Georgia Tech $
169 Kennesaw State $
155 Mercer $
________________________________________________________________________________
SECTIONAL#29 total miles=888 average distance between teams=89 pairs=10 teams=5
162 Eastern Kentucky $
167 Kentucky $
178 Louisville $
143 Marshall $
151 Morehead State $
________________________________________________________________________________
SECTIONAL#30 total miles=746 average distance between teams=50 pairs=15 teams=6
168 Cincinnati $
159 Dayton $
170 Miami-Ohio $
146 Ohio State $
158 Wright State $
165 Xavier-Ohio $
________________________________________________________________________________
SECTIONAL#31 total miles=1116 average distance between teams=112 pairs=10 teams=5
184 Alabama State $
175 Auburn $
160 Florida A&M $
161 Florida State $
180 Troy $
________________________________________________________________________________
SECTIONAL#32 total miles=987 average distance between teams=99 pairs=10 teams=5
171 Central Michigan $
172 IPFW $
166 Michigan State $
183 Notre Dame $
177 Western Michigan $
________________________________________________________________________________
SECTIONAL#33 total miles=971 average distance between teams=97 pairs=10 teams=5
188 Alabama A&M $
173 Chattanooga $
185 Middle Tennessee $
157 Tennessee $
176 Tennessee Tech $
________________________________________________________________________________
SECTIONAL#34 total miles=958 average distance between teams=64 pairs=15 teams=6
174 Ball State $
181 Butler $
182 IUPUI $
187 Indiana $
198 Indiana State $
195 Purdue $
________________________________________________________________________________
SECTIONAL#35 total miles=485 average distance between teams=32 pairs=15 teams=6
197 Austin Peay $
190 Belmont $
191 Lipscomb $
194 Tennessee State $
193 Vanderbilt $
186 Western Kentucky $
________________________________________________________________________________
SECTIONAL#36 total miles=816 average distance between teams=82 pairs=10 teams=5
200 Alabama $
179 Jacksonville State $
214 Mississippi State $
189 Samford $
192 UAB $
________________________________________________________________________________
SECTIONAL#37 total miles=334 average distance between teams=22 pairs=15 teams=6
201 Chicago State $
203 DePaul $
202 Illinois-Chicago $
204 Loyola-Chicago $
205 Northwestern $
196 Valparaiso $
________________________________________________________________________________
SECTIONAL#38 total miles=932 average distance between teams=93 pairs=10 teams=5
208 Marquette $
213 Northern Illinois $
207 Wis.-Green Bay $
206 Wis.-Milwaukee $
219 Wisconsin $
________________________________________________________________________________
SECTIONAL#39 total miles=936 average distance between teams=94 pairs=10 teams=5
199 Evansville $
220 SE Missouri State $
224 SIU-Edwardsville $
228 Saint Louis $
217 Southern Illinois $
________________________________________________________________________________
SECTIONAL#40 total miles=837 average distance between teams=84 pairs=10 teams=5
222 Bradley $
209 Eastern Illinois $
211 Illinois $
216 Illinois State $
232 Western Illinois $
________________________________________________________________________________
SECTIONAL#41 total miles=1078 average distance between teams=108 pairs=10 teams=5
231 Arkansas State $
223 Memphis $
221 Mississippi $
212 Murray State $
215 Tennessee-Martin $
________________________________________________________________________________
SECTIONAL#42 total miles=1297 average distance between teams=130 pairs=10 teams=5
234 Alcorn State $
227 Jackson State $
229 MVSU(Mississippi Valley St.) $
210 South Alabama $
218 Southern Mississippi $
________________________________________________________________________________
SECTIONAL#43 total miles=726 average distance between teams=48 pairs=15 teams=6
235 LSU $
225 New Orleans $
233 Nicholls State $
230 SE Louisiana $
236 Southern University $
226 Tulane $
________________________________________________________________________________
SECTIONAL#44 total miles=1342 average distance between teams=134 pairs=10 teams=5
252 Drake $
237 Iowa $
251 Iowa State $
241 Missouri $
244 Northern Iowa $
________________________________________________________________________________
SECTIONAL#45 total miles=1198 average distance between teams=120 pairs=10 teams=5
242 Ark.-Little Rock $
239 Ark.-Pine Bluff $
255 Arkansas $
243 Central Arkansas $
250 Missouri State $
________________________________________________________________________________
SECTIONAL#46 total miles=555 average distance between teams=55 pairs=10 teams=5
253 Centenary $
246 Grambling $
245 Louisiana Tech $
240 Louisiana-Monroe $
247 Northwestern State $
________________________________________________________________________________
SECTIONAL#47 total miles=1230 average distance between teams=123 pairs=10 teams=5
254 Lamar $
238 Louisiana-Lafayette $
248 McNeese State $
263 Sam Houston State $
257 Stephen F. Austin $
________________________________________________________________________________
SECTIONAL#48 total miles=586 average distance between teams=39 pairs=15 teams=6
259 Houston $
262 Houston Baptist $
267 Prairie View A&M $
261 Rice $
268 Texas A&M $
260 Texas Southern $
________________________________________________________________________________
SECTIONAL#49 total miles=1178 average distance between teams=118 pairs=10 teams=5
265 Creighton $
258 Kansas $
269 Kansas State $
270 Nebraska $
256 UMKC $
________________________________________________________________________________
SECTIONAL#50 total miles=2156 average distance between teams=216 pairs=10 teams=5
249 Minnesota $
276 North Dakota $
273 North Dakota State $
274 South Dakota $
271 South Dakota State $
________________________________________________________________________________
SECTIONAL#51 total miles=961 average distance between teams=96 pairs=10 teams=5
283 Oklahoma $
275 Oklahoma State $
266 Oral Roberts $
264 Tulsa $
280 Wichita State $
________________________________________________________________________________
SECTIONAL#52 total miles=1976 average distance between teams=132 pairs=15 teams=6
277 Baylor $
279 North Texas $
272 SMU $
282 TCU $
288 Texas Tech $
278 Texas-Arlington $
________________________________________________________________________________
SECTIONAL#53 total miles=1491 average distance between teams=149 pairs=10 teams=5
284 Texas $
281 Texas A&M-CorpusChristi $
285 Texas State $
286 Texas-Pan American $
287 Texas-San Antonio $
________________________________________________________________________________
SECTIONAL#54 total miles=1085 average distance between teams=72 pairs=15 teams=6
290 Air Force $
293 Colorado $
292 Colorado State $
291 Denver $
289 Northern Colorado $
294 Wyoming $
________________________________________________________________________________
SECTIONAL#55 total miles=2358 average distance between teams=236 pairs=10 teams=5
298 Arizona $
305 Arizona State $
296 New Mexico $
297 New Mexico State $
295 UTEP $
________________________________________________________________________________
SECTIONAL#56 total miles=1959 average distance between teams=196 pairs=10 teams=5
300 BYU $
301 Northern Arizona $
308 Southern Utah $
304 Utah $
302 Utah Valley State $
________________________________________________________________________________
SECTIONAL#57 total miles=2047 average distance between teams=205 pairs=10 teams=5
311 Boise State $
307 Idaho State $
299 Montana State $
303 Utah State $
306 Weber State $
________________________________________________________________________________
SECTIONAL#58 total miles=910 average distance between teams=91 pairs=10 teams=5
318 Eastern Washington $
317 Gonzaga $
312 Idaho $
309 Montana $
314 Washington State $
________________________________________________________________________________
SECTIONAL#59 total miles=582 average distance between teams=58 pairs=10 teams=5
320 CS Fullerton $
315 San Diego $
313 San Diego State $
319 UC Irvine $
316 UC Riverside $
________________________________________________________________________________
SECTIONAL#60 total miles=300 average distance between teams=20 pairs=15 teams=6
325 CS Northridge $
321 Long Beach State $
323 Loyola Marymount $
326 Pepperdine $
322 Southern California $
324 UCLA $
________________________________________________________________________________
SECTIONAL#61 total miles=1730 average distance between teams=173 pairs=10 teams=5
327 CS Bakersfield $
331 Cal Poly-SLO $
328 Fresno State $
330 UC Santa Barbara $
310 UNLV $
________________________________________________________________________________
SECTIONAL#62 total miles=10407 average distance between teams=1041 pairs=10 teams=5
347 Hawaii $
329 Nevada $
332 Pacific $
333 Sacramento State $
334 UC Davis $
________________________________________________________________________________
SECTIONAL#63 total miles=401 average distance between teams=27 pairs=15 teams=6
339 California $
337 Saint Mary's-Cal. $
342 San Francisco $
335 San Jose State $
336 Santa Clara $
338 Stanford $
________________________________________________________________________________
SECTIONAL#64 total miles=1921 average distance between teams=128 pairs=15 teams=6
345 Oregon $
346 Oregon State $
344 Portland $
343 Portland State $
341 Seattle $
340 Washington $
________________________________________________________________________________
Explanation of Data
It's really hard to get the exact solution other than literally
checking every possible combination but with 402 Indiana high school(347 in NCAA division I) teams,
you can see that the factorial issue soon makes that impossible
on even the best PC's and in fact for even the fastest super-computers.
In fact, for the Indiana High Schools, based on there being 46 sectionals with 6 teams and 18 sectionals with 7 teams,
I think it would be:
402!
____________________________________
(6!)^46 * (7!)^18 * 64!
And the natural logarithm of the answer is 1351.224517650 and thus the logbase10 of the answer is 586.8293232143
which means the number of possible combinations is 10^586.8293232143 which is an amazingly large number!
It would be actually be an exact integer, but I don't have access to a computer that can handle such large integers.
I think only the latest "super computers" can.
For college basketball, it would be
347!
____________________________________ ˜ 10^489.2824010963 which is also a number beyond the capacity to
(5!)^37 * (6!)^27 * 64! simply "try all the possibilities".
Remember, there are 31.5576 million seconds in an average year or ˜ 3.16 * 10^7 seconds.
Thus for even a super-computer to check every possibility in one-year, it would have to evaluate about more than 10^578 possibilities per second
for the high schools and 10^481 per second for the colleges.
So the simulated annealing technique is used to avoid the impossible task of literally checking every possibility.
It was first developed in the early 1950s at Los Alamos by guys like Edward Teller when they
(Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M.; Teller, A. H.; and Teller, E.
"Equation of State Calculations by Fast Computing Machines." J. Chem. Phys. 21, 1087-1092, 1953. )
were working on the hydrogen bomb. I was in my early oughts then
and so was only consulting with them on an informal basis :-) (just kidding!) at that time.
http://scienze-como.uninsubria.it/bressanini/montecarlo-history/index.html
> Equation of State Calculations by Fast Computing Machines
> N. Metropolis, A.W.Rosenbluth, M.N.Rosenbluth, A.H.Teller and E. Teller 1953
> The Journal of Chemical Physics, 21, 1087 (1953)
> THE paper. Probably THE most important algorithm of the 20th century, often called the M(RT)2 algorith from the names of the authors.
(the paper begins in the middle of page 1087)
The exact same technique could be used to redistrict
for congressional and state districts if you were to treat each
voter as a "team". And each district would end up with very close to the
same number of voters and those voters would all be close to each other
geographically. No more ludicrously shaped districts created totally for the
purpose of giving an advantage to the party drawing the lines.
Of course you'd need to use the supercomputers at the major universities
because even using simulated annealing would take a very long time with
several million "teams(voters)" in each state :-D
But the neat thing about is that each party could have a mathematician
who liked them run the program and they'd either both get the same answer,
or the one with the lower total distance would have the better answer.
The fact that this would literally "take politics out of the equation", is probably
the reason that it wouldn't be adopted; each party would worry that an honest
redistricting could take away "their" safe seats.
Enjoy!
Jeff Sagarin
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