Tuesday, January 31, 2012

NBA Team Alignment


Seven years ago, the NBA added its newest team, the Charlotte Bobcats, and realigned the league with three divisions in each conference. A few years after the Grizzlies moved to Memphis from Vancouver, Seattle lost its basketball team to Oklahoma City, and the new Northwest Division featured only one team truly in the northwest quadrant of the nation and one in Minnesota and another in Oklahoma. There are other oddities as well -- the aforementioned Grizzlies are in the "Southwest" Division, the desert city Phoenix is in the Pacific Division despite being hundreds of miles away, and likewise Toronto is in the Atlantic Division even though the closest they get to the Atlantic is raptor-filled lake Ontario. Next year will feature the New Jersey Nets moving to Brooklyn, and although it's akin to moving to a house down the block in the NBA landscape it hints at future developments.

Isn't there a better arrangement of NBA teams? What happens when a city like Charlotte or Sacramento loses their team? Or Seattle and Las Vegas gain one? With even a modest computer like mine, these questions can be tested with a little ingenuity.


Teams typically play the same schedule for 58 out of 82 games because they play every other team at least twice. They also have to play division opponents four times, and other teams in the division three to four. There is a significant difference between conferences, but divisions are essentially meaningless in terms of scheduling. Therefore, for computational simplicity, I will set the conferences first, and then organize the divisions.

In order to find the most efficient alignment of teams, first I had to calculate the distance between every NBA arena. This was surprisingly easy thanks to the internet and programming. The latitude and longitude of every arena was found online, and the distance between was calculated using the spherical law of cosines -- basically I found the length of an arc on two different points on a sphere using trigonometry. Teams, obviously, don't travel on perfect paths from on arena to the next because they need to access airports and hotels, but over large distances the effects are small. With some nice logical statements and programming, I compiled a 30 by 30 matrix of distances between teams. I used the Brooklyn arena instead of New Jersey, although it doesn't significantly matter, and for computational efficiency the calculation was for 29 teams and the Lakers' results were copied for the Clippers.

The next part was surprisingly difficult. A fifteen team conference doesn't sound all too large, but there are 15,510,000~ unique combinations possible. (As a side note, combination is a specific mathematical term to describe the scenario. A permutation cares about order, so a conference with teams X, Y and Z would be different than X, Z, and Y, while combination only cares about unique groupings.) I wouldn't be able to crank out that many calculations with ease without the aid of the Jeopardy! computer, so I locked in 11 teams on each coast to their respective conferences and left eight teams in conference purgatory, waiting for their ultimate judgement from the program.

Next I found the minimum number of total league miles between arenas giving a 1.8 weight to teams in the same conference and a weight of 1.0 to those in different conferences. The weighting is basically just saying teams play 2 games against teams in the other conference, but because one of those will be at home they only travel once, and at least an average of 3.6 games within conferences, but again they only travel on average 1.8 games. Note this simulates 80.4 games, which makes it apparent how little divisions matter, and essentially that any efficient method of organizing divisions only affects 1.6 games half of which are at home.

I also included scenarios for teams relocating to different cities -- the Grizzlies, for example, moving back to Vancouver and restoring the sanity of the team's name. Looking at the smallest metropolitan areas and which cities already have many professional teams along with population growth, another candidate for relocation is the Bucks in Milwaukee, and I had them move to Seattle, which is currently the largest metro area without an NBA team. The last scenario tested was a major relocation with the New Orleans Hornets moving to Seattle to become the Sonics, the Sacramento Kings to Las Vegas, and the Memphis Grizzlies to Tampa and hopefully changing their name (Tampa Panthers after the elusive Florida cat?) Those three organizations are having trouble attracting enough fans and keeping their business above the profit line.

Las Vegas doesn't have a pro team, has hugely growth, and the Las Vegas Kings is a great name; Seattle deserves its team back and New Orleans had negative population growth last decade and became the smallest market besides Salt Lake City; and Memphis is another small market while Tampa is another warm weather city athletes love with plenty of people and growth. These choices, however, were mainly an exercise in seeing how the league's alignment would change if teams moved. For the geographical coordinates, the old arenas were used for Vancouver and Seattle, while the city's coordinates were used for Las Vegas and Tampa Bay.


For reference, using the same conference alignment presently except with the Nets in Brooklyn, if every team traveled to every other team in its conference on average 1.8 times and once to teams in the other conference, they would have traveled 1,272,633 miles in total for an average of 42,421 miles per team. This does not include the extra games within one's conference, but again that amount is negligible for these purposes (only 1.6 games more on average per team per season.)

The most efficient conference alignment from the algorithm was Minnesota, Memphis, Oklahoma City, and New Orleans in the west, and Milwaukee, Chicago, Atlanta, and Indiana in the east ... which is exactly how it's already set-up. Basically, the conferences were already efficiently arranged. The entire operation wasn't fortuitous, however, since there's nothing wrong with proving the current system is best, and the best results are usually surprising.

The first alternate scenario also did not feature a shift in the conferences when the Grizzlies migrated back to Canada in Vancouver, although it's not surprising because Memphis was already in the Western Conference.  The total miles for the league was 1,329,218 miles, which was an increase of 4.4 percent from the present order, and an average of 44,307 miles per team. The Blazers would be happy for another team in the northwest, but the overall effects are fairly significant because Memphis is a centrally located city and Vancouver would be the furthest team from the center of the country. For minimizing travel, the most efficient placement of a single team would be near the center of mass, so to speak, of the conference teams and shifted to the center of the country for all the travel to the other conference. Memphis is in an ideal location in this respect.

The Bucks to Seattle scenario had a shift in the conferences, which is obvious since the Bucks were in the east and Seattle's a ways from the Mississippi. Only two teams would change conferences -- the Bucks to Seattle, and the Memphis Grizzlies to the east. The total miles from the league was 1,323,964 increased by 3.9 percent and an average of 44,132 miles. Once again, a team moving from the central of the country to one of the coasts would increase the total miles traveled by every team.

The Kings to Kansas City, predictably, decreased the total number of miles league-wide by 4.0 percent for 1,221,952 and an average of 40,732 miles. Surprisingly, teams didn't change conferences, as the Kings would stay in the west. Kansas City is geographically the center of the country, and even in the Western Conference it would still have a center location with Minnesota, Memphis and New Orleans east. A rule of thumb seems to be if you move switch teams from the center of the country to the outer edges league-wide traveling between cities would change by roughly 4 percent.

The most radical scenario with the Kings to Las Vegas, the Grizzlies to Tampa Bay, and the Hornets to Seattle featured again only one conference switch. The Hornets would stay in the west by going to Seattle, the Kings would also stay in the west because of Las Vegas, but Milwaukee would interestingly join the western conference while the Grizzlies in Tampa would join the east. The total miles increased by 3.7 percent to 1,320,134 with an average of 44,004 miles. Divisions would have to be completely changed, but that's a story for another section.


The NBA schedule is heavily dependent on which conference team occupies, but little about the specific division. 80.4 games are comprised of two with each opposing conference team and an average of 3.6 within the conference. The extra 1.6 games are added to teams within your division. The present conference alignment is the most efficient system in terms of minimizing travel distance between cities tested against seventy different combinations of eastern and western conference teams. The total travel distance of every team in the league was 1,272,633 miles (not including the extra 1.6 games.)

A few different team relocation scenarios were tested, and the results suggested that moving a team from the middle of the country to the outer edge would increase total travel distance by a fair amount but nothing too large (around 4 percent based on where the teams are.) Pairing a lonely city like Portland with Seattle wouldn't decrease travel distances for the league because of how far Seattle is from every other city. All four scenarios tested only switched at the most one pair of teams from each other's conferences, suggesting that the NBA's current alignment is mostly static.

This article is a little late in arriving, but I had trouble sorting the teams by division and as a result part two of this article will test for the most efficient alignments for divisions, their importance, and how relocations change those.

1 comment:

  1. Hey Justin. I find these optimization problems very interesting, and I am working on a similar project in school. Do you think you could send me your 30 x 30 matrix of the distances between teams? Of course I could do it myself, but I thought I could contact you to maybe save me the headache. Any help would be much appreciated Justin! You can reach me at oechsli@wisc.edu if you're interested. Thanks