The University of Washington's Math 381 project aims to develop an efficient routing model for snowplows in downtown Seattle, addressing the challenges posed by one-way streets. Utilizing graph theory, particularly the postman problem, the project seeks to minimize travel distance while ensuring all streets are cleared. Future optimizations include incorporating real-time traffic data and refining cost estimations for improved snowplow efficiency.

1. SEATTLE’S SNOWPLOWS
THE POSTMAN PROBLEM
University of Washington
MATH 381

2. PROBLEM STATEMENT
The City of Seattle has hired us to find an efficient way to route
snowplows around downtown
Why does this matter?
DIRECTOR: GRACE CRUNICAN

3. WE KNOW WHAT HAPPENS WHEN IT SNOWS
IN SEATTLE
DIRECTOR: GRACE CRUNICAN

4. PROBLEM STATEMENT
This problem goes beyond funny news reels
The City of Seattle has hired us to find an efficient way to route
snowplows around downtown
The natural follow up question: what does it mean to be efficient?
DIRECTOR: GRACE CRUNICAN

5. GIVEN ASSUMPTIONS
The plow is based at a location of your choosing
Plow(s) begin and end their route at the home base location
One way streets are plowed in one direction
Two way streets are plowed in both directions

6. PROBLEM STATEMENT
Rather than search for a single “perfect” solution, find a model that
is efficient
Efficiency: the minimization of wasted driving of snowplows
­ Saves labor costs and fuel
­ Allows for faster removal of snow
­ Overall public safety

7. SPECIFICATIONS / ASSUMPTIONS
MADE
Specs:
­ Make a (big!) graph of downtown Seattle
­ Assure that distances are accurate
Assumptions:
­ Traffic – for simplicity, assume traffic is constant
­ There are usually fewer cars out when snowplows are up and running,
making this assumption justifiable

8. THE CHALLENGE
One way streets prevent us from completing a simple Eulerian
circuit
We need a model that can account for one-way streets, and
traverses all edges (twice for 2-ways, once for one-ways)
Key insight : note that you may consider one-way streets as a single
directed edge, and two-way streets as two directed edges

9. MATHEMATICAL MODEL
In graph theory terms, we want to find a minimum length closed
walk that traverses each edge at least once
This is problem is known as the ‘Postman Problem’ or ‘Route
inspection problem’
SOURCE: HTTP://XLINUX.NIST.GOV/DADS//HTML/CHINESEPOSTMAN.HTML
How to start at any vertex and hit every edge
while minimizing the number of repeated edges?

10. POSTMAN PROBLEM
Alan Goldman of the U.S. National Bureau of Standards first coined
the name ‘Chinese Postman Problem’
Studied by Chinese mathematician Kwan Mei-Ko in 1960 to address
the problem of a rural postman attempting to most efficiently
complete his delivery route
Chinese paper was translated into English in 1962
SOURCE: HTTP://XLINUX.NIST.GOV/DADS//HTML/CHINESEPOSTMAN.HTML

11. THE ALGORITHM
SOURCE: HTTP://WWW3.CS.STONYBROOK.EDU/~ALGORITH/IMPLEMENT/CPP/DISTRIB/SPAECPP.PDF

12. THE ALGORITHM, EXPLAINED
Construct a shortest distance matrix e.g.: row i., column j is the shortest
distance from vertex i to vertex j
Floyd-Warshall’s algorithm
Find the vertices with an ‘excess’ of outgoing/ingoing paths (e.g. the
start of a one-way) from the vertices with an ‘deficit’ of incoming paths
The Postman Tour must walk extra paths to ‘join up’ these unbalanced
vertices
The optimal tour will take a combination of extra paths with minimal
‘cost’ (i.e. least distance)

16. CLIP: THE ALGORITHM IN ACTION

17. PREPARING INPUT:
GRAPHING DOWNTOWN
1. Take a large cross-section of Downtown Seattle via Google
Maps.

18. PREPARING INPUT:
GRAPHING DOWNTOWN
1. Take a large cross-section of Downtown Seattle via Google
Maps.

19. PREPARING INPUT:
GRAPHING DOWNTOWN
2. Convert it to black and white to distinguish the streets.

20. PREPARING INPUT:
GRAPHING DOWNTOWN
2. Convert it to black and white to distinguish the streets.

21. PREPARING INPUT:
GRAPHING DOWNTOWN
3. Extract streets and use intersections to create nodes.

22. PREPARING INPUT:
GRAPHING DOWNTOWN
3. Extract streets and use intersections to create nodes.

23. PREPARING INPUT:
GRAPHING DOWNTOWN
4. Use a built-in Mathematica function to convert this image into a
weighted undirected graph.
• The weights are proportional to the streets’ distances.
5. Transform each undirected edge into two opposing directed
edges, each with the same weight as the original edge.
6. Account for one-way streets by removing the extra edges
created in the previous step.

24. • The red edges correspond to one-way streets.
• Note that this is only an approximation of downtown Seattle’s streets.
PREPARING INPUT:
GRAPHING DOWNTOWN

25. PREPARING INPUT:
GRAPHING DOWNTOWN
7. Created a function to map our graph’s edge weights to the
physical distances of their corresponding streets (obtained via
Google Maps).

26. PREPARING INPUT:
A COUPLE OF COMPLICATIONS
1. One way streets were added manually
2. Interstate 5 was manually removed as it is not part of the
domain of the problem

27. MATHEMATICAL SOLUTION
Observations
1. Graph edited to convert two-way streets (gray) into two distinct
di-edges, and one-way streets into one-directional di-edges (in
red)
2. It is possible to obtain marginally different solutions by choosing
a different starting point
3. Similarly, one may accommodate additional vehicles by
partitioning the graph and running the algorithm for each sub-
graph

28. MATHEMATICAL SOLUTION
Color corresponds to the
number of road traversals:
Purple: once
Gold: twice
Red: thrice

29. HOW MUCH?
The optimization condition minimizes the extra distance that is
needed to cover all of the streets
Quantifying cost (marginal and total, in both time and $) is a
matter of assumption / data
Example: A safe assumption for the speed of a snowplow is 30km/
hr ≈ 18.64 mi/hr (safe low end speed for snowplow), would give a
slight overestimate on the time cost
SOURCE: HTTP://WWW.HIGHWAYS.GOV.SK.CA/ADX/ASPX/ADXGETMEDIA.ASPX?DOCID=759,104,81,1,DOCUMENTS&MEDIAID=6221&FILENAME=SNOW+PLOW+SAFETY+FACT+SHEET+2011-12.PDF

30. HOW MUCH TIME?
Example: Find the total time cost of having 1 snowplow make one
run through downtown Seattle: 179.915mi
• Simply multiply the total ‘cost’ (in terms of distance) by 1hr/
18.64km to obtain 9.652 hours for one snowplow to clear
downtown
• Extra information: The total distance of the streets (double weight
for two way) 171.337 mi
• ‘Wasted driving’ total: 8.578 mi
• Efficiency of route: 1.05

31. FUTURE OPTIMIZATIONS
1. Incorporating live data on traffic to more accurately weigh the
‘cost’ of clearing a given street (helps particularly to partition
the routes for more than one snowplow)
2. Collect more accurate data for estimating costs; e.g. data on the
average speed of snowplows, the costs / benefits of adding
additional snowplows
3. Implementing code to account for DoT policy (certain streets are
given priority, some only require one lane on each side of the
street to be plowed)

32. FUTURE OPTIMIZATIONS
4. Exploiting further mathematical properties of the graph, such as
the fact that it is planar (since every intersection is interpreted
as a vertex, so no edges intersect).
• E.g., using the A* algorithm instead, the obvious choice of an admissible
heuristic being the Euclidean distance between vertices.
5. Account for the finer details that would affect real operational
costs and time; e.g., idling due to stoplights, acceleration/
deceleration over streets of varying lengths and incline
gradients, etc.

33. FUTURE OPTIMIZATIONS
HTTP://WWW.SEATTLE.GOV/TRANSPORTATION/DOCS/WW2015MAP.PDF

34. THE END