snow plows in iowa city
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Snow plows in Iowa city. Graph theory at work. Project. Examine the procedure utilized by snow plows in Iowa City Systemize and minimize routes Review mathematical concepts involved Look into how math concepts apply to this problem Model Apply to example section - PowerPoint PPT PresentationTRANSCRIPT
Graph theory at work
Examine the procedure utilized by snow plows in Iowa City
Systemize and minimize routes Review mathematical concepts involved Look into how math concepts apply to
this problem Model Apply to example section Conclusions/Recommendations
Winter of 2007-2008 Unplowed areas Results
Uneven roadsUnable to plowCracks and potholes
Environmental Reduce Gas Consumption
Greenhouse gas emissions Save money
Public Complaints
City website Safety
System Current process
Downtown/Bus routes Steep slopes Flat secondary roads
Easy to teach Little confusion
Seven Bridges of Könisburg Euler
Traverse each edge exactly once Circuits exist if all vertices of even
degree Digraph: indegree equals outdegree for all
vertices
Use here If one exists, will be optimal route More than one truck
Multigraph Vertices - intersections of roads Edges – bidirectional streets Directed arcs – one-way streets For snow plows
Must traverse each lane of each road at least once
Digraph Vertices – intersections of roads Arcs – directed lanes
Kwan Mei-Ko 1960’s Goal: traverse every street in least
distance More general than bridge problem
If contains eulerian circuit, this is the shortest route
If not, solution can be found
If not using city’s current priorities Weights represent distance Want Mininimum Find degrees of all vertices in graph
Must be even number of vertices of odd degree Handshake lemma
Find shortest weighted paths between these vertices Draw duplicate edges along path
Will then have all even degrees Create Eulerian Circuit
If I choose to comply with current process
Assign weights to streets Weights represent grade of street
Find maximal weighted paths first Represent steep slopes
Follow by lower weighted paths Flatter streets
Square matrix Each row and column represents a vertex ‘1’ in Xij if arc (edge) exists from i to j ‘0’ otherwise Will be used to find degree of vertex in
multigraph Sum of ones in vertex row/column (digraph)
Traverse bus routes Simple because already circuits
Divide city into sections (10) Within each section, split roads into phases Maintain city’s current priorities Each phase
Create adjacency matrix For vertices of odd degree, create connected graph with
weights of shortest distance between Find perfect matching New edges along path Find Eulerian circuit Repeat for steep roads, flat roads
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Assign weights (shortest distance) Find minimal matching
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Duplicate red edges
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Divide the city into sections Determine which streets fall into which
phase Determine distances between vertices Create computer program
Takes in vertices/edges Forms adjacency matrices Finds degrees Forms weighted matrix for vertices of odd degree Minimizes matching Duplicates these edges Results in minimal distance path for each phase