Suwon Report
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Page | 1 PATHFINDING FOR GARBAGE TRUCKS AND STREET SWEEPERS NGUYEN THANH TRUNG – 2011711405 NGUYEN HUU HUNG – 2011711406 ROCHAK SACHAN – 2011711404 I. Introduction In the 18th century, there is a town called Königsberg, in which there were seven bridges that spanned a forked river that flows past an island. One day a man who wishes to walk along each bridge, but that bridge must only traverse exactly once. The man think: "Does there exist such a continuous tour which satisfies the requirement?" This long-standing problem was solved in 1735 in an ingenious way by the Swiss mathematician Leonhard Euler (1707-1782). His solution, and his generalization of the problem to an arbitrary number of islands and bridges, gave rise to a very important branch of mathematics called Graph Theory. This problem is similar to “Pathfiding for garbage trucks and street sweepers” problem: Design efficient routes for garbage trucks and street sweepers, and the routes must assure that each street is travelled exactly once without having service vehicles retrace positions of their route.
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Transcript of Suwon Report
8/6/2019 Suwon Report
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PATHFINDING FOR GARBAGE TRUCKS AND STREET SWEEPERS
NGUYEN THANH TRUNG – 2011711405
NGUYEN HUU HUNG – 2011711406
ROCHAK SACHAN – 2011711404
I. Introduction
In the 18th century, there is a town called Königsberg, in which there were
seven bridges that spanned a forked river that flows past an island. Oneday a man who wishes to walk along each bridge, but that bridge must
only traverse exactly once. The man think: "Does there exist such a
continuous tour which satisfies the requirement?" This long-standing
problem was solved in 1735 in an ingenious way by the Swiss
mathematician Leonhard Euler (1707-1782). His solution, and his
generalization of the problem to an arbitrary number of islands and
bridges, gave rise to a very important branch of mathematics called Graph
Theory.
This problem is similar to “Pathfiding for garbage trucks and street
sweepers” problem: Design efficient routes for garbage trucks and street
sweepers, and the routes must assure that each street is travelled exactly
once without having service vehicles retrace positions of their route.
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Page | 2
→
→
II. Solution
1. Definition
An Eulerian trail or Euler walk or Euler path in an undirected graph isa path that uses each edge exactly once. If such a path exists, the graph is
called traversable or semi-eulerian. An Eulerian cycle or Eulerian circuit or Euler tour in an undirectedgraph is a cycle that uses each edge exactly once. If such a cycle exists,the graph is called unicursal. While such graphs are Eulerian graphs, notevery Eulerian graph possesses an Eulerian cycle.
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For directed graphs path has to be replaced with directed path and cyclewith directed cycle.
The definition and properties of Eulerian trails, cycles and graphs are validfor multigraphs as well.
2. Properties
a. A connected undirected graph is Eulerian if and only if every graphvertex has an even degree.
b. A directed graph is Eulerian if it is strongly connected and every vertexhas equal in degree and out degree.
c. An Eulerian trail exists in a directed graph if and only if the graph'sunderlying undirected graph is connected, at most one vertex has outdegree-in degree=1, at most one vertex has in degree-out degree=1and every other vertex has equal in degree and out degree.
d. An undirected graph is traversable if it is connected and at most twovertices in the graph are of odd degree.
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3. Constructing Eulerian circuits
References
1. http://www.people.vcu.edu/~gasmerom/MAT131/euler.html 2. http://en.wikipedia.org/wiki/Eulerian_path
