graph_lecture-Sam_Brody

graph_lecture-Sam_Brody - Euler Cycle A cycle (path that...

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Euler Cycle A cycle (path that starts and ends at the same node) which traverses each edge exactly once. Bridges of Konigsberg (http://en.wikipedia.org/wiki/Seven_Bridges_of_Konigsberg): Sufficient and Necessary condition: all nodes have an even degree Intuition – each node you reach, you have to be able to leave. That's impossible with odd an odd number of edges – you reach a dead end the last time you get there. Eulerian Path – don't have to stop and end at the same node – at most two nodes have uneven degree (one for start, one for end). Hamiltonian Path A path covering all nodes exactly once. Considered a “hard” problem – no know solution that is more efficient than trying all the possibilities.
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Traveling Salesman Problem (TSP) Now looking at graphs where edges have weights. These can be distance, cost, time, etc. This adds a new dimension of complexity. Naturally, we want to minimize the cost. The problem is – find the minimal path that covers all the nodes. Repeats are allowed. In the example
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This note was uploaded on 02/20/2012 for the course 790 373 taught by Professor Boros during the Fall '09 term at Rutgers.

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graph_lecture-Sam_Brody - Euler Cycle A cycle (path that...

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