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CYCLE OF EULER

A Eulerian cycle is one way that runs through all vertices (nodes) of a graph from one and only once for each arc (edge) of the graph, being a necessary condition to return to the initial vertex output (cycle = path in a graph where vertex match departure and initial or final vertex or goal). A formal definition defines it as "one cycle containing all edges of a graph only once."

In the picture, c = {1,2,3,4,6,3,5,4,1} is an Eulerian cycle, then a graph is Eulerian.

A graph is a representation, a model consisting of a number of vertices (nodes) and a number of arcs (edges) that relate to each edge or arc has the ability to connect two nodes. The word cycle is used in graph theory to indicate a closed path in a graph, ie the starting node and end node are the same, as a counterpart a Hamiltonian path is a path through all vertices of a graph without passing twice on the same vertex. If the road is closed call a Hamiltonian cycle.

If a graph admits an Eulerian cycle, Eulerian graph is called. Determines

Euler cycle in the following figure:

Why is it useful?

in everyday life we \u200b\u200buse very often when we visit all our friends and ultimately return home or when we are in a museum, we visit all the rooms and back out through the same entrance. is more than anything to analyze all the way and see if there is an error and improvement.

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DIJKSTRA ALGORITHM

Dijkstra's algorithm, also called shortest path algorithm is an algorithm for determining the shortest path given a source vertex to other vertices in a directed graph with weights in each edge. Its name refers to Edsger Dijkstra, who first described it in 1959.

ALGORITHM Given a weighted directed graph of N nodes are not isolated , x is the initial node, a vector D of size N stored at the end of the algorithm the distances from x to all other nodes.

1. Initialize all distances in D with an infinite value on because they are unknown at first, except of x to be placed at 0 because the distance from x to x would be 0.
2. Let a = x (we take as the current node).
3. We travel all the adjacent nodes, except nodes marked, call these vi.
4. If the distance from x to vi stored in D is greater than the distance from x to a plus the distance from a to vi, this is replaced with the second named, ie
if (Di> Da + d (a, vi)) then Di = Da + d (a, vi)
5. We mark the node as complete a.
6. We as the next current node with the lowest D value (can be done by storing the values \u200b\u200bin a priority queue) and go back to step 3 while there are unmarked nodes. Once completed
the algorithm, D is completely full.

For what is useful?

often in everyday life we \u200b\u200bfind the most efficient way to reach our destination, this algorithm is perfect, as it evaluates all possible paths and choose the shortest to the destination. Thereby reducing the most efficient route. Now compared to my career is really useful, since making a program that solves the problem in the shortest time is the best. Putting it another way, you'll be hired, not.

Calculate the shortest path from "1" to any node

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Hamilton Highway

In the mathematical field of graph theory, a Hamiltonian path in a graph is a path , a succession of adjacent edges, which visits all vertices of the graph once. If also the last vertex visited is adjacent to p irst, the path is a Hamiltonian cycle .
The problem of finding a cycle (or path) Hamiltonian an arbitrary graph is known to NP-complete .
The Hamiltonian paths and cycles were named after William Rowan Hamilton , inventor of the game Hamilton, threw a toy that involves finding a Hamiltonian cycle on the edges of a graph of a dodecahedron . Hamilton solved this problem by using quaternions , but this solution does not generalize to all graphs.

A Hamiltonian path is a path that visits each vertex exactly once. A graph containing a Hamiltonian path is called a Hamiltonian cycle or circuit Hamiltonian if it is a cycle that visits each vertex exactly once (except the apex of which party and which arrives). A graph containing a Hamiltonian cycle graph is called Hamiltonian .
can also say that Hamiltonian graphs are when they meet:
-Hamiltonian Circuit -must be related, must be closed.

to summarize: hamilton Road is a walkway that goes all the edges only 1 time, arriving at the same point.

Exercise:
there is a problem for such famous examples:

a chess board try to make that horse, scroll all the board squares

Neither algorithm guarantees an optimal solution. However, usually both are good solutions, close to the optimal.

THEOREM: Let K

* n a complete directed graph, ie K * n is nv 'erotic
and for any pair of v' erotic different x, and at least the edge (x, y) or (y, x)
est 'to K * n. This graph always contains a Hamilton path .

PROOF:

Let m ≥ 2 with Tm a simple path that contains the m-1 edges (v1, v2), (v2, v3 ),...,( vm -1, vm). If m = n is finished, otherwise v is a v 'erotic does not appear in tm.

If (v, v1) is an edge of K * n can be extended tm adding this edge. In another case
(v1, v) should be an edge. Suppose now that (v, v2) is in the graph. Then there is the greatest path (v1, v), (v, v2), (v2, v3 ),...,( vm-1, vm).

If (v, v2) is not an edge K * n then (v2, v) should be . According

continues this process there are only two possibilities :

1. for 1 ≤ k ≤ m-1 edges ( vk, v), (v, vk +1) are replaced K * n ( vk, vk +1) for this pair of edges

2. ( vm, v) is in K * n and adds this edge to tm.

In any case, we obtain results in a simple way tm including m +1 +1 has m vertices and edges. This process can be repeated until there is a path of n vertices.

BIBLIOGRAPHY :
http://es.wikipedia.org/wiki/Camino_hamiltoniano
http://docs.google.com/viewer?a=v&q=cache:JNDPlNjUeggJ:www.matap.uma.es/profesor/magalan/MatDis/material/GrafosTema5_1_MatDiscreta.pdf+caminos+de+hamilton& hl =es& gl = mx & pid = bl & srcid = ADGEESjhF 5 pcBVaMbrj 8 kWCfuPdH 4 CVND 5 lKIyifHirJg 6 Wi 48 mGQJPdZAnrvC 3y5 aU 6 zwOJstFridndiLhVsNnEZmHNiT 4 Dq 8 eyp 4 XnqA 6 tEjWcK 9 wj 3 hffTsTdMiynAwAKVx 1 boWKm & sig = AHIEtbTAHM 1- WypcKHRi 25 gFE 92 alt 3 ofQ
http://docs.google.com/viewer?a=v&q=cache:7CjbQrO-eUIJ:www.fing.edu.uy/tecnoinf/cursos/mdl2/material/teo/teorico2.pdf+caminos+de+hamilton& hl =es& gl = mx & pid = bl & srcid = ADGEESgw 7 LRDpcJCpSyzGyrAJuGasfz 72 trbe 0_ DB 6 And _ CA 9o- tIqipWnMUho - aEnjfg 5C0 ev 4 FPU 3B2 OvQJ __ szzb 8 ICDE -K4e- QOxk - idphdOji 1T_ hOiEzTHd 9 USgf 8 ysHBTyDU 7s & themselves = AHIEtbRANt 1P1 qlH 2C9 kBAV 0A2 ck 182 FTA

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