hamiltonian cycles and paths bin zhou. definitions hamiltonian cycle (hc): is a cycle which passes...
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Hamiltonian Cycles and paths
Bin Zhou
Definitions
• Hamiltonian cycle (HC): is a cycle which passes once and exactly once through every vertex of G (G can be digraph).
• Hamiltonian path: is a path which passes once and exactly once through every vertex of G (G can be digraph).
• A graph is Hamiltonian iff a Hamiltonian cycle (HC) exists.
History
• Invented by Sir William Rowan Hamilton in 1859 as a game
• Since 1936, some progress have been made
• Such as sufficient and necessary conditions be given
History
• G.A. Dirac, 1952, If G is a simple graph with n(>=3) vertices, and if the
degree of each is at least 1/2n, then G is Hamiltonian
• O.Ore , 1960 If G is a simple graph with n(>=3) vertices, and if the sum
of the degrees of each pair of non-adjacent vertices is at least n, then G is Hamiltonian
History
• Bondy and Chvatal , 1976
For G to be Hamiltonian, it is necessary and sufficient that [G]n be Hamiltonian. ([G]n is gotten from G by adding edges joining non-adjacent vertices whose sum of degrees is equal to, or greater than n)
History
• Fraudee, Dould, Jacobsen, Schelp (1989)
If G is a 2-connected graph such that for every pair of nonadjacent nodes u and v,
|d(u)+d(v)|>=(2n-1)/3
then G is Hamiltonian
Application
• Hamiltonian cycles in fault random geometric network
• In a network, if Hamiltonian cycles exist, the fault tolerance is better.
Hamiltonian problem is NPC
• This is a well known NP complete problem
• For general graph, we can not find an exactly linear time complexity algorithm to find a Hamiltonian cycle or path
HC algorithms• For general graphs, no efficient algorithm NP-complete for perfect graphs, planar bipartite graphs,
grid graphs, 3-connected planar graphs
• For some special graphs, exist efficient algorithms.
N. Ghiba, T. Nishizeki (1989)
Polynomial algorithm for 4-connected planar graphs.
G.Gutin (1997)
Polynomial algorithm for quasi-transitive digraphs
Some Algorithms for HC
• L. Pósa (1976)
Rotational transformation
• B. Bollobás, T.I.Fenner, and A. M. Frieze
Cycle extension (HAM) (1987)
• Silvano Martello
Algorithm 595 (1983)
Two classes of algorithms
• Heuristic algorithm
Pósa, UHC, DHC, HAM, etc
• Backtrack algorithm
595HAM, KTC, MultiPath
Backtrack Algorithm
• Recurse(Path p, endpoint e)• While (e has unvisited neighbors) { GetNewNode x; (add x node to P)
PruneGraph. (Prune graph. If result graph does not permit a HC forming, remove x from P and continue)
FormCycle (If P includes all nodes then try to form cycle. Fail, remove x and continue; Succ, return success)
BackTrack: Recurse(P,x)
}
Return fail.
Backtrack Algorithm
• Search all the potential solutions
• Employ pruning of some kind to restrict the amount of researching
• Advantage:
Find all solution, can decide HC exists or not
• Disadvantage Worst case, needs exponential time. Normally, take
a long time
Heuristic Algorithm
Initialize path P
While {
Find new unvisited node.
If found { Extend path P and pruning on the graph. If this choice does not permit HC, remove the extended node.
} else
Transform Path. Try all possible endpoints of this path
Form cycle. Try to find HC
}
Heuristic Algorithm
• Advantage:
Fast. Linear or low-order polynomial time
• Disadvantage
Maybe can not find the HC
Ham heuristic algorithm
• Try to extend existing path and never decrease the path length
• Do cycle extension
• Do rotational transformation
Ham algorithmStart from a random node and find a neighbor to get a path P. |P|=2
Do {
Change partial path array A. oldlength=|P|.
While |P|==oldlength {
Find neighbors of P’s endpoints.Try to extend P.
For (each neighbor) do {
If Extendable
Extend and continue;
Else
Do cycle extension or rotational transformation; }
Check termination condition and change P
} }}
Cycle Extention
x1 xi xi+1
u
xk
Path P:
xi x1 xk xi+1u
Path P’:
Rotational transformation
x1 xi xi+1 xk
Path P:
xi x1 xi+1 xk
Path P’:
Results
Node 20
0
20
40
60
80
100
0 2 4 6 8
Mean degree
% H
am
ilto
nia
n c
ycle
Series1
Results
Nodes 50
-
20
40
60
80
100
0 2 4 6 8 10
Mean degree
% H
ave H
am
ilto
nia
n c
ycle
Series1
ResultsNode 100
0
20
40
60
80
100
0 5 10 15
Mean degree
% H
am
ilto
nia
n C
yc
le
Series1
Problems
• The program can not check large graph due to the memory restriction
• May be need more conditions to decide the probability of HC exists
• We can solve large problem using parallel computing