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A light-based device for solving
the Hamiltonian path problem
Mihai Oltean
Department of Computer Science,
Faculty of Mathematics and Computer Science,
Babes¸-Bolyai University, Koga˘lniceanu 1
Cluj-Napoca, 400084, Romania.
moltean@cs.ubbcluj.ro
www.cs.ubbcluj.ro/∼moltean
Abstract. In this paper we suggest the use of light for performing useful
computations. Namely, we propose a special device which uses light rays
for solving the Hamiltonian path problem on a directed graph. The device
has a graph-like representation and the light is traversing it following the
routes given by the connections between nodes. In each node the rays are
uniquely marked so that they can be easily identified. At the destination
node we will search only for particular rays that have passed only once
through each node. We show that the proposed device can solve small
and medium instances of the problem in reasonable time.
1 Introduction
Using the light to perform computations is an exciting idea whose applications
can be already seen on the market.
An important step was made by Intel researchers who have developed the
first continuous wave all-silicon laser using a physical property called the Raman
Effect [7,16,17,18]. The device could lead to such practical applications as op-
tical amplifiers, lasers, wavelength converters, and new kinds of lossless optical
devices.
Another solution comes from Lenslet [13] which has created a very fast pro-
cessor for vector-matrix multiplications. This processor can perform up to 8000
Giga Multiple-Accumulate instructions per second. Lenslet technology has al-
ready been applied to data analysis using k-mean algorithm [15] and video com-
pression.
In this paper we suggest a new way of performing computations by using
some properties of light waves. The idea may be used within a special device for
solving the Hamiltonian path problem.
We are not taking into account the quantum properties of light which have
been used for solving the Traveling Salesman Problem [4,10].
The paper is organized as follows: The Hamiltonian path problem is briefly
described in section 2. The proposed device is presented in section 3. Mathe-
matical background of the labeling system is described in section 3.2. The way

in which the proposed device works is given in section 3.3. A list of components
required by the proposed device is given in section 3.4. Complexity is discussed
in section 3.5. Suggestions for improving the device are given in sections 3.8 and
3.9. Further work directions are suggested in section 4.
2 The Hamiltonian path problem
The description of the Hamiltonian Path (HP) problem for a direct graph is the
following:
Given a directed graph G = (V, E) with |V | = n nodes and a start node
(vstart) and a stop node (vstop), the problem asks to compute is there is a simple
path, beginning with node vstart and ending with node vstop, containing all
nodes exactly once. The output for this decision problem is either YES or NO
depending on whether the Hamiltonian path does exist or not.
The Hamiltonian path problem arises in many real-word applications [3,6].
The problem belongs to the class of NP-complete problems [9]. No polynomial
time algorithm is known for it.
A small instance of this problem was also the first problem solved using a
DNA computer [1].
3 The proposed device
Our idea is based on two properties of light:
– The speed of light has a limit. The value of the limit is not very important
at this stage of explanation. What is important is the fact that we can delay
the ray by forcing it to pass through an optical fiber cable of a certain length.
– The ray can be easily divided into multiple rays of smaller intensity/power.
Initially a light ray is sent to the start node. Generally speaking two opera-
tions must be performed when a ray passes through a node :
– The light ray is marked uniquely so that we know that it has passed through
that node.
– The ray is divided into a number of rays equal to the external degree of that
node. Each obtained ray is directed toward one of the nodes connected to
the current node.
At the destination node we will search only for particular rays that have
passed only once through each node.
This section deeply describes the proposed system. First step is to find a
way to mark the signals which passes through nodes such that the interesting
signals can be easily identified at the destination node. The mathematical back-
ground required for this operation is described in section 3.2 and the hardware
implementation of the labeling system is described in section 3.4.

3.1 Labeling system
At the destination node we will wait for a particular ray which has passed
through all nodes of the graph exactly once. This is why we need to find a
way to label that particular ray so that it could be easily identified.
Actually we are interested in marking all rays which pass through a particular
node with a unique label, such that Hamiltonian path is uniquely identified at
the destination node (vstop).
In the solution proposed in this paper, the rays passing through a node are
marked by delaying them with a certain amount of time. This delay can be easily
obtained by forcing the rays to pass through an optical fiber of a certain length.
Roughly speaking, we will know if a certain ray has traversed a Hamiltonian path
only if its delay (at the destination node) is equal to the sum of delays of all
nodes in that graph. We will also know the particular moment when the expected
ray (the one which has completed a Hamiltonian path) will arrive. In this case
the only thing that we have to do is to ”listen” if there is a fluctuation in the
intensity of the signal at that particular moment. Due to the special properties
of the proposed system we know that no other ray will arrive, at the destination
node, at the moment when the Hamiltonian path ray has arrived.
The delays, which are introduced by each node, cannot take any values. If
we would put random values for delays we might have different rays (which are
not Hamiltonian paths) arriving, at the destination node, in the same time with
a ray representing a Hamiltonian path.
We need only the ray, which has traversed a Hamiltonian path, to arrive in
the destination node at the moment equal to the sum of delays of each node (the
moment when the ray has entered in the start node is considered moment 0).
Thus, the delaying system must have the following property:
Property of the delaying system
Let us denote by d1, d2, ..., dn the delays introduced by each node of the
graph. A correct set of values for this system must satisfy the condition:
d1 + d2 + ... + dn 6= a1 · d1 + a2 · d2 + ... + an · dn,
where ai (1 ≤ i ≤ n) are natural numbers and cannot be all 1 in the same
time.
If a given value ai is strictly greater than 1 it means that the ray has passed
at least twice through node 1.
3.2 Mathematical background for the labeling system
Finding the appropriate labeling system was two steps process. First of all we
have written a computer program which generates this numbers by using a back-
tracking procedure [5]. We also wanted to generate numbers such that the highest
number in a system is the smallest possible. This will ensure that the network is
constructed in an efficient way. The labeling systems generated by our computer
programs are given in Table 1.
From Table 1 it can easily seen these numbers follow a general rule:

Table 1. The labeling system generated by our backtracking procedure. First
column contains the number of nodes of the graph. The second column represents
the labels applied to nodes.
n Labels (delays)
1 1
2 2, 3
3 4, 6, 7
4 8, 12, 14, 15
5 16, 24, 28, 30, 31
6 32, 48, 56, 60, 62, 63
For a graph with n nodes one of the possible labeling systems is:
2n − 2n−1,
2n − 2n−2,
2n − 2n−3,
... ,
2n − 20.
As the second step we have to prove that the property of delaying system
(see section 3.1) holds for this sequence of numbers.
Actually we have to prove that the equality:
2n − 2n−1 + 2n − 2n−2 + 2n − 2n−3 + ... + 2n − 20 =
a1 · (2n − 2n−1) + a2 · (2n − 2n−2) + a3 · (2n − 2n−3) + ... + an · (2n − 20)
(1)
is not possible unless all ai are equal to 1.
The left part of the equality is:
2n − 2n−1 + 2n − 2n−2 + 2n − 2n−3 + ... + 2n − 20 =
n · 2n − (2n−1 + 2n−2 + 2n−3 + ... + 20) =
n · 2n − 2n + 1 =
(n − 1) · 2n + 1.
First of all we have to see that the equality does not hold if all ai numbers
are at least 1 and at least one number is strictly greater than 1. If this happens
the 2n term will be represented at least n times. But, it must be represented
only n − 1 times (see above).
Thus, if at least one number ai is strictly greater than 1, it means that other
numbers aj must be 0. We will prove that the equality (1) does not hold in this
case too.

As discussed above, if one of the coefficients is 0, at least one of the other
coefficients must be strictly greater than 1. For instance, if a1 is set to 0, it
means that a2 must be set to 3 in order to compensate the missing 2n−1 term.
This is a direct consequence of the fact that 2n−1 = 2 · 2n−2. Of course, we also
have to take into account that 2n−2 must be represented once. This is why we
have to set a2 to value 3.
If a2 is also 0 we have to set a3 to value 7 (we need 4 · 2n−3 in order to
compensate the missing term 2n−1, we also need 2 · 2n−3 to compensate the
missing term 2n−2 and, of course, the term 2n−3 must be represented 1 time).
As a general idea: if a particular term (2n−j) is missing (the corresponding
coefficient aj is set to 0), it can be compensate by setting one of the next coef-
ficients to a value of at least 3. But, a coefficient set to 0 means that the term
2n is missing once, and by setting another coefficient to at least 3 we will get
at least 2 extra representations for 2n. This will mean that right part of the
equation (1) will be at least (n +1) · 2n +1. But, the left part of the equation is
only (n − 1) · 2n + 1.
With this we have shown that all ai(1 ≤ i ≤ n) must be 1 in order to have
equality in equation (1). For any other values of ai the equality (1) does not
hold.
An important question is whether this system is the minimal possible (the
biggest number is the minimal possible). A partial answer to this question is
given in section 3.5.
Fig. 1. A directed graph with 5 nodes. Start node is 1 and the destination node
is 4. The list of arcs is: (1,3), (1, 5), (2, 1), (2, 3), (2, 4), (3, 4), (5, 2)

3.3 How the system works
An schematic example of a graph-like system is given in Figure 1.
In the graph depicted in Figure 1 the light will enter in node 1. It will be
delayed with a certain amount of time and then it will be divided into 2 rays
which will be sent to the nodes 3 and 5. In node 3 the ray will be delayed (with
the amount of time corresponding to node 3) and then it will be sent to node 4.
However, this is not a Hamiltonian path because it has visited only the nodes 1,
3 and 4. The other ray which was sent to node 5 (from node 1) can generate a
Hamiltonian path by following the route 1, 5, 2, 3 4.
Note that there is also a cycle in the graph: 1, 5, 2. The cycle will make
that some particular rays to be trapped within the system. The rays which have
passed once through the previously described cycle are not considered Hamilto-
nian paths because the moments when they arrive at the destination node are
greater than the sum of delays introduced by each node.
The partial paths traversed by rays are given below:
1
1, 3
1, 3, 4
1, 5
1, 5, 2
1, 5, 2, 1
.............{paths starting with 1, 5, 2, 1}
1, 5, 2, 3
1, 5, 2, 3, 4
1, 5, 2, 4
3.4 Hardware implementation of the labeling system
For implementing the proposed device we need the following components:
– a source of light (laser),
– Several beam-splitters for dividing light rays into multiple subrays. A stan-
dard beam-splitter is designed using a half-silvered mirror. For dividing a
ray into k subrays we need k − 1 beam-splitters.
– A high speed photodiode for converting light rays into electrical power. The
photodiode is placed in the destination node.
– A tool for detecting fluctuations in the intensity of electric power generated
by the photodiode (oscilloscope),
– A set of optical fiber cables having certain lengths. These cables are used for
connecting nodes and for delaying the signals within nodes. The length of
the cables must obey the rules described in section 3.1. A practical example
is given in section 3.6.

3.5 Complexity
This section answers a very important question: Why the proposed approach is
not a polynomial-time solution for the HP problem?
At the first sight one may be tempted to say that the proposed approach
provides a solution in polynomial time to any instance of the HP problem. The
reason behind such claim is given by the ability of the proposed device to provide
output to any instance by traversing only once all nodes (O(n) complexity).
This could mean that we have found a polynomial-time algorithm for the HP
problem. A direct consequence is obtaining solutions, in polynomial time, for all
other NP-Complete problems - since there is a polynomial reduction between
them [9].
However, this is not our case. As can be seen from Table 1 the delay time
increases exponentially with the number of nodes. Even if the ray has to traverse
only n nodes (resulting a complexity of O(n)), the total time required by the ray
to reach the destination node increases exponentially with the number of nodes.
There are two direct consequences which are derived from here:
– The length of the optical fibers, used for delaying the signals, increases ex-
ponentially with the number of nodes,
– The intensity of the signal decreases exponentially with the number of nodes
that are traversed.
These two issues are discussed in sections 3.6 and 3.7.
3.6 Problem size
We are interested in computing the size of the cables required to solve a certain
instance of the problem in a small amount of time. This will give as a rough
indication on the size of the graphs that can be solved using our system in
reasonable time.
This size heavily depends on the accuracy of the measurement tools. The rise-
time of the best oscilloscope available on the market is in the range of picoseconds
(10−12 seconds). This means that we should not have two signals that arrive at
2 consecutive moments at a difference smaller than 10−12 seconds.
Knowing that the speed of light is 3 · 108m/s we can easily compute the
minimal cable length that should be traversed by the ray in order to be delayed
with 10−12 seconds. This is obviously 0.0003 meters.
This value is the minimal delay that should be introduced by a node in order
to ensure that the difference between the moments when two consecutive signals
arrive at the destination node is greater or equal to the measurable unit of 10−12
seconds. This will also ensure that we will be able to correctly identify whether
the signal has arrived in the destination node at a moment equal to the sum of
delays introduced by each node. No other signals will arrive within a range of
10−12 seconds around that particular moment.
Once we have the length for that minimal delay is quite easy to compute the
length of the other cables that are used in order to induce a certain delay.

Recall from section 3.2, Table 1 that a graph with 5 nodes has the following
delaying system:
16, 24, 28, 30, 31.
From the previous reasoning line we have deduced that the smaller indivisible
unit is 0.0003. So, we have to multiply these numbers by 0.0003. We obtain:
0.00048, 0.00072, 0.00084, 0.0009, 0.00093.
These numbers represent the length of the cables that must be used in graph’s
nodes in order to induce a certain delay.
Note that the delay introduced by the cables connecting the nodes was not
taken into account in this example. This is not a limitation of our system. The
cables connecting nodes can be set to have some length which must obey the
property of delaying system (see section 3.1). Note that all cables must have the
same length. In this case if we have a graph with 4 nodes the length of every
cable connecting the nodes must be set to 16 units (the shortest possible - in
order to reduce the costs). The length of cables within the nodes should be 24,
28, 30 and 31 units.
The largest length in this sequence is 0.00093 meters. This length is not very
big, but for larger graphs the length of the cables within nodes can be a problem.
Once we have the length for that minimal delay is quite easy to compute the
maximal number of nodes that a graph can have in order to find the Hamiltonian
path in one second. We know the facts:
– the largest delay has the form 2n − 1 (see equation 1),
– the distance traversed by light in 1 second is 3 · 108 meters,
– the shortest delay possible is 0.0003 meters.
We simply have to solve the equation:
2n · 0.0003 = 3 · 108 (2)
This number is about 40 nodes. However, the length of the optic fibers used
for inducing the largest delay for this graph is huge: about 8 · 108 meters. We
cannot expect to have such long cables for our experiments.
However, shorter cables (of several hundreds of kilometers) are already avail-
able in the internet networks. They can be easily used for our purpose. Assuming
that the longest cable that we have is about 300 kilometers we may solve in-
stances with about 26 nodes. The amount of time required to obtain a solution
is about 10−6 seconds.
Note that the maximal number of nodes can be increased when the precision
of our measurement instruments (oscilloscope and photodiode) is increased.
Also note that this difficulty is not specific to our system only. Other major
unconventional computation paradigms, trying to solve NP-complete problems
share the same fate. For instance, a quantity of DNA equal to the mass of Earth
is required to solve HP instances of 200 cities using DNA computers [11].

3.7 Amplifying the signal
Beam splitters are used in our approach for dividing a ray in two or more subrays.
Because of that, the intensity of the signal is decreasing. In the worst case we
have an exponential decrease of the intensity. For instance, in a graph with n
nodes, each signal is divided (within each node) into n − 1 signals. Roughly
speaking, the intensity of the signal will decrease nn times.
This means that, at the destination node, we have to be able to detect very
small fluctuations in the intensity of the signal. For this purpose we will use
a photomultiplier [8] which is an extremely sensitive detector of light in the
ultraviolet, visible and near infrared range. This detector multiplies the signal
produced by incident light by as much as 108, from which even single photons
can be detected.
3.8 Improving the device by reducing the speed of the signal
The speed of the light in optic fibers is an important parameter in our device.
The problem is that the light is too fast for our measurement tools. We have
either to increase the precision of our measurement tools or to decrease the speed
of light.
It is known that the speed of light traversing a cable is significantly smaller
than the speed of light in the void space. Commercially available cables have
limit the speed of the ray wave up to 60% from the original speed of light. This
means that we can obtain the same delay by using a shorter cable.
However, this method for reducing the speed of light is not enough. The order
of magnitude is still the same. This is why we have the search for other methods
for reducing that speed. A very interesting solution was proposed in [12,14] which
is able to reduce the speed of light by 7 orders of magnitude. This could help our
mechanism significantly. However, is still a question how to use this idea for our
device because of the complex equipment involved in those experiments [12,14].
By reducing the speed of light by 7 orders of magnitude we can reduce the
size of the involved cables by a similar order. This will help us to solve larger
instances of the problem.
3.9 Improving the performance of the device for particular graphs
The labeling system proposed in section 3.1 is a general one. It can be used
for any kind of graph (with any number of nodes and arcs). We have shown
that this system has a big problem: the value of the involved numbers increase
exponentially with the number of nodes in the graph being solved.
But, for particular graphs we can find other labeling systems which are not
exponential. For example, the linear graph (see Figure 2 can be solved by our
device by using virtually no delays. In this case the moment when the signal
arrives in the destination node is equal to sum of delays introduced by the
cables connecting the nodes.
Finding the optimal labeling system for a particular graph is an interesting
problem which will be investigated in the near future.

Fig. 2. A linear graph with 7 nodes. No delays are required for the nodes of this
graph in order to find an Hamiltonian path
3.10 Technical challenges
There are many technical challenges that must be solved when implementing the
proposed device. Some of them are:
– Cutting the optic fibers to an exact length with high precision. Failing to
accomplish this task can lead to errors in detecting a ray which has passed
through each node once.
– Finding a high precision oscilloscope. This is an essential step for solving
larger instances of the problem (see section 3.6),
– Finding cables long enough so that larger instances of the problem could
be solve. This problem might have a simple solution: the internet networks
connecting the world cities. It is easy to find cables of hundreds of kilometers
connecting distant cities. This will help us to solve instance of more than
10 nodes. However, this solution introduces a difficulty too: cables of certain
lengths must be found or the system must be rescaled in order to fit the
existing lengths.
4 Conclusions and further work
The way in which light can be used for performing useful computations has been
suggested in this paper. The techniques are based on the massive parallelism of
the light ray.
It has been shown the way in which a light-based device can be used for
solving the Hamiltonian path problem. Using the today technology we can build
a light-based device which can solve small and medium size instances in several
seconds.
Further work directions will be focused on:
– Implementing the proposed hardware,
– Finding optimal labeling systems for particular graphs. This will reduce the
length of the involved cables significantly,
– Finding other non-trivial problems which can be solved by using the pro-
posed device,
– Finding other ways to introduce delays in the system. The current solution
requires cables that are too long and too expensive,

– Using other type of signals instead of light. A possible candidate would be
electric power,
– Finding other ways to implement the system of marking the signals which
pass through a particular node. This will be very useful because the currently
suggested system, based on delays, is too time consuming.
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