Droplet-Trace-Based Array Partitioning and a Pin
Assignment Algorithm for the Automated Design of Digital
Microfluidic Biochips*
Tao Xu and Krishnendu Chakrabarty
Department of Electrical and Computer Engineering
Duke University, Durham, NC 27708, USA
{tx, krish}@ee.duke.edu
Abstract
Microfluidics-based biochips combine electronics with biology
to open new application areas such as point-of-care medical
diagnostics, on-chip DNA analysis, and automated drug discovery.
Bioassays are mapped to microfluidic arrays using synthesis tools, and
they are executed through the manipulation of sample and reagent
droplets by electrical means. Most prior work on CAD for biochips
has assumed independent control of electrodes using a large number
of (electrical) input pins. Such solutions are not feasible for low-cost
disposable biochips that are envisaged for many field applications. A
more promising design strategy is to divide the microfluidic array into
smaller partitions and use a small number of electrodes to control the
electrodes in each partition. We propose a partitioning algorithm
based on the concept of “droplet trace”, which is extracted from the
scheduling and droplet routing results produced by a synthesis tool.
An efficient pin assignment method, referred to as the “Connect-5
algorithm”, is combined with the array partitioning technique based
on droplet traces. The array partitioning and pin assignment methods
are evaluated using a set of multiplexed bioassays.
Categories & Subject Descriptors
B.7.2 [Integrated Circuits]: Design Aids; J.3 [Computer Applications]:
Life and Medical Sciences—Biology and genetics; health
General Terms:
Algorithms, Design, Performance
Keywords:
Connect-5 algorithm, droplet-based microfluidics, droplet trace,
pin-constrained biochip design, synthesis
1.Introduction
Microfluidics-based biochips constitute an emerging technology
area that can potentially open up several exciting applications. These
devices enable precise control of microliter and nanoliter volumes of
biological samples. They combine electronics with biology, and they
integrate various bioassay operations, such as sample preparation,
analysis, separation, and detection [1, 2, 3], in a single miniaturized
platform. It has been predicted that, by providing miniaturization,
automation and integration, microfluidic biochips will revolutionize
laboratory procedures in molecular biology with applications to
point-of-care diagnostics, DNA analysis, and automated drug
discovery [2, 3].
Currently, most commercially-available biochips are either
based on microarrays [4] or they rely on continuous fluidic flow in
etched microchannels [1]. An alternative design approach utilizes
droplets with microliter and nanoliter volumes, thereby obviating
the need for cumbersome micropumps and microvalves. Droplets
are actuated using on-chip electrodes and moved under the control
of a system clock; this microfludic system is similar in operation to
a digital microprocessor. Thus, this novel technology is referred to
as “digital microfluidics”. The “digital” structure also offers
reconfigurability and a scalable system architecture based on a
two-dimensional array [5, 6].
A typical biochip consists of a two-dimensional patterned metal
electrode array (e.g., chrome or indium tin oxide), on which droplets
containing biological samples are dispensed, transported, mixed,
incubated, separated or detected. As bioassays increase in complexity,
e.g., for high-throughput DNA sequencing [7] and large-scale protein
assays for drug discovery [8], design tools are needed to map and
execute them on the digital micofludic platform. In the next few years,
biochip integration and design complexity level are expected to
increase significantly. Automated design therefore becomes necessary
for this emerging marketplace. An appropriate addressing scheme
must be used to activate individual electrodes (unit cells) in the array.
Design and CAD research for digital microfluidic biochips has
mostly been focused on directly-addressable arrays [9, 10, 11, 12, 13,
14]. In such schemes, each cell of the patterned electrodes can be
accessed directly and independently via a dedicated control pin. This
method is adequate for small/medium-scale microfluidic electrode
arrays (with fewer than 10×10 electrodes). However, the number of
pins for a design based on direct addressing can be prohibitively high
for a large array. For example, a total of 10
4
pins are needed to
independently control the electrodes in a 100×100 array. Multi-layer
electrical connection structures and wire routing solutions are
complicated by the large number of independent control pins in such
arrays. Product cost, however, is a major marketability driver due to
the one-time-use (disposable) nature of most emerging devices.
Hence, simpler routing solutions are necessary so that the electrical
wiring can be easily incorporated in a low-cost implementation.
In this paper, we propose an automated digital microfluidic
biochip design method based on the partitioning of the microfluidic
array, and the assignment of a small number of control pins to a large
number of electrodes. The partitioning algorithm is based on the
concept of “droplet trace”, which is extracted from the scheduling and
droplet routing results produced by a synthesis tool.
The organization of the rest of the paper is as follows. In Section
2, we provide an overview of digital microfluidic biochips. Section 3
discusses related prior work on biochip design automation and
pin-constrained system design. Section 4 describes the proposed
_
________________________________________
*This work was supported in part
b
y the National Science Foundation
under grants IIS-0312352 and CCF-0541055.
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112
partitioning and pin assignment algorithm for a large microfluidic
array. Section 5 evaluates the proposed method using a set of
real-life bioassays. Finally, conclusions are drawn in Section 6.
2. Digital Microfluidic Biochips
A digital microfluidic biochip utilizes the phenomenon of
electrowetting to manipulate and move microliter or nanoliter droplets
containing biological samples on a two-dimensional electrode array
[5]. A unit cell in the array includes a pair of electrodes that acts as
two parallel plates. The bottom plate contains a patterned array of
individually controlled electrodes, and the top plate is coated with a
continuous ground electrode. A droplet rests on a hydrophobic surface
over an electrode, as shown in Figure 1. It is moved by applying a
control voltage to an electrode adjacent to the droplet and, at the same
time, deactivating the electrode just under the droplet. This electronic
method of wettability control creates interfacial tension gradients that
move the droplets to the charged electrode. Using the electrowetting
phenomenon, droplets can be moved to any location on a
two-dimensional array.
By varying the patterns of control voltage activation, many
fluid-handling operations such as droplet merging, splitting, mixing,
and dispensing can be executed in a similar manner. For example,
mixing can be performed by routing two droplets to the same location
and then turning them about some pivot points. The digital
microfluidic platform offers the additional advantage of flexibility,
referred to as reconfigurability, since fluidic operations can be
performed anywhere on the array. Droplet routes and operation
scheduling result are programmed into a microcontroller that drives
electrodes in the array. In addition to electrodes, optical detectors such
as LEDs and photodiodes are also integrated in digital microfluidic
arrays to monitor colorimetric bioassays [3].
To address the need for low-cost, PCB technology has been
employed recently to inexpensively mass-fabricate digital
microfluidic biochips. Using a copper layer for the electrodes, solder
mask as the insulator, and a Teflon AF coating for hydrophobicity, the
microfluidic array platform can be fabricated by using an existing
PCB manufacturing process [15]. This inexpensive manufacture
technique allow us to build disposable PCB-based microfluidic
biochips that can be easily plugged into a controller circuit board that
can be programmed and powered via a standard USB port; see Figure
2. However, a large number of independent control pins necessitates
multiple PCB layers, which adds significantly to the product cost.
3. Related Prior Work
Recently years have seen growing interest in the design of
microfluidic biochips and CAD methods for system design [10, 12, 14,
16, 17]. In [10], classical architectural-level synthesis is adapted for
automated biochip design based on bioassay protocols. The problem
of microfluidic module placement, where array area and fault
tolerance serve as the placement criteria, is discussed in [12]. A
unified synthesis method, which combines operation scheduling,
resource binding, and module placement, is proposed in [16]. A
drawback of these CAD techniques is that they assume a
direct-addressing scheme, which requires a very large number of
independent control pins for large-scale biochips. Thus these methods
are unlikely to be useful in practice for low-cost disposable devices.
Pin-constrained design of digital microfluidic biochips was
recently proposed and analyzed in [3]. The number of control pins for
a fabricated electrowetting-based biochip is minimized by using a
multi-phase bus for the fluidic pathways. Every nth electrode in an
n-phase bus is electrically connected. Thus, only n control pins are
needed for a transport bus, irrespective of the number of electrodes
Figure 1: Schematic diagram of a digital microfluidic biochip.
Figure 2: The concept of a commercial disposable microfluidic
biochip.
that it contains. Although the multi-phase bus method is useful for
reducing the number of control pins, it is only applicable to a
one-dimensional (linear) array.
An alternative method based on a cross-reference driving scheme
is presented in [18]. This method allows control of an N×M grid array
with only N+M control pins. The electrode rows are patterned on both
the top and bottom plates, and placed orthogonally. In order to drive a
droplet along the X-direction, electrode rows on the bottom plate
serve as driving electrodes, while electrode rows on the top serve as
reference ground electrodes. The roles are reversed for movement
along the Y-direction. This cross-reference method facilitates the
reduction of control pins. However, it requires a special electrode
structure (i.e., both top and bottom plates containing electrode rows),
which results in increased manufacturing cost for disposable
microfluidic chips. Moreover, this design is not suitable for
high-throughput assays because droplet movement is inherently slow.
More recently, a promising design method based on array
partitioning has been proposed for pin-constrained biochips [17]. The
microfluidic array is divided into several partitions and sets of pins are
determined, where each set of pins correspond to a partition and all
the sets are of the same size. For example, if a biochip of arbitrary size
is divided into six partitions and five pins are allocated per set, only
5×6 = 30 pins are needed to independently address the individual unit
cells of the array. By carefully controlling the number of partitions,
the total number of pins is reduced significantly compared to the
direct-addressing scheme.
However, the design method presented in [17] suffers from
several drawbacks. First, the array partitioning in [17] is ad-hoc and
no systematic algorithm has thus far been presented. Secondly,
microfluidic modules such as mixers, splitters, and detectors are not
considered in the ad-hoc partitioning method; an additional design
step is needed to handle these modules separately. Moreover, the
partitioning method assumes a priori that partitions do not overlap;
this restriction can be a limitation for many bioassays. Finally, no
pin-assignment algorithm is presented in [17].
113
4. Array Partitioning and Pin Assignment
Figure 3: An example to illustrate electrode interference.
We first review the problem of electrode interference in
microfluidic arrays. This problem can appear if multiple electrodes
are controlled using a single pin. For example, assume that a droplet
rests on an electrode (unit cell) and two of its neighbors are
connected to the same pin. Recall that to move the droplet to one of
two neighbors (i and j) that share the same pin, we must deactivate
the electrode where the droplet rests and activate the destination
electrode i. However, when electrode i is activated, the other
neighbor electrode j is also activated since it shares the same pin
with electrode i. In this case, the droplet undergoes a split, instead of
being moved to electrode i. This problem can be solved by
addressing each electrode and its neighbors with distinct pins. Since
one electrode can have at most four neighbors in a two-dimensional
array, the minimum number needed is five. Recent experimental
studies have shown that five independent pins are adequate to route
a droplet to any place on the chip for single droplet manipulation
[17].
When multiple droplets are manipulated simultaneously on the
chip, a pin-constrained layout may also result in unintentional
droplet movement or other unintended consequences. For the
example in Figure 3, electrode interference will occur if we attempt
to move Droplet D
i
and let Droplet D
j
stay where it is. To move D
i
one cell downwards, we need to activate Pin 8 and deactivate Pin 1.
To hold Droplet D
j
, we need to activate Pin 3. However, since both
Pin 3 and Pin 8 are charged, D
j
will be split unintentionally. This
type of problem is referred to as electrode interference.
Electrode interference can be solved by “virtually” partitioning
the array into regions, with each of them having only one activated
cell at any point in time. Mutually-exclusive sets of pins are utilized
for manipulating the droplets in different regions. The partitions can
be viewed as subarrays that can contain at most one droplet. Recall
that regardless of size, a two-dimensional array only needs five
independent pins to ensure full control of a single droplet. By using
different sets of five pins for electrode control in different partitions,
electrode interference among partitions can be avoided. Therefore,
for the partitioned array, the number of droplets that can be
simultaneously transported without stall cycles is equal to the
number of partitions, and the total number of control pins needed is
equal to five times the number of partitions. The above partitioning
solution was proposed recently in [17].
However, both array partitioning and the assignment of control
pins to electrodes in [17] are done in an ad-hoc manner. No
systematic algorithms have been proposed thus far to implement the
partitioning-based pin-assignment method and incorporate it in
automated design tools. Here we propose an algorithm based on the
concept of droplet trace, which unifies array partitioning and pin
assignment.
4.1. Trace-Based Partitioning Algorithm
As discussed above, partitioning can effectively avoid
electrode interference if each partition includes only one droplet.
Hence, the partitioning criterion here is to ensure at most one
Detector1(x,y) Detector2(x,y) Detector3(x,y)
Droplet 1 (8, 3) (8, 9) (5, 9)
Droplet 2 (3, 2) (3, 6) (5, 6)
(a)
(b)
Figure 4: (a) Detectors used in bioassay; (b) Routing result and
array partitions.
droplet is included in each partition. However, partitions with no
droplets (at any point in time) should be avoided because no droplet
manipulation is done in this region with the additional set of pins
assigned to it. Hence it is best to ensure that each partition has exactly
one droplet in it.
Based on this requirement, we find that the droplet trace, defined
as the set of cells traversed by a single droplet, serves as a good tool
for generating the array partitions. Since we view pin assignment as
the last step in system synthesis, information about module placement
and droplet routing is available a priori. The droplet trace can be
easily extracted from the droplet routing information and the
placement of the modules it is routed to. A trace extraction example is
shown in Figure 4, where two droplets are to be manipulated on the
microfluidic array. Both of these are required to be detected by an
optical sensor three times in a specific bioassay. The placement of
these detectors is shown in Figure 4(a). The droplet routes, i.e., the
path taken by droplets, are shown by the arrows in Figure 4(b). The
connected arrows illustrate the traces of the two droplets. For each
droplet, we create a partition composed of all the cells on its trace as
well as the cells adjacent to the trace. The adjacent cells are included
to form a “guard ring” along the trace to avoid inadvertent mixing and
movement. The guard rings are a consequence of the fluidic constraint
described in [19].
Note that in Figure 4(b), there are two “white” regions that
belong to neither partition. They are referred to as “don’t-care”
regions because they are similar to the “don’t-care” terms in logic
synthesis; they can either be assigned to any partition or they can
together form an additional partition if multi-droplet-operation
modules, e.g. mixers, can be positioned in them.
In order to reduce the number of partitions, we introduce a
time-division pin-sharing method. The basic idea is to merge
partitions that have no overlapping time spans, where a time span for a
partition is defined as the period of time during which it contains a
droplet. The time spans for all the partitions can be easily calculated
from the operation schedule, module placement and droplet routing
results [19]; the overlaps can then be readily determined. Partitions
with non-overlapping time spans are merged to form a larger partition.
This check-merge procedure continues until all partition pairs overlap
in their time spans. By reducing the number of partitions, we can
reduce the number of control pins needed for the array. Note that
droplet traces may have spatial overlap, i.e., they may intersect at one
or more unit cells on the array. In this case, the requirement of one
droplet per partition is not met and electrode interference may occur.
This problem is handled by simply modifying the partitioning result.
Droplet Trace
Partition 1
Partition 2
x
1 2 3 4 5 6 7 8 9
9
8
7
6
5
4
3
2
1
y
D
i
D
j
1 2 3 8
8 7 6 5
5 4 9 1
1 2 3 8
114
(a) (b)
Figure 5: (a) Routing result and partitioning (b) Time-span table
for the droplets.
We next study the case where droplets traces intersect on the
array. This implies that partitions derived by the proposed method
overlap in some regions. Sets of pins from an “overlapping” partition
cannot be used in the overlapped region since the reuse of the pins
leads to electrode interference. One solution to this problem is to
make the overlapping region a new partition, referred to as the
overlapping partition, and use direct-addressing for it. Again,
time-division pin-sharing (TDPS) can be used to reduce the number of
pins since pin sets of the other (non-overlapping) partitions can be
candidates for direct-addressing in the overlapping partition.
An example of this approach is shown in Figure 5. The droplet
traces are first derived from the droplet routing information. Partitions
1, 2, 3, and 4 are assigned accordingly. Partition 2 and Partition 3
overlap with each other as shown. Thus a new Partition 23 is created.
From the scheduling result in Figure 5(b), the time span for Partition
23 is found to be 10-14s. Next the time spans for Partitions 1 and 4 are
checked and it is seen that their time spans do not overlap with that for
Partition 23. Hence the two set of pins (a total of 2x5=10 pins) in
Partitions 1 and 4 can be used to directly address the nine electrodes in
Partition 23.
Partitions that share pins with the overlapping partition are empty
while droplets are manipulated in the overlapping partition. Therefore,
the sharing of pins in these cases does not lead to electrode
interference. By introducing the concept of TDPS, we can
significantly reduce the number of pins required for independent
addressing. The concept of TDPS can also be applied in the spatial
dimension to the operations inside the overlapping region to further
reduce the number of control pins.
Once a spatially overlapping region is found, we determine if
there are temporally overlapping droplets in this region. Depending on
the outcome of this procedure, a spatial overlap region can be then
divided into two groups—a spatially overlapping but temporally
non-overlapping (SOTN) region, and a spatially overlapping as well
as temporal overlapping (SOTO) region. For SOTO regions,
direct-addressing is used. For SOTN regions, even though droplets
traces cross each other, different droplets are sequenced in time (one
after the other), i.e., at any point in time, there is at most one droplet
inside the region. In this case, a pin set with the minimum size (k = 5)
for single droplet manipulation is assigned to this SOTN region.
Again, we use the above example of Figure 5 for illustration.
Table 1 shows the schedule information needed for carrying out the
temporal check for the overlapping region.
Partitions 23.2 and 23.3 represent the manipulation of Droplet 2
and Droplet 3 in Partition 23 respectively. Table 1 shows that the time
spans for these partitions do not overlap, thus five pins (in contrast to
the nine pins needed for direct-addressing) are adequate for the
overlapping partition.
4.2. Extended Partitioning Algorithm
In this subsection, we present an extension of the partitioning
algorithm that does not require module placement information. In
Section 4.1, we needed the placement information for modules that
Table 1: Time-span table with detailed scheduling results for the
overlapping region.
(a) (b)
Figure 6: Pin assignment example for (a) a mixer and (b) a
splitter.
handle multiple droplets, such as mixers and splitters to determine the
droplet traces. Here we rely only on the schedule of operation and
droplet routing results to indirectly determine module placement. For
example, the mixing operation can be viewed as two droplets being
routed together along an identical path simultaneously with the start
point in the mixer region. Similarly, droplet splitting can be viewed as
two droplets sharing the same start point in both the time and space
domains. We can therefore identify mixer regions by checking
whether droplet traces exactly overlap instead of just intersecting each
other in the same time span; a splitter can be recognized in a similar
manner. As a result, overlapping partitions can be assigned to mixers
and splitters. Note that splitting and mixing can both be viewed as
deliberate (desired) electrode interference. Thus though multiple
droplets are manipulated in mixer or splitter regions, five control pins
are sufficient, as shown in Figure 6. In this way, the number of pins
can be further reduced.
4.3. Pin assignment using the Connect-5 algorithm
In Section 4.1 and Section 4.2, we have described an automated
partitioning method for digital microfluidic arrays. Each partition is
assigned a pin set. In this section, we address the problem of how to
map control pins to the electrodes in a partition. An efficient and
easy-to-implement algorithm is presented. The algorithm is based on a
strategy of the Connect-5 (Gomoku) board game [20], thus it is
referred to as the Connect-5 algorithm.
The sets of pins assigned to the partitions belong to two groups
according to their cardinality, i.e., the minimum for single droplet
manipulation (k = 5) or the number of pins required for
direct-addressing. Here we focus on the pin assignment problem for
the first case, since pin assignment for direct-addressing is
straightforward (there exists a simple one-to-one mapping between
pins and electrodes).
Our goal is to ensure that any five adjacent unit cells (a central
cell and its four neighbors) that form a “cross” are assigned distinct
pins. We refer to the above constraint as the “cross constraint”. The
pin assignment problem under cross constraints can be mapped to the
famous vertex coloring problem in graph theory [21]. The problem
here is to obtain a 5-coloring of the graph derived from a partition, as
shown in Figure 7. The unit cells in the partition are mapped to
vertices and any two cells that belong to a “cross” are
connected by an edge. The graph corresponding to a partition is
referred to as the partition graph.
Partition Time Span
1 1-7
2 5-16
3 7-14
4 17-20
23.2 10-11
23.3 13-14
Partition Time Span
1 1-7
2 5-12
3 7-23
4 17-20
23 10-14
1 2 3
4 5 1
1 2 1
1 4 1
1 3 1
Partition 1
Partition 2
Partition 23
Partition 3
Partition 4
x
y
115
Figure 7: Mapping of an array to an undirected graph.
Figure 8: A single Bagua structure (the tilted square) and its
repetition in a square partition.
Figure 9: Covering a partition by shifting Bagua repetition along
rows.
The graph coloring problem, which involves the determination
of the chromatic number χ(G) for a graph G, is known to be
NP-complete [21]. However, if χ(G) or the number of colors to be
used is known, as in the case here, there exists efficient algorithm for
graph coloring. However, the regular structure of the partitions
can be used to solve the problem more efficiently using tiling. This
approach allows us to use a regular distribution of pins, a layout
feature that is not directly obtained from graph coloring. The tile (or
template) used here is referred to as “Bagua”, a Chinese game strategy
for the Connect-5 board game [20]. A Bagua is a tilted square, as
shown in Figure 8. By repeating placing Bagua structures next to each
other until the partition boundaries are reached, a Bagua repetition is
derived as shown in Figure 8. The tiling using Bagua repetitions forms
the basis for the Connect-5 algorithm.
Five copies of Bagua repetitions are sufficient to cover a partition
of any size. This is because of the following property of a Bagua
repetition: vertices connected to the same (shared) pin appear after
exactly five cells in the same row or column of the partition. The
partition can be covered with Bagua repetitions by simply taking a
Bagua repetition and shifting it one cell along an arbitrary direction,
e.g., upwards, then assigning it to another control pin and repeating
this step four times, as shown in Figure 9. Note that, although the
shifting direction is arbitrarily selected at the start of the tiling process,
once chosen it must be consistent over the four shifting steps.
As shown in Figure 9, the pin assignment that results from the
shifting of Bagua repetition satisfies a cyclic property, i.e., each row is
a cyclic repetition of an ordered sequence, and it is also a shifted copy
(shift by two cells) of the previous row. This cyclic property provides
an easy way to implement the Connect-5 algorithm.
To start, the first row of a partition is selected. Pins are assigned
in a fixed cyclic order until the boundary of the partition is
reached. Then in the next row, the same order is used for
but with a 2-cell-shift to the left/right. The procedure continues until
all cells in the partition have been assigned pins. Recall that the
Figure 10: A demonstration that the “cross constraints” are met.
Figure 11: A wiring example for the pin assignment obtained
using the Connect-5 algorithm. For each partition, two pins can
be wired in one layer.
shifting direction, once chosen, must remain fixed during the
assignment procedure for a given partition.
Next we show that control pins assigned to the electrodes this
method in a partition allow free movement of a single droplet, i.e., the
“cross constraint” is met. To demonstrate this, we consider the cell
which is hatched in Figure 10. If the cell is assigned Pin 1, we cannot
assign the same pin to the unit cells that are shaded. Otherwise, we
will violate the cross constraint in some cases. It can be found that all
the unit cells in the Bagua tile and its repetitions stay out of the
forbidden area. Thus for each pin assigned to cells in a Bagua
repetition, the cross constraint is not violated. Since this is true for any
Bagua repetitions and any partition can be tiled by five copies of
Bagua repetitions, the “cross constraint” is automatically met for
every cell in our pin assignment method.
Compared to the graph coloring approach, the Connect-5
algorithm offers the important advantage that it allows wiring to be
done easily on a 3-layer PCB; see Figure 11. The graph coloring
approach does not lend itself to this simple pin layout because of the
likelihood of irregular vertex coloring.
5. Evaluation Example: Multiplexed Bioassays
To show how partitioning and pin assignment work for
pin-constrained microfluidic biochips, we use a real-life experiment of
a multiplexed biochemical assay consisting of a glucose assay and a
lactate assay based on colorimetric enzymatic reactions. These assays
have been demonstrated recently [3]. The digital microfluidic biochip
contains a 15
×15 microfluidic array, as shown in Figure 11. The
schedule for the set of bioassays, if a microfludic array with 225
control pins is available, is listed in Table 2; one iteration of the
multiplexed assays takes 25.8 seconds [3]. The movement of droplets
is controlled using a 50 V actuation voltage with a
switching frequency of 16 Hz. A depiction of the droplet paths for
multiplexed glucose and lactase assays is shown in Figure 12.
When the partitioning and pin assignment algorithm starts, six
partitions are first assigned to the four droplet traces of Reactants 1,
1
1
1
1 1
1
1
1
1
1
1 1
1
1
1 2 3
2 3 1 2
1 2 3
1 2 3 1
3 1 2 3
1 2 3
5 1 2 3 4 5
2 3 4 5 1 2
4 5 1 2 3 4
1 2 3 4 5 1
3 4 5 1 2 3
5 1 2 3 4 5
1
1
1
1 1
1
1
1 2 3
2 3 1 2
1 2 3
1 2 3 1
3 1 2 3
1 2 3
116