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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 851751, 6 pages
Compression of Meanders
1Department of Informatics, University of Piraeus, Karaoli & Dimitriou 80, 18534 Piraeus, Greece
2Department of Informatics, Ionian University, Plateia Tsirigoti 7, 49100 Corfu, Greece
Received 21 June 2012; Revised 29 August 2012; Accepted 12 September 2012
Academic Editor: Annalisa De Bonis
Copyright © 2013 A. Panayotopoulos and P. Vlamos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper refers to the algorithmic transformation of a meander to its uniquely defined compression. We obtain this directly from meandric permutations, thus creating representations of large classes of meanders of different orders. We prove basic properties, give arithmetic results, and produce generating procedures.
A closed meander of order is a closed self-avoiding curve crossing an infinite horizontal line times . In this paper, we obtain the compression as the determination of a unique simple meander, directly from its permutation. The meanders as planar permutations were introduced by Rosenstiehl  and they have been studied with nested sets [3, 4].
More specifically, in Section 2, we define the flow of a meander consisted by its traces and corresponding blocks. In Section 3, we create a specific form of meanders: the simple ones, we study the properties of their numbers of cuttings and cutting degree and we use them in order to introduce the compression. In Section 4, we determine the flow of the meandric permutations and we achieve also numerical results for the classification of the meanders of the compressions according to their order. Finally, in Section 5, we establish the compression of meanders directly from their meandric permutations divided in suitable blocks. Thus, we change their interpretation and produce a simplified procedure for generating the compressions.
The following definitions and notation are necessary for the rest of the paper .
A set of disjoint pairs of such that and for any we never have is called nested set of pairs on . Each pair of a nested set consists of an odd and an even number. We denote the set of all nested sets of pairs on by . Two nested sets define a permutation on , such that and , for every . The sets are -matching if and only if has cycles. In the case where , are simply called matching. This definition is equivalent to the one given in .
We call short pair of any pair of consecutive numbers that belongs to , and outer pair of any pair such that there is no pair with . Each nested set of pairs contains at least one outer and one short pair.
A meander of order is equivalently defined  as a cyclic permutation on , for which the following properties hold true: , and the sets are both nested and matching.
We take all numbers . It is clear that is odd if and only if is odd. In the corresponding geometrical representation, the nested arcs correspond to nested pairs. A pair of nested sets , should be matching, in order to generate a meander. For example, the meander of Figure 1 is of order 14, with
The set of all the meanders of order is denoted by . Let be a meander crossing a horizontal line. Following , for any we consider the vertical line, which shall be called the -line, passing through the middle point of the segment of the horizontal line. The numbers of those arcs of the meandric curve which are intersected by the -line and lie above and beneath the horizontal line of , are called the numbers of cuttings and , respectively .
The sum of the number of those arcs of the meandric curve which are intersected by the -line is called the cutting degree of the meander at . We notice that and are of the same parity; hence, is always even .
The meandric curve always has points of intersection with the -line, which we call traces. Obviously, the number of the traces is equal to . Starting below the horizontal line, we label the traces with the numbers , knowing that (resp., ) of them are lying above (resp., beneath) the horizontal line. From now on, we will consider that the traces are identical to their corresponding labels. For the meander of Figure 1 and for , we have , , and .
Beginning from trace 1 and moving clockwise upon the meandric curve, following its “natural flow,” we obtain a shuffle of the permutation of the traces and the meandric permutation, see Figure 2 where the circled elements are the traces.
In the general case, we have the shuffle with being the traces of the meander at and the parts of consecutive elements of the meandric permutation , lying between consecutive traces, called blocks of the meander; that is, the block , , is the set of the consecutive elements of the permutation , which are lying between the traces and . These two traces are called the “entrance” trace and the “exit” trace of the block, respectively, while the shuffle is called flow of the meander from the trace of the -line, or for simplicity -flow.
For every , corresponds to the part of the meandric curve starting from the trace and ending at the trace , which we denote by . If is odd (resp., even), then this curve lies on the left (resp., right) of the -line. In Figure 1, we denote the curve by .
3. Simple Meanders and Compression
Let and its flow . To each block , , we correspond a number such that
The set can be partitioned into three classes , , and , where the set consists of the blocks with and the set consists of the blocks with and , while the set consists of the blocks with and .
A meander is called simple at (i.e., simple referring to the -line) if , . Hence, every block of the sets (resp., , ) has exactly one element (resp., two elements). We notice that the following necessary and sufficient condition holds true.
A meander is simple at if and only if every triple of consecutive terms of its permutation contains elements from both of the sets and .
For example, the meander of Figure 3 is simple at .
We note that a meander can be simple at more than one point. For example, the meander is simple at and .
Let be a meander that is not simple at . Further on, we will study every meander according to its -line, so for simplicity we omit the index from the notation. Let be its already defined set of blocks, where and the blocks , for (resp., ), are the ones placed at the left (resp., right) of the -line. Each block lies between the trace and the trace of the flow .
We denote by , , the closed interval of with ends the traces , . Given a pair of blocks , or , with , then the block is called internal of the block , and the block is called external to the block . We can easily deduce that the number of the internal blocks of a block is equal to .
If we replace the blocks , , of the meander by blocks having one element (resp., two elements) whenever (resp., ), then we obtain a meander of order (since its blocks have one or two elements, corresponding to crossing points with the horizontal line) and simple at (counting the points of intersection with the horizontal line to the left of the -line).
The result of the above replacement is the set of blocks , where for simplicity. The set is partitioned into the classes , , and corresponding to the classes , , and of the set .
When we put a dash upon any existing notation, we refer to the elements of the deduced simple meander . We easily obtain that(i), are of the same parity,(ii), , , ,(iii)if (resp. ) and , then its corresponding block (resp., ) contains one short pair of (resp., ). If , then its corresponding block is the pair , .
The pair is called central compression or simply compression of the meander . The simple at meander has as same invariants with the meander the traces and the flow of curves. For example, the compression of the meander of Figure 1 is the pair , where is the meander of Figure 3 and .
4. The Flow
The traces and the blocks of the flow of a meander can be found from its permutation with the help of the subsets and , containing the pairs satisfying the relation
These pairs have one element belonging to the set and the other belonging to the set , with and .
According to the absolute value , we place the elements of in decreasing order, while of in increasing order. If , then we correspond the numbers to the ordered pairs of , and the numbers to the ordered pairs of . If (resp., ), then we correspond to the ordered pairs of (resp., ) the numbers . It is obvious that the previous numbers coincide with their corresponding traces.
In order to start the -flow from the first trace, we choose the pair which corresponds to the trace . If , then this pair is the outer pair of with odd and less than . If , then this is the pair of with the smallest value of .
For example, for the meander of Figure 1, we have that Hence, we choose the pair of to correspond to the trace .
Since the permutation is cyclic, we do not change the notation, that is, which also defines a partition of with classes including its consecutive elements, which belong, respectively, to the sets and . This partition gives the classes of the set .
Practically, at first we partition the permutation of the meander into classes including the consecutive terms of , which are less or equal to (resp., greater than) . Thus, we have the partition of into blocks, putting at the beginning the last remaining elements and marking the traces.
The placement of the traces follows the change of parity of the elements of the permutation, if we have an odd (resp., even) element followed by an even (resp., odd) element, then their intermediate trace is lying above (resp., beneath) the horizontal line. From the above partition, we obtain the flow: see Figure 2.
The meanders of the compression of the set are partitioned into classes, with elements belonging to the sets .
In the methods of generating planar permutations , we can also include the way to find the blocks of meanders, their corresponding numbers , and consequently the orders of the meanders of their compressions.
These meanders can be used as generators for the reverse problem of ‘‘decompression.’’ We can use them to extend a meander simple at to all possible meanders , with .
Table 1 presents the cardinalities of those classes for , without taking into account the corresponding -line.
The zeros of the first (resp., second) column verify the fact that if the order of the meander is even (resp., odd), then there do not exist meanders of order with compression of order 1 (resp., 2). We can easily prove that the values of the first column express that there exist meanders of order with compression of order 1. For meanders of larger order, we should try to calculate the number of different blocks of given orders.
5. Determining the Compression
We shall find the compression of a meander with the help of its -flow . We recall that each block consists of one or two elements. Obviously, the elements of belong to the sets (resp., ), when (resp., ). The relative position of these points defines a relation of preceding for the blocks of the set ; hence, the block precedes the block (), iff .
This relation is defined by the following conditions.(a) If , , with and (resp., ), then (resp., ), since the block is internal of the block .(b) If , , with and (resp., ), then (resp., ), since this is imposed by the nature of the meander.(c) If , with and or , then , since the block is internal of the block .(d) If , with , and (resp., ), then (resp., ), since this is imposed by the nature of the meander. The case where is similar. Obviously, the above results do not cover the cases of two blocks, the one belonging to the set , and the other to the set , where none of them is internal to the other. In order to obtain a unique solution, we have to make the following choice.(e) If , , with and or , then (resp., ), where , is the first element of , is the last element of and .
From the above conditions (a)–(e), we obtain an ordering for the blocks of the set , concerning their relation of preceding, that is, .
For example, applying the above conditions for the meander of Figure 1, we obtain that and . Indeed, due to the condition (a), due to the condition (b), due to the condition (e), due to the condition (a), and finally due to the condition (b). Similarly, we obtain the ordering for the rest of the blocks. Hence, .
In the general case, the ordering is deduced from a Hamiltonian path of the directed graphs with vertices the elements of the set (resp., ) and arcs the pairs (resp., ) such that .
We note that iff , given that defines the position of the block at the total order of . For our example, we have that .
Practically, the whole procedure of finding the compression can be presented in a table, where the second row refers to the flow , which includes all the elements necessary for the conditions (a)–(e), while from their application we deduce in the third row the total order of the set . The last row refers to the flow by assigning the numbers of the set to their corresponding blocks of , according to the following remarks for the blocks .(i)Their elements have the same ordering (ascending or descending) with those of .(ii)When their elements are not consecutive, then .
We have introduced the compression of a meander, directly with the use of blocks of its permutation. The uniqueness of the compression is established by the ordering of the blocks, which is deduced according to its flow.
Various open questions can arise by the above meanders, when they are used as representatives of large classes of meanders as shown in Table 1. Yet, the main open problem is the reverse procedure of compression. The decompression of a meander to others of larger order having the same traces, number of cuttings, and flow seems to be the final step for integrating the procedures of cutting and compressing meanders, and in parallel being very promising for enumeration results and applications in physical phenomena.
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