A. Amir and G. Benson, Two-dimensional periodicity and its applications, Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 1992, 440--452.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 Introduction Pattern matching in compressed text is one of the most interesting topics in the combinatorial pattern matching. Several researchers tackled this problem. EilamTzore # and Vishkin [8] addressed the run length compression, and Amir, Landau, and Vishikin [6] and Amir and Benson [2, 3] and Amir, Benson, and Farach [4] addressed its two dimensional version. Farach and Thorup [9] and Gasieniec, et al. 11] addressed the LZ77 compression [18] Amir, Benson, and Farach [5] addressed the LZW compression [16] Karpinski, et al. 12]andMiyazaki,et al. 15] addressed the straight line ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Symposium on Discrete Algorithms, page 440, 1992.

....are often stored in compressed forms. We therefore need a fast pattern matching technique for searching the compressed text directly. Several researchers tackled this problem. Eilam Tsoreff and Vishkin[6] addressed the run length compression, and Amir, Landau, and Vishikin[5] and Amir and Benson[2, 3] addressed its twodimensional version. Farach and Thorup[7] and Gasieniec, et al. 8] addressed the LZ77 compression[14] Amir, Benson, and Farach[4] addressed the LZW compression[13] Karpinski, et al. 9] and Miyazaki, et al. 12] addressed the straight line programs. For a fast pattern matching ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Symposium on Discrete Algorithms, page 440, 1992.

....In other words, the text compression can speed up the pattern matching. In this framework, it is required to develop an efficient pattern matching algorithm for searching directly the compressed text without decoding. The compressed pattern matching problem has been studied by many researchers [6, 5, 1, 2, 3, 7, 9, 4, 10, 13, 11], mainly from theoretical viewpoints. Most of the compression methods dealt with are the adaptive compression methods such as the LZ77 compression [15] and the LZW compression [14] Since in such compression methods the encoding of text substring depends on the previous part of the text, it is ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 440--452, 1992.

....are often stored in compressed forms. We therefore need a fast pattern matching technique for searching the compressed text directly. Several researchers tackled this problem. Eilam Tsore# and Vishkin[6] addressed the run length compression, and Amir, Landau, and Vishikin[5] and Amir and Benson[2, 3] addressed its twodimensional version. Farach and Thorup[7] and Gasieniec, et al. 8] addressed the LZ77 compression[14] Amir, Benson, and Farach[4] addressed the LZW compression[13] Karpinski, et al. 9] and Miyazaki, et al. 12] addressed the straight line programs. For a fast pattern matching ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Symposium on Discrete Algorithms, page 440, 1992.

.... introduced the notion of bidimensional periodicity (in particular the notions of symmetry and periodicity vector) and proved theorems analogous to the periodicity lemma (cf. 13] After them, many researchers have worked on bidimensional periodicity and its applications (cf. for instance, [4], 5] 6] 15] 12] and references therein) In particular Galil and Park developed in [8] and [9] this line of research to obtain the first totally alphabet independent bidimensional pattern matching algorithm. In [10] and [11] Giancarlo and Mignosi considered similar periodicity problems ....

A. Amir and G. E. Benson, Two-dimensional periodicity and its applications, Proc. 3rd ACM-SIAM Syrup. on Discr. Algorithms (

....classes appear linear in the dimension. Also, one provides a full characterization of sources positions, including the degenerated ones that are essential to the design and correctness of 2D pattern matching algorithms. This considerably refines and achieves the previous classification by [1], and even the extended results in [4] and allows for a classification of space coverings, where non degenerated periodicities appear essential. One exhibits relationship between the periods of a pattern and the possible space coverings by the same pattern. This is relevant both to the derivation ....

....the free zone, that creates border effects . This additive property allows to use general results on biperiodic functions on Z 2 and prove a lattice distribution of almost all invariance vectors. Notice this vectorial approach provides a very short proof of the previous results in [1, 4]. Many proofs rely on the Factorisation Theorem [5] equation ab = ba implies that a and b are powers of a same primitive word. For example, in Theorem 2, equation (3) implies that, for any , js j divides L. Otherwise, for some j, one has L mod js j j = ff 6= 0. With a = s j [1] s j [ff] ....

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Amir (A.) and Benson (G.). -- Two-dimensional periodicity and its application. In SODA'92. -- 1992. Proceedings of the 3rd Symposium on Discrete Algorithms, Orlando, FL.

....divided into two point of view: algorithmic and image processing ones. In computer algorithm, a texture can be viewed as a two dimensional string array consists of a finite number of alphabets. Several algorithms for analyzing the translation symmetry of two dimensional strings have been proposed [21, 22, 23]. Although these algorithms can determine the translation symmetry in O(mn) time [21, 22] where m Theta n is the size of the texture, they are much complex to implement. Further, they do not provide any mechanism to handle noisy textures because the basic operation on the algorithms are based ....

A. Amir and G. Benson. Two dimensional periodicity and its application. In 3rd Annual ACM-SIAM Symposium On Discrete Algorithms (SODA), pages 440--452, 1992.

....1 Introduction Pattern matching in compressed text is one of the most interesting topics in the combinatorial pattern matching. Several researchers tackled this problem. EilamTzore and Vishkin [8] addressed the run length compression, and Amir, Landau, and Vishikin [6] and Amir and Benson [2, 3] and Amir, Benson, and Farach [4] addressed its two dimensional version. Farach and Thorup [9] and G asieniec, et al. 11] addressed the LZ77 compression [18] Amir, Benson, and Farach [5] addressed the LZW compression [16] Karpinski, et al. 12] and Miyazaki, et al. 15] addressed the ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Symposium on Discrete Algorithms, page 440, 1992.

....in Park and Galil [1992] See Amir, Landau and Vishkin [1992] for a description of the scaled matching algorithm of Section 10.2, and Amir and Calinescu [1996] for an alphabet independent and dictionary scaled matching algorithm. The compressed matching algorithm of Section 10.3 appeared in Amir and Benson [1992], and an improvement can be found in Amir, Benson and Farach [1994b] See Aho and Corasick [1975] for a description of the the automaton based dictionary matching algorithm. Suffix trees and their contruction are described in Weiner [1973] and McCreight [1976] The first optimal algorithm for ....

A. Amir and G. Benson [1992]. "Two dimensional periodicity and its application", Proc. of 3rd Symoposium on Discrete Algorithms, Orlando, Florida, 440--452.

....limited, we cannot exploit such data structures. 2 Table 1: History of compressed pattern matching. compression method compressed pattern matching algorithms run length T. Eilam Tzoreff and U. Vishkin [16] run length (two dim. A. Amir, G. M. Landau, and U. Vishkin [7] A. Amir and G. Benson [3, 4]; A. Amir, G. Benson, and M. Farach [6] LZ77 M. Farach and M. Thorup [17] L. G asieniec, M. Karpinski, W. Plandowski, and W. Rytter [21] LZW A. Amir, G. Benson, and M. Farach [5] T. Kida, M. Takeda, A. Shinohara, M. Miyazaki, and S. Arikawa [26] T. Kida, M. Takeda, A. Shinohara, and S. Arikawa ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. of the 3rd Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 440-- 452, 1992.

....BYR93, ABF92, GP92, RR93, GM94] It is interesting to notice that the now classical duel paradigm [Vis85] allows a drastic improvement on string searching optimal parallel algorithm [Gal92] In the following, we use equivalently the terms of self overlapping patterns and periodic patterns. [AB92] introduces the notion of sources, the locations where an overlap can originate. A geometric regularity of this pattern of candidates to be a source has been proved and fully characterized in [AB92, GP92, RR93] Additionally, patterns are classified according to the location of sources. It is ....

....the following, we use equivalently the terms of self overlapping patterns and periodic patterns. AB92] introduces the notion of sources, the locations where an overlap can originate. A geometric regularity of this pattern of candidates to be a source has been proved and fully characterized in [AB92, GP92, RR93]. Additionally, patterns are classified according to the location of sources. It is also shown in [RR93] that a periodic pattern actually is the repetition of a smaller pattern. This extends a well known 1D property. It leads to fully describe the characters repetitions in all periodic patterns. ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. In SODA'92, 1992. Proc. 3-rd Symposium on Discrete Algorithms, Orlando,FL.

....in Rocquencourt, France. The author wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for a generous support. In addition, support was provided by NSF Grants NCR 9206315 and CCR 9201078 and INT 8912631, and from Grant AFOSR 90 0107, and in part by NATO Collaborative Grant 0057 89. on words [2, 4, 8, 9, 23, 24]. The parameters of interest to these applications are: the number of phrases, the number of phrases of a given size, the size of a phrase, the length of a sequence built from a given number of phrases, etc. By all means the most important parameter is the number of phrases which is studied by us ....

A. Amir and G. Benson, Two-Dimensional Periodicity and Its Application, Proc. 3-rd Symposium on Discrete Algorithms, Orlando, 1992.

....an efficient pattern matching algorithm for searching a compressed text directly. The problem of pattern matching in compressed text is of not only practical interest but also of theoretical interest. It has been studied recently by several researchers for several compression methods. For example, [1, 2, 3, 5, 6] are for the run length coding, 4] for the LZW coding, 7, 8, 9] for the LZ77 coding. A straight line program is a compact representation of string. It is a context free grammar in the Chomsky normal form that derives only one string. The length of the string represented by a straight line ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Symposium on Discrete Algorithms, page 440, 1992.

....Verification of a candidate could then be done in the naive character by character comparison, but the time would still be linear because the candidates do not overlap. The problem with implementing this idea is that there is no guarantee that the pattern is non periodic. Indeed it has been shown [2] that there are four different types of two dimensional periodicity and that a pattern may contain many locations where it can superimpose on itself without mismatch. Moreover, it is not possible to subdivide all patterns into non periodic subunits, as is the case with one dimensional strings. In ....

....in time O(n ) The paper is organized as follows. The pattern analysis is described in section 2. Section 3 consists of the text scan. We conclude with some open problems. 2 Pattern Preprocessing The idea of array overlap or periodicity and the pattern preprocessing algorithm are given in [2]. For completeness, we review the algorithm here. Our goal is to determine where two copies of an array A can overlap without conflict. Such sites are called sources (figure 1) For each location that is not a source, there exists a witness that proves that the overlapping copies of A mismatch. A ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, Jan 1992.

....we would like the number of duels to be proportional to the size of the compressed text. In this subsection we describe how this is possible. In the following subsection we describe how the actual duel is performed on the compressed text. We consider the four periodicity classes de ned in [3]. We show separately for each periodicity type that no more than O(jcompressed(T )j) duels will be necessary. Following we describe, for each periodicity class, the way the pattern starts may appear in an m=2 m=2 text block B. 1. Non periodic: There is at most one pattern occurrence in B. 2. ....

....are ordered monotonically if they are non decreasing in both row and column indices or non increasing in row index and non decreasing in column index. 11 4. Lattice Periodic: The pattern starts in B fall on the nodes of a lattice. The lattice is de ned by the basis vectors of the pattern (see [3, 9]) 1) Non periodic: If the pattern is non periodic, then one candidate is killed in every duel. Since at the start there are at most jcompressed(T )j candidates, the number of duels is no more than jcompressed(T )j. 2,3) Line and Radiant Periodic: As in [4] we perform the duels within each ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. SIAM J. Comp., 27(1):90-106, February 1998.

....of the algorithm is also greatly reduced as no cache misses will occur [11] De nition 1 (inplace) We say that a compressed matching is inplace if the extra space used is proportional to the input size of the pattern. Note that this de nition encompasses the compressed matching model (e.g. [2]) where the pattern is input in uncompressed form, as well as the fully compressed model (e.g. 7] where the pattern is input in compressed form. The inplace requirement allows the extra space to be the input size of the pattern, whatever that size may be. In this paper, we present an inplace ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, pages 440-452, Jan 1992.

....element in this multimedia effort is the equivalent of text string searching in an image database. Most of the results in multidimensional pattern matching (e.g. 9, 8, 11, 5, 6] have been a reduction of the multidimensional problem to some string matching problem. Recently, Amir and Benson [2] introduced an inherently two dimensional tool, that of two dimensional periodicity. The two dimensional periodicity idea has been instrumental in several interesting two dimensional results. Amir, Benson and Farach [3] introduced the first alphabet independent two dimensional text scanning ....

....table was first introduced for strings by Vishkin [15] The witness table indicates, for every possible translation of the pattern, whether the overlap of the pattern with the translation matches or not. If it does not match, the table gives a location that witnesses a mismatch in the overlap. In [2], a O(m 2 log oe) algorithm was presented for witness table construction, where m 2 is the size of the pattern, and oe = min(m; j Sigmaj) j Sigmaj is the alphabet size) In [14] an alphabet independent witness table construction was presented. The main contribution of this paper is a new ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, pages 440--452, Jan 1992.

....and Vishkin [11] They showed that all appearances in text T of a pattern P scaled to any discrete size, can be found in linear time for fixed finite alphabets. That solution gave rise to the compressed matching problem [2, 7] which, in turn led to the development of two dimensional periodicity [3]. Two dimensional periodicity turned out to be the most important tool in two dimensional matching. Its development led to an alphabet independent two dimensional matching algorithm [6, 4, 18] and to optimal parallel two dimensional matching algorithms [5, 13] Much progress has been made with ....

A. Amir and G. Benson. Two-dimensional periodicity and its application. SIAM J. Comp., 27(1):90--106, February 1998.

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A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, pages 440--452, Jan 1992.

....that the compression achieve a good ratio and that the compression algorithm be fast. We now need algorithms for pattern matching in time and space proportional to the compressed size, i.e. without the need to decompress. The compressed matching problem was formally defined by Amir and Benson [2, 1] as follows: Let oe = s 1 1 1 1 s u be a text string of length u over alphabet 6 = fa 1 ; a q g. Let oe:c = t 1 1 1 1 t n be a compression of oe of length n u. INPUT: Compressed text oe:c = t 1 1 1 1 t n , and pattern P = p 1 1 1 1 pm . OUTPUT: The first text location i such that ....

....: a q g. Let oe:c = t 1 1 1 1 t n be a compression of oe of length n u. INPUT: Compressed text oe:c = t 1 1 1 1 t n , and pattern P = p 1 1 1 1 pm . OUTPUT: The first text location i such that there is a pattern occurrence at s i , i.e. s i j01 = p j ; j = 1; m. Amir and Benson [2, 1] also defined a compressed matching to be efficient if its time complexity is o(u) almost optimal if its time complexity is O(n log m m) and optimal if it runs in time O(n m) The first compressed matching algorithms were side effects of papers by Eilam Tsoreff and Vishkin [6] and Amir, Landau ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, pages 440--452, Jan 1992.

....this data has given rise to a new paradigm in searching, that of compressed matching. Previously, the main thrust in the study of data compression has been to achieve compressions that are efficient in packing while also being practical in time and space usage. This is no longer sufficient. In [1, 2], a new goal for compression was introduced. The compression must have the additional property of allowing pattern matching in the compressed data without the need to decompress. In traditional pattern matching, all occurrences of pattern P in text T are sought. The pattern and text are explicitly ....

....model in which the alphabet is unordered and only equality of characters can be tested. Thus it matches the extremely weak model assumed by e.g. 12] 3 Two Dimensional Periodicity and Witness Tables 3.1 Two Dimensional Periodicity Our algorithm uses the periodicity class of the pattern. In [2], periodicity in two dimensional arrays is defined based on the ability of an array A to overlap itself without mismatch. For simplicity, let A be an m Theta m array (although A can be any rectangular array) Definition 2 Each corner of A is included in a subarray called a quadrant (see figure ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of the Third Ann. ACM-SIAM Symp. on Discrete Algorithms, Jan 1992.

....type, thus the model is weaker than in the above mentioned two dimensional matching algorithms. As opposed to previous algorithms, our algorithm is inherently two dimensional, and uses a novel technique in two dimensional matching two dimensional periodicity, as introduced by Amir and Benson in [2]. The two dimensional periodicity idea: A periodic pattern contains locations, other than the origin, where the pattern can be superimposed on itself without mismatch. Suppose our pattern is non periodic, i.e. there are no such locations, other than the origin. We could then narrow down the ....

....Verification of a candidate could then be done in the naive character by character comparison, but the time would still be linear because the candidates do not overlap. The problem with implementing this idea is that there is no guarantee that the pattern is non periodic. Indeed it has been shown [2] that there are four different types of two dimensional periodicity and that a pattern may contain many locations where it can superimpose on itself without mismatch. Moreover, it is not possible to subdivide all patterns into non periodic subunits, as is the case with one dimensional strings. In ....

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A. Amir and G. Benson, Two-dimensional periodicity and its application, Proc. of 3rd Symposium on Discrete Algorithms, Orlando, FL, (1992).

....though in a different way. The one dimensional string algorithms rely on the fact that periodic strings can be broken down into aperiodic substrings. The algorithm can then proceed by finding occurrences of the smaller aperiodic string. We take a different approach. Amir and Benson showed in [AB92] that periodicity in two dimensions has a much richer structure than does periodicity in strings. In particular, there are four classes of periodicity in rectangular arrays. Here, we achieve our efficient algorithm by dividing the four classes into two groups and showing how each group is ....

....number of potential locations in which the pattern may occur. Finally, in section 6 we describe how to verify which of the surviving candidates represent actual occurrences of the pattern in the text. 2 Preliminaries Central to our method are ideas about two dimensional periodicity developed in [AB92] There, periodicity in two dimensional arrays was defined. It was shown that there are four classes of periodicity for rectangular arrays. For the present work, we do not need four classes. Rather we separate arrays into two classes, those that are dense lattice periodic and those that are not. ....

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A. Amir and G. Benson. Two-dimensional periodicity and its application. Proc. of the Third Ann. ACM-SIAM Symp. on Discrete Algorithms, Jan 1992.

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A. Amir and G. Benson, Two-dimensional periodicity and its applications, Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 1992, 440--452.

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A. Amir and G. Benson. Two-dimensional periodicity and its application. In Proc. 3rd Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 440--452, 1992.

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