Patent Application
20140082021
This disclosure relates to ordering strings in a multi-dimensional space and comparing the strings.
Today, data is collected about many different types of activities more frequently than ever before. The advancement of such data collection capability has been accompanied by an evolution in data storage capability, as the enormous amount of data that is collected has to be stored and made accessible. While a lot of data is now available due to the advancement in data collection capability and data storage methods, it is not always easy to translate the enormous amount of available data into useful information.
Approximate string matching remains an active area of research due to the need to find computational techniques that scale beyond small dictionaries and provide fast approximate matching on very large datasets. Examples of current state of the art methods include the deletion-neighborhood approach, and table-driven finite state automata. However, there is much room for new and improved ways to process queries against a large corpus of data, often in the Terabyte range.
In one aspect, the inventive concept pertains to a computer-implemented method of comparing strings. The method entails mapping a first string and a second string in a multi-dimensional space where each axis represents a character in the first and/or second strings. The mapped positions of the first and second strings in the multi-dimensional space are used to generate first and second one-dimensional representations of the first and second strings, respectively. The degree of similarity between the first string and the second string is determined based on a difference between the first and second one-dimensional representations.
In another aspect, the inventive concept pertains to mapping a string in a multi-dimensional space that includes a first axis that represents a first character. The method includes dividing the first axis into a first region and a second region, assigning the string to the first region if a first character is absent from the string, and assigning the string to the second region if the string includes the first character. The second region may be further subdivided into a third region and a fourth region and the string assigned to one of the third and fourth regions based on position of the first character in the string. Each sub-region may continue to be further divided, such that the number of divisions equals the number of digits in the string.
An additional axis may be added to the space to represent each character in the string. A string will have a location in the space that is defined by the type, frequency, and position of the characters in the string. The location may be converted to a one-dimensional representation, such as an integer.
The one-dimensional number generated in the above manner may be used to process queries. For example, a query string may be matched with one or more strings in a corpus of data by comparing the one-dimensional representation of the query string with one-dimensional representations of strings in the corpus.
In another aspect, the inventive concept pertains to a non-transitory computer-readable medium storing instructions that, when executed, cause a computer to perform a method for placing a string on a one-dimensional space filling curve by mapping the string in a multi-dimensional space, logically assigning positions for each character in the string in the multi-dimensional space, and converting the positions into a one-dimensional representation.
A method and system for comparing strings (e.g., to process a query) is provided. More specifically, a method and system for matching a set of strings that most closely match a query string is presented. In one aspect, the inventive concept provides a fast, approximate string matching that is useful for discrete words and scales up to be used with a group of words (e.g., sentences).
In one aspect, the disclosure includes a method of ordering strings in a multi-dimensional space such that similar strings end up within a predefined Levenshtein distance of one another.
In another aspect, the inventive concept includes a method of performing approximate string matching in which an input string is used to retrieve output strings that have low Levenshtein distances to the input string. This method may be useful for spell checking, among other functions.
In another aspect, the inventive concept includes a method of organizing an inverted index so that the posting lists′ keys are compatible with the approximate string matching method.
A “string,” as used herein, is one or more characters. For example, a string may be a word (“doggy”) or a sentence (“where is the doggy?”). Characters include letters, numbers, and symbols.
Data is usually stored in the form of strings. When a query is received, usually in the form of strings, it is desirable to be able to find the best-matching data efficiently, without heavily taxing the memory or the computational capability of the computing system. The method disclosed herein avoids having to store a huge amount of data in memory, and also avoids having to execute a large amount of computation per query. In one aspect, the inventive concept converts the query to a vector and maps it on a space-filling curve to generate a one-dimensional index. As the one-dimensional index can be traversed with a single-range scan, finding a match to the query string is efficient both in terms of cost and speed. The method converts the words in a corpus to a list of simple ordered integers, which may be stored in well-known disc-based structures such as a B-tree without the need for a RAM-resident main structure.
A point in a k-dimensional space is sometimes represented as an integer in which successive bits of the integer divide the axes of the space. There are techniques for converting between a point on a space-filling curve such as a Hilbert curve and the integer-point representation in a k-dimensional space. The Hilbert curve represents a re-ordering of the integer representation of points in k-dimensional space such that points near each other by metrics such as Euclidean distance tend to be near each other in a one-dimensional natural ordering of the Hilbert numbers. This disclosure includes a method of creating an integer representation of a string in a k-dimensional space so that known methods of Hilbert encoding may be applied. In so doing, an ordering will be created such that strings near each other in the one-dimensional ordering will be similar to each other by a Levenshtein distance metric. This facilitates fast approximate string matching.
To determine how similar the strings are to one another, the differences between them are calculated, for example in the form of distances. Hence, of the three strings shown in the example, String 1 and String 2 represented by integers p1 and p2 are the most similar (because distance p2-p1 is the shortest), String 2 and String 3 are not too similar (as the distance p3-p2 is not short, qualitatively speaking), and String 1 and String 3 are very different (as indicated by the long distance p3-p1).
To map an integer representation of a string onto a space-filling curve, a hierarchical division of a logical “space” is performed such that strings are mapped in that space. In the method to be disclosed, an integer represents the hierarchical division of the string space as a series of logic tests. Various techniques—such as known techniques for converting an integer representation of a point in k-dimensional space to a point on the Hilbert curve—may be used to convert an integer representation of a point in a k-dimensional space to a point on the Hilbert curve.
Strings may be represented as points in k-dimensions in the following manner:
Given a string such as “CAT” or “DOG,” the objective is to represent this string as a single point in 26 dimensions (i.e., a value in the range [0, MAX] on each of the 26 axes). Since each axis encodes some information about a particular character in the alphabet, some simple strings can be examined, such as strings built using just one character of the alphabet, e.g., “A,” “AA,” and “AAA.” The integer value of the string “A” on the A-axis should be closer to the value of “AA” than it is to “AAA” if there is to be a relationship between Euclidean distance and Levenshtein distance. Simply using the number of occurrences of A—the cardinality of A—as the value might make sense for strings consisting only of a single character of the alphabet; however, it can break down when more than one member of the alphabet is introduced. For example, this simple number-of-occurrence method would make the values on the A-axis for “A,” “AA,” and “AAA” 1, 2, and 3, respectively. However, this simple method would represent strings “ABAB” and “AABB” by the value 2 on the A-axis and the value 2 on the B-axis, thus placing them both at {2,2,0,0,0, . . . , 0} in 26 dimensions and failing to differentiate them. Therefore, a simple system using primarily the cardinality of a particular character in a string would give two dissimilar strings a Euclidean distance of zero from each other (and a Levenshtein distance of 2 (one deletion, one insertion)), which is insufficient for accurate integer conversion.
A hierarchical, logic-based mapping of the string to Euclidean space that facilitates a mapping to the Hilbert curve is presented in this disclosure. In the method that is disclosed, the spatial organization of strings places similar strings near each other with regard to Euclidean distance. With the hierarchical approach, nested regions of increasing similarity can be created as the division of the space proceeds from coarse to fine.
By using the value MAX/2 as the dividing point, and by making the top-level division a binary division based on the presence and absence of a character, a hierarchical integer representation of strings is created. As such, the use of single bits to represent the divisions of each axis is logical. The most significant bit (MSB) of the encoding would divide the axis into two halves: if the MSB were zero, then the point would fall in the range [0, MAX/2], whereas if the MSB were 1, then the point would fall in the range [MAX/2, MAX]. Given this encoding, it is convenient to let the absence of a particular character in a string be encoded as a value of zero on the given axis. For example, for the string DOG, all 26 axes will have a zero value except the D-axis, the O-axis, and the G-axis.
It should be noted that although the examples are provided in terms of a single axis (the A-axis), this is done for clarity of explanation and the method can be expanded to multiple-axes situations where the same logic applies to each of the multiple axes.
Each string has a “character index,” which is the right justified ordinal position of a character in the string. Reformulating the top-level division discussed above (“does the string possess the letter A?”) in terms of the character index, the question now becomes “does the string possess the letter A at character index greater than or equal to zero?” Since any string having the letter A would have a character index for A greater than or equal to zero, the two questions are substantially equivalent. The division of the axis in terms of character indices is now reformulated.
Given an encoding of the value on an axis as a sequence of bits, it becomes apparent that the method disclosed herein uses as many bits as characters in the strings it is desired to encode, and can produce sequences of zero or more 1-bits followed by zero bits. As a standard integer is 32 bits, 32 characters are sufficient to capture common words in English. For words longer than 32 characters, the method works because questions such as “Does this string possess the letter A at character index 31 or greater?” are still applicable. Therefore, the method still works with strings longer than 32 characters.
Encoding of a value that falls on a single axis is now examined. For example, for the string “AAA,” its encoding as a 32-bit integer value would be 111000 . . . where the total number of zeros is 29 (there are three 1-bits, since A occupies the first and only three characters of the string and there are 29 zeros which are essentially fillers to make 32 bits). The first zero indicates that “the string does not have an A at character-index 3 or greater,” the second zero indicates that “the string does not have the letter A at character-index 4 or greater,” etc. Given a 32-bit representation, MAX would equal to 232−1. Therefore, the base-10 numeric value of “AAA” on the A-axis is 231+b 230+229.
The conversion between an integer bit string and a Hilbert number in the desired number of dimensions can be performed using a well-known method. When re-ordered using the Hilbert number, strings that are similar tend to be ordered near each other. However, a more rigorous definition of similarity uses a Levenshtein distance. Organization by Hilbert re-ordering produces a sufficient reordering that by choosing a window of J entries, all indexed strings within a desired Levenshtein distance can often be captured. In practice, J can be sufficiently small (say 1000) that J represents a small fraction (e.g., less than 0.5%) of the total dictionary of indexed strings. A post filtering step is then applied to the window of J strings in which the Levenshtein distance to the input string is calculated for each of J strings in the window, and the J strings are then re-ordered by ascending Levenshtein distance to the input string. The application is then free to pick a window size J such that execution speed and recall performance can be tuned to the needs of the application.
The hierarchical encoding described for the two dimensional (A, B) example above takes the most significant bit of each D dimensions as the most significant D bits of the integer q. Therefore, where each dimension has two possible values (the 0 and the 1) correlating with two respective regions on an axis, the most significant bit of each of D dimensions determines the broadest region (herein also referred to as the “first region”) in which the final result will reside along a particular dimension. The second bit of each of D dimensions determines which region of the aforementioned first region the final result resides in, the third bit determines which region of the second region the final result resides in, and so on. The regions get narrower with each layer, such that the last bit has only one of two index locations to choose from on its respective dimension.
Therefore, in order to describe a string as a position in a k-dimensional hierarchical grid, a string is encoded in terms of hierarchical regions. Consider each character axis as being hierarchically divided as in a k-d tree. One can then consider the MSB for axis A to answer the question “does this string contain the character A?” A value of zero in the MSB means “no, there is no character A in this string.” A value of 1 in the MSB means “yes, there is a character A in this string.” One can then consider the second most significant bit to answer the question “does this string contain a second occurrence of A?” Notice that the value of the second most significant bit cannot be 1 if the MSB was zero. One can then consider the third most significant bit to answer the question “does this string contain a third occurrence of A?” A string of 1 bit is accreted, such that the length of the string is equal to the number of occurrences of A in the string.
In many languages, the first half of a word is often semantically more significant than the second half of a word. Therefore, two strings that are different in the first half are more likely to be different words than two strings that differ only in the second half. To weigh the difference properly, the following code may be implemented which differentiates between the first and second half of the strings in formulating the sequences of 1-bits described earlier for each axis:
For every character in the first half of the string, a number of bits proportional to the position of the character is set along the axis that corresponds to the character. The bits that get set first are the most significant bits (MSBs) as opposed to the least significant bits (LSBs) of the integer. If the same character appears more than once in the word, then for each additional time the character occurs, one additional bit is turned on along the corresponding axis.
A similar operation is applied for every character in the second half of the string, with one main modification. Instead of running the bits at the leftmost MSB, it starts at the second MSB, or the bit that is one bit less significant than the MSB. In the hierarchical division of the space described earlier, this corresponds to the range (0, MAX/2), a space that would otherwise never be occupied.
For the indexing process 40, the words in the corpus are mapped to the space-filling curve as described above in the Mapping process. Different techniques, including but not limited to the Interleaving technique described above, may be used to convert the points in 26-dimensional space to a single integer in one-dimensional space (step 44). Step 44 converts the list of strings to a list of simple ordered integers and produces a one-dimensional index with an integer representing each word in the corpus (step 46). This one-dimensional index does not require a RAM-resident main structure for storage. For example, the index can be stored in well-known disc-based structure such as a B-tree.
Upon receiving a query string (step 52), the query string is mapped to the space-filling curve (Mapping process 10) and converted to a single integer q (step 56). To find a match for the query string, the index is scanned to find the J closest integers to the integer q (step 58). The one-dimensional index makes the matching process efficient as only a single-range scan is needed to find the match. With the one-dimensional index, the matching function, e.g. “find words like ‘doggy’,” is reduced to “determine the J closest integers to the integer q,” wherein the integer q represents the point on the Hilbert curve that corresponds to the query word. Only a single-range scan may be used to process the query (i.e. “find the largest J/2 integers less than q, and find the smallest J/2 integers greater than q”. In some implementations this requires 2 range queries.).
The integer conversion in steps 44, 56 may be done using the Hilbert space-filling curve, but this is not a limitation of the inventive concept. In one implementation, the Interleaving method described above may be used.
In one implementation of the method (a sum-based mapping method), the weights assigned to a letter are summed when the same letter appears multiple times in a string. Applying the sum-based mapping method to the string “doggy” would make the value along the g-axis g=3+2=5 where m=1. Hence, “doggy” is represented as a vector with non-zero components {5, 5, 4, 1} on four axes {d, g, o, y}, respectively. In this manner, the word “doggy” is represented in vector space as a point (step 16). After a plurality of strings is represented as points in the 26-dimensional space, distances between the points are computed (step 20). The assumption is that the strings that are closer together in the 26-dimensional space are more likely to be a match than the strings that are far apart. It should be appreciated that a distance metric other than Euclidean distance (such as Manhattan distance) could be applied as well.
The simplified character-index-based mapping method of
The sum-based mapping method described above may be used to map a string to a vector space. Using the sum-based mapping method, the letter “g” in the word “doggy” was assigned the value of 5 on the g-axis. An alternative method uses an average value instead of the sum when the same letter appears multiple times in a word.
In double-letter words, the letter that appears twice in a row tends to be the dominant axis value. As a result, misspelling a word by omitting one of the double letters may have the effect of causing the misspelled variation to be too far from the intended word in the mapping space. Using the average instead of the sum for the double letters mitigates the dominating effect of the double letters, thereby bringing the misspelled version closer to the correctly-spelled version in the vector space. This way, a user who inputs “dogy” instead of “doggy” will be presented with approximately the same match result as if the word had been spelled correctly.
It should be understood that the average-based mapping method and the sum-based mapping method may be combined selectively to achieve optimal result. For example, the average-based mapping method may be used only for the specific case of double-letter words where the same letter appears in adjacent positions. Hence, in one implementation, the average-based mapping method may be applied to the word “tennis” (where the two “n”s are immediately next to each other) but not to the word “nine.”
The vector space identified in step 18 may be mapped onto various lower dimensional spaces, for example, a one dimensional Hilbert space-filling curve. In the case of a 26-letter alphabet, 26 dimensions are mapped to a Hilbert curve which encodes each of 26 dimensions to a resolution of 32 divisions per dimension (i.e., 5-bit resolution per dimension). However, use of a Hilbert curve is not a limitation of the inventive concept and other alternative space-filling methods may be used.
The assignment of weights as a function of the length and position of the letter in a string is based on two assumptions. The first assumption is that important words generally contain more characters than unimportant words. This assumption is based on a distribution of word length vs. TF-IDF in a large corpus. The second assumption is that endings of English words (such as “ed,” “ing,” “ment,” ‘es”) are generally less important than beginnings By applying the highest weight to the letters in the beginning of the word and assigning weights in a descending fashion to subsequent characters, the methodology described above captures the notion that beginnings of words are more important than endings. The methodology may be modified and adapted for other languages based on the trends for the specific language.
For words in a corpus to be placed in a vector space, the vector is created as shown below:
Once a query is received, Euclidean distance is used as a measure of word similarity to find words that may be a “match.” Upon receiving the word “doggy,” the distance between words “doggy” and “dog” is computed by comparing the distances along each axis individually, as follows:
Distance(doggy, dog)=sqrt[(5−3)2+(4−2)2+(5−1)2+(1−0)2]=5
In a similar manner, the distance between words “doggy” and “cat” is calculated as follows:
As the word “dog” is closer to the query word than the word “cat,” it would be a closer match to the word “doggy.”
Interleaving is a process for combining the individual binary-number representation of each character in a string into an integer that represents the entire string. There are a number of Interleaving techniques that are known. For simplicity, examples will be given in the context of an alphabet that consists of two characters or letters, A and B. The words would then be mapped in a space with two axes, one for A and one for B. Let's examine the word “ABAB.” Where n=4 and m=1, the character index for each character would be as follows: the leftmost A=4, the left B=3, the right A=2, and the rightmost B=1. Using the Sum-based Mapping described below, the word “ABAB” would be converted to a vector as follows: A=4+2=6, B=3+1=4, and mapped on the two axes. The numbers 6 and 4, when represented in binary form, are 0110 and 0100, respectively. When the two binary numbers are written vertically, top to bottom, next to each other, a chart such as what is shown below is created with the MSB highest, and LSB lowest:
Rewriting the number going from left to right and row by row, one gets the binary number 00111000 which interleaves the two binary numbers. Converted to a decimal integer, the number is 112. Hence, this would be number that is used for the indexing and the matching.
As for the word “BBBB,” it would be converted to a vector as A=0 and B=4+3+2+1=10, and mapped on the two axes accordingly. The binary numbers would be 0000 and 1010.
Rewriting the number going from left to right and row by row, the pattern of numbers above results in the binary number 01000100, which is equivalent to the number 68. In the same manner, strings AAAA and BBAB can be converted to integers 136 and 132, generating a one-dimensional index as below:
Based on the one-dimensional index, the distance between two words can be calculated. For example, the distance between ABAB and AAAA is 136−112=24. The distance between ABAB and BBBB is 112−68=44, and the distance between ABAB and BBAB is 132−112=20. The distances indicate how close of a match a word is to the query word. The shorter the distance between the two words, the closer of a match they are. The match result may be made as narrow or as broad as desired; for example, the match result may say “find J integers that are close to the integer q” wherein J=10. If J is set to 1000, many more match results will turn up, increasing the likelihood that the result that the user is looking for will be ensnared. However, as J increases, matches that are not as close to the integer q may also be included in the result. Post filtering of the results by ranking the K integers by their Levenshtein distance to the input string may be used to rank the J integers.
Each horizontal “slice” across the chart corresponds to slicing off the most significant bits of a binary representation of each axis value. The larger the number, the “higher” is the slice where the first “1” is encountered. When the ones and zeros are written vertically top to bottom, the highest “1”s are placed in the most significant bit of a binary number. In two dimensions (A, B), each pair of bits (i.e., each horizontal slice) corresponds to one of four quadrants of a quad tree (00, 01, 10, 11). When a long string of is and Os are written out, what is generated is a “quad tree” ordering of the bits which, although arguably not as good as a Hilbert curve at preserving locality, still has the property that for many points that fall inside the same quads, their decimal numbers will be close by each other. While the quad tree (or k-d tree when you have k dimensions instead of two dimensions) may not be as good at the Hilbert curve at preserving this property of spatial locality corresponding to integer-numbering locality, it may be an adequate alternative. However, converting the hierarchical interleaved integer to a Hilbert integer results in one-dimensional organization in which points near each other in k dimensions tend to be nearer each other in 1 dimensional space than they would be if we simply considered the interleaved integer as a one-dimensional space.
Suppose two parts of the word “doggy” are being weighted according to the Sum-based Mapping method described above. The binary axis value would be as follows:
An interleaved integer 1000011100100000 is then formed. The conversion between the interleaved integer representation and the Hilbert integer can then be performed using one of several methods.
Although examples are provided herein to weigh two different parts of a string, it would be appreciated that the technique can be adjusted to weigh different parts of a string that is divided up into more than two regions.
When the query is a multi-string query, such as a sentence, the individual word vectors are added to generate a vector for the string. For example, if the query is “how much is that doggy in the window,” the vector for the sentence would be computed as follows:
V(“how much is that doggy in the window”)=v(how)+v(much)+v(is)+v(that)+v(doggy)+v(in)+v(the)+v(window)
Some examples of sentences that represent a contiguous range of Hilbert number using the above method are provided below:
“*i was crazy about him the first party i saw him”
“on january 15, 1990, a historic new law was passed”
“*on january 320, 1990, a historic new law was passed”
“in january 1990, a historic new law was passed”
“i am ready and eager to go to the party”
“*almost three years, i saw ruth again”
“the nearest drug store is about ¾ of a mile away”
“they are the number 3 auto maker and a fortune 500 company”
“*who alice did really annoyed me”
“what alice did really annoyed me”
“her career lasted almost thirty years”
“*her career lasted almost thirty books”
“*we had an argument at whether it was a good movie”
“*almost three years for our first date, i saw ruth again”
“the data on file will be used for the project at hand, which is already under way”
“*he told them about the accident presumably”
“i saw her again a year and a half later”
“the former astronaut was alone and afraid”
“almost three years later, i saw ruth again”
“almost three years after our first date, i saw ruth again”
“almost three years after i first met her, i saw ruth again”
“the graduating of fred changes the situation”
“the 7-11 is half a mile up the road, but the supermarket is a long way away”
“*we argued adding new features to the program”
“included in our paper is a summary of the features of our program”
“*such flowers are found mainly particularly in europe”
“*we like to eat at restaurants, fortunately on weekends”
“your house and garden are very attractive”
“we like to eat at restaurants, usually on weekends”
“he is apparently an expert on dogs”
“*he knows apparently an expert on dogs”
“the apparently angry man walked out of the room”
The vectors for sentences would be processed in the same manner as the vectors for words.
Sometimes, there may be undesirable crosstalk between words. A “crosstalk” happens when two words that happen to have similar arrangement of letters get grouped together during the matching process 50 even though the two words actually are not close. For example, suppose there is a first vector including the words “xxx” and “xingyu” (the latter being a non-English word) and a second vector including the words “zzz” and “you.” In a 26-dimensional space, there may be crosstalk between the vectors because the word “xingyu” and the word “you,” both of which have the arrangement “y . . . u,” cause the vectors to have similar values on the y-axis and the u-axis.
This type of crosstalk may be avoided by dividing the vector space into multiple sets of 26-axis space. For example, two sets of 26 axes may be used, effectively creating 52 axes or dimensions a1-z1 and a2-z2. With this setup, each character has a primary dimension and a secondary dimension.
In an example embodiment where the string “aaabc aaa” is being indexed, the string is first subdivided, or tokenized (e.g., into “aaabc” and “aaa”). The tokens (the subdivided strings) are then lexicographically sorted (e.g., “aaa” is before ‘aaabc”) and a zero-based index is assigned (e.g., aaa=0, aaabc=1). The vectors for the tokens are added to generate an overall vector for the entire string, as described above. However, for each character in the token, either the primary axis or the secondary axis is selected by taking the token's zero-based index. In the case of “aaa,” since it has a token index of 0, its vector would use the primary axis. On the other hand, in the case of “aaabc” with token index 1, the secondary axis would be used.
One generalization of this multi-axes technique entails selecting a set of axes by taking the token index and modulo-dividing it by the number of axis sets. For example, the string “asaa aaabc aaabcd aaazz” may be indexed with corresponding token indices {0, 1, 2, 3}, respectively and, by modulo division, the respective sets of axes would be {0, 1, 01}.
Optionally, one could build two spellcheck indexes, one for the regular string (e.g., “doggy”) and one for the reverse string (e.g., “yggod”). This dual-index approach improves accuracy when the head of the string contains a severe typographical error but the tail of the string remains intact. Generally, a spellchecking function may be implemented by scanning a set number (x) of Hilbert integers before and after the computed Hilbert integer (e.g., x=1000).
Any appropriate number axis sets may be used.
While use of multiple sets of axes may be helpful for spell checking, it may not be as helpful for accurate matching. Some empirical studies show that the greater the number of axis-sets that are used, the more the long strings (sentences) tend to align around having the same number of words, diluting the closeness of the match. By using a single set of axes, a more accurate match may be obtained.
On the other hand, when matching index keys (i.e., in a keyword search with a small number of discrete tokens), the proximity of words to their misspelled variations improves with the number of sets of axes.
Hence, the method may entail selectively combining a single-set axis mapping and multi-set axis mapping to yield the optimal result depending on the task that the user is accomplishing. For example, if the user is if the user's goal is to take 1000 sentences and process them against a corpus of documents, a single set of axes may be used. On the other hand, if the goal is to support keyword search with powerful spellchecking, a larger number of axes sets may be desirable.
Various embodiments of the present disclosure may be implemented in or involve one or more computer systems. The computer system is not intended to suggest any limitation as to scope of use or functionality of described embodiments. The computer system includes at least one processing unit and memory. The processing unit executes computer-executable instructions and may be a real or a virtual processor. The computer system may include a multi-processing system which includes multiple processing units for executing computer-executable instructions to increase processing power. The memory may be volatile memory (e.g., registers, cache, random access memory (RAM)), non-volatile memory (e.g., read only memory (ROM), electrically erasable programmable read only memory (EEPROM), flash memory, etc.), or combination thereof. In an embodiment of the present disclosure, the memory may store software for implementing various embodiments of the present disclosure.
Further, the computer system may include components such as storage, one or more input computing devices, one or more output computing devices, and one or more communication connections. The storage may be removable or non-removable, and includes magnetic disks, magnetic tapes or cassettes, compact disc-read only memories (CD-ROMs), compact disc rewritables (CD-RWs), digital video discs (DVDs), or any other medium which may be used to store information and which may be accessed within the computer system. In various embodiments of the present disclosure, the storage may store instructions for the software implementing various embodiments of the present disclosure. The input computing device(s) may be a touch input computing device such as a keyboard, mouse, pen, trackball, touch screen, or game controller, a voice input computing device, a scanning computing device, a digital camera, or another computing device that provides input to the computer system. The output computing device(s) may be a display, printer, speaker, or another computing device that provides output from the computer system. The communication connection(s) enable communication over a communication medium to another computer system. The communication medium conveys information such as computer-executable instructions, audio or video information, or other data in a modulated data signal. A modulated data signal is a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired or wireless techniques implemented with an electrical, optical, RF, infrared, acoustic, or other carrier. In addition, an interconnection mechanism such as a bus, controller, or network may interconnect the various components of the computer system. In various embodiments of the present disclosure, operating system software may provide an operating environment for software's executing in the computer system, and may coordinate activities of the components of the computer system.
Various embodiments of the inventive concept disclosed herein may be described in the general context of computer-readable media. Computer-readable media are any available media that may be accessed within a computer system. By way of example, and not limitation, within the computer system, computer-readable media include memory, storage, communication media, and combinations thereof.
Having described and illustrated the principles of the inventive concept with reference to described embodiments, it will be recognized that the described embodiments may be modified in arrangement and detail without departing from such principles. It should be understood that the programs, processes, or methods described herein are not related or limited to any particular type of computing environment, unless indicated otherwise. Various types of general purpose or specialized computing environments may be used with or perform operations in accordance with the teachings described herein. Elements of the described embodiments shown in software may be implemented in hardware and vice versa.
While the exemplary embodiments of the inventive concept are described and illustrated herein, it will be appreciated that they are merely illustrative.
This application claims the benefit of U.S. Provisional Application No. 61/702,578 filed on Sep. 18, 2012, the content of which is incorporated by reference herein.
| Number | Date | Country | |
|---|---|---|---|
| 61702578 | Sep 2012 | US |
