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Journal of Computer Science

A fast approximate string searching algorithm

A fast approximate string searching algorithm

Mahmoud Mhashi

Abstract: In both approximate and exact string searching algorithms, the shift distance at the skipping step plays a major role in the performance of string matching algorithms. A new algorithm called the Modified Character-Weight Algorithm (MWA) has been developed to test the effect of the shift distance on the performance of approximate string matching. An experiment was performed comparing the performance of the MWA with that of Mhashi’s Character-Weight Algorithm (WA) using English text of size 1,005,077 characters. Using the average number of comparisons and the clock time as evaluation criteria, the MWA algorithm used only about 4% to 15% as many comparisons as the WA algorithm and about 10% to 35% of the clock time.

Key words: Approximate String Matching, Searching, Shift Distance, Condition Types, Character Access

INTRODUCTION

String searching algorithms that are used to retrieve information based on a search criterion form one of the most important topics in computer science. One example of a widely used application for approximate string matching algorithms nowadays is the Internet. People using the Internet usually search for information by typing keywords (string patterns) into a search engine and then waiting for information to be returned. The matching process between the keyword input and the text to be searched can be accomplished in two ways: (I) Exact match [1-4], meaning that the passages returned will contain an exact match of the keyword input. (II) Approximate match [5-7], meaning that the passages returned will contain some part of the keyword input. More information may be returned and users must then filter information accordingly to find what they need. The contribution of this study falls under the category of approximate string matching algorithms.

In the present study, we are interested in designing an algorithm that finds all the occurrences of Pat = [P.sub.1] … [P.sub.m] in Text = [T.sub.1] … [T.sub.n] where m < n with at most k differences of type (a), (b) and (c) below. Different authors [8-10] have studied this problem. Three types of differences are distinguished [11]:

* The [i.sup.th] character in Text does not exist in Pat.

* The [k.sup.th] character in Pat does not exist in Text.

* A character of the pattern corresponds to a different character of the text (i.e., a mismatch). An intensive search for information is taking place on the Internet all around the word and one of the most crucial issues associated with approximate string searching algorithms is performance. How efficiently information can be retrieved becomes an important issue. Mhashi [12] introduced an improved algorithm, known as the Character-Weight Algorithm (WA). The WA algorithm introduced improvements over a previous counterpart algorithm in terms of both relevance and performance. This study focuses on the performance issue. We have enhanced the performance of an existing algorithm, in terms of the clock time required for the search, the number of character comparisons and the number of positions shifted while moving on with the comparison process. The new developed algorithm will work for any character set. The organization of the study is as follows: Section 2 describes the earlier algorithm WA and shows how it works through an example. Section 3 describes the new MWA algorithm and illustrates how it works using the same example. Section 4 contains a discussion of the two algorithms and illustrates the performance improvement of MWA over WA. Tables showing the performance improvement achieved in this research support the evaluation process. Finally, Section 5 contains the conclusions of our research work and outlines future work in this area.

DESCRIPTION OF THE CHARACTER-WEIGHT ALGORITHM (WA)

The WA algorithm concentrates on both the relevance and the performance of retrieving all the occurrences of the pattern in the text. In this section, we address the performance issues (for more details of the full algorithm see [12]. The WA algorithm has three steps:

1. The preprocessing step: In this step, the pattern is preprocessed to determine the important and unimportant parts. For our experiment, the first two-thirds of the characters are considered to be the important part while the rest are considered the unimportant part. The result of this step is a table called the weight table. Theweight table expresses the importance of the different characters in the pattern Pat. In our case, the weight for each character in the important part is k_mismatches + 1 (i.e., if there is a mismatch at any position in the important part, then the algorithm decides that there is no occurrence at that location). The weight for each character in the rest of the pattern (unimportant) is one.

2. The checking step: The pattern is checked in the following order:

* Last and first characters in the important part,

* The rest of the characters in the important part from right to left and

* The rest of the characters in Pat (unimportant) from left to right.

3. The skipping step: At this step, the straightforward algorithm is applied and the pattern is shifted forward one position in relation to the text when a match/mismatch is found.

A pseudo code description of the WA algorithm follows:

int new_k_mismatches (Text, n, Pat, m, weight, kmis)

char *Text; // Text: Text[0] … Text[n-1]

int n; // Text length

char *Pat; //Pat: Pat[0] … Pat[m-1]

int m; // Pat length

int *weight; //weight: weight [0] … weight[m-1]

int kmis; // kmis: The maximum number of k_mismatches allowed

int mis; // the total current number of mismatches

int i, j, k;

//Preprocess

(1) pref inf = m/3 * 2 -1 ; suf = m-pref inf – 1;

// prepare the weight table

for(i = 0; i< = pref inf; i++) weight[i] = kmis + 1;

for(i = pref inf+1; i

i = m-1;

// Searching Process.

While ( i < = n-1 + kmis) { 11 Outer-loop, as long there is text > =

m-kmis, continue searching

mis = 0; j = 0;

// test the last and first characters in the important part

first

(4) if ( Text[ i + suf] != pat[pref inf] II text [ i – m + 1 ] != pat

[j])

(5) mis += weight[j];

(6) else //test remaining important part characters from right to left

(for j=pref_inf-1, k=i-suf-1; j>0; k–, j–)

(8) if ( text[k] != pat[j] ) {

(9) mis += weight[]]; break; //a mismatch found, no need to

continue

}

(10) if ( !mis) //test unimportantpart characters from left to right.

(11) for (j = pref inf +1;k = i-suf +1;j

(12) if ( text[k] != pat[j] ) {

(13) mis + = weight[j];

(14) if (mis > kmis) break; // There is no occurrence

}

(15) if (j == m) report that there is an occurrence at i-m -l to i

(16) i++; //move to the next location

}

The following example demonstrates how the WA works:

Example: Let the number of mismatches kmis = 1. Let Text and Pat be as follows:

Text: PPEEPPPIIDPPIDPPIISE Pat: PPEESS

Assume that “PPEE” is the important-part and “SS” is the unimportant-part. The comparison starts at character ‘E’ at [Text.sub.3] (the last character in the important part), then Texto (first character) followed by [Text.sub.2] then [Text.sub.1] respectively (remaining characters compared right-to-left). Since there is no mismatch for the important part the algorithm moves to the unimportant part, [Text.sub.4] and [Text.sub.5] (the unimportant part is compared from left to right). The algorithm detects two mismatches (making mis = 2), which is greater than the specified criterion (kmis = 1). Six character comparisons are needed. Thus the algorithm moves forward one position.

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

The algorithm detects one mismatch at character ‘P’ at Text4, in one character comparison. Thus it will move ahead one position.

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

Another mismatch is detected at [Text.sub.5], in a one-character comparison. Moving ahead two positions, we get:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

Another mismatch is detected at [Text.sub.6], in four character comparisons. After moving one location, the result as follows:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

As in the first step above, two mismatches detected at [Text.sub.9] and [Text.sub.10] and six character comparisons are needed. Thus moving ahead four positions (one movement at a time), we get:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

One mismatch is detected at [Text.sub.9] and five character comparisons are needed. There is a mismatch at the next three positions, thus we can move ahead four positions. We get:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

Up to this point, five character comparisons are needed to detect one mismatch at [Text.sub.13], thus we can move ahead one position.

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

At this point, each character in Text at the important part is matched with the corresponding character in Pat. There is only one mismatch at character ‘E’ at [Text.sub.l9] (i.e., mis [less than or equal to] kmis), thus we have found one occurrence of Pat in Text. Six character comparisons are needed. Moving forward one position, we get:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

One character comparison is needed to detect one mismatch at [Text.sub.18]. Moving forward one position ends the searching process, because the number of characters left in Text is less than m – kmis, where m is the number of characters in the pattern. The total number of character comparisons needed is 35.

THE MODIFIED CHARACTER-WEIGHT ALGORITHM (MWA)

Like any approximate string matching algorithm, MWA finds all the relevant occurrences of a pattern [Pat.sub.0], …, [Pat.sub.m-1] in the text [Text.sub.0] … [Text.sub.n-1] with a maximum of k mismatches. It preprocesses the pattern to divide the pattern into two parts (important and unimportant) and to produce three different arrays skip, position and sign. The length of the first part is called pref_inf. It is equal to m / 3 * 2 – 1. In other words, it equals to two-thirds of m, where m is equal to the Pat length. The length of the second part is called suf. It is equal to m-pref_inf – 1. In this study, we assumed that the first part (with length pref_inf) is the important part. Generally speaking, the important part might be any segment of the pattern.

The skip array records how much the pattern is to be shifted forward in relation to the text when a match/mismatch is found and the reference character in the text exists in the pattern. In this algorithm, there is only one dynamic reference character.

Let 10 = [Text.sub.j], where Text is the character that corresponds to the last character in the pattern. Let base = j – suf be the position of the base character in the text that corresponds to the last character in the important part in Pat. From the base character, the position of the dynamic reference character (ref) of MWA can be calculated. If the base character occurs in Pat, then ref is calculated according to the following formula:

ref = (base + 1) + (pref_inf + 1) – pT, where base is defined above, pref_inf is the length of the important part in Pat and pT = pos[Text[base]], the position of the base character in the Text.

Example: In order to motivate these formulas, let us consider this example.

Text: ABCDEFGH Pat: CDEFGH

In this example, j = 5 at character ‘F’ in Text. The value of pref_inf is 3 and the value of suf is 2. The base reference equals 3 at character ‘D’ in Text and it occurs at position pT = 2 in Pat. The position of the reference character ref = 3 + 3 + 1 – 2 = 5. So, the character ‘F’ in Text is the reference character. The location of ref ranges from base to base + pref_inf + 1.

The shift distance d depends on the occurrence of ref in Pat. If ref does not occur in Pat, then d = 2 * (pref_inf +1) – pT + 1 (i.e., d ranges from pref_inf +1 to 2 * pref_inf_+1). If ref occurs in Pat, then the shift distance d =skip[Text[ref]] – pT +1. In other words, d ranges from 1 to 2 * pref_inf – 1. The sign array tells whether the reference character in Text exists in the given pattern or not. Each location in this array holds one value, either zero (i.e., ref does not occur in Pat and there is a mismatch) or one (i.e., ref character occurs in Pat, but it is not necessary that there is a match). So, the algorithm works as follows:

* Checking step: The existence of base in Pat is checked first. If it does not exist, move to the skipping step. Otherwise, there is a possibility of an occurrence. In such a case, the last and first characters of the important part will be checked first followed by the rest of the important part from right to left. If there is any mismatch at the important part, then move to the skipping step. Otherwise, the unimportant part will be checked from right to the left. The number of mismatches will be counted. If the number of mismatches is less than or equal to k_mismatches, then an occurrence will be reported. Otherwise, there is no occurrence and we move to the skipping step.

* After the checking step, the reference character is determined according to the character that is next to the base character. Next, the shift distance is determined according to the occurrence of ref in Pat. Alignment is carried out and followed by the checking step. The pseudo code for the MWA algorithm that reflects the above ideas follows:

void MWA (char *Text, int n, char *Pat, int m)

/* where:

n is the Text length

Text: Text[0] … Text[n-1] m is the Pat length

Pat: Pat[0] … Pat[m-1] */

int sign[ALPHABET SIZE]; /*sign: sign[0] … sign[ALPHABET SIZE – 1] */

int skip [ALPHABET SIZE]; /* skip: skip[0] … skip [ALPHABET

SIZE – 1] */

int pos[ALPHABET SIZE]; pos: pos[0] … pos[ALPHABET SIZE – 1] */

{

pref inf=m/3 *2-1;

suf=m-pref inf-1;

// initialize the arrays pos, skip and sign

for(j=0; j

pos[j]=0; skip[j] = 2 * (pref inf + 1); sign[j]=0;

}

//Update the arrays according to Pat

for(j=0; j<=pref inf; j++) {

pos[pat[j]]=j+l; skip[pat[j]] = 2 * (pref inf + 1) – j – 1;

sign[pat[j]] = 1;

}

// Start Searching Here

i0 = m – 1; base = i0 – suf;

while(i0 <= n - 1 + kmis) {

mis = 0;

if (!sign[text[base]]) { // skip if base character doesn’t occur in Pat

pT= pos[text[base]];

ref = base + (pref inf + 1) – pT;

if (!sign[text[ref]])

i0 += 2 * (pref inf + 1) – pT;

else

i0 += skip[text[ref]] – pT;

}

else { /* Test the possibility of occurrence */

// test the last and the first characters

if( text[base] !=pat[pref_inf] II text[i0 – m_minus-1]

!=pat[0] ) goto

MOVE;

else {

// test the rest of important part

for (j = pref inf -1 , k = base -1; j > 0; k–, j–)

if( text[k] != pat[j] ) goto MOVE;

// Test the unimportant part

for (j=pref_inf +1,k=base +1;j

if ( text[k] !=pat[j] ) rnis += wight[j];

if ( mis > kmis ) goto MOVE;

}

/*Report a match at i0 – m … i0-1 */

}

MOVE:

pT=pos[text[base+ 1]];

ref = (base + 1) +(pref_inf + 1)-pT;

if (!sign[text[ref]])

i0 += 2 * (pref_inf + 1) – pT + 1;

else

i0 += skip[text[ref]] – pT + 1;

} //End while

} // Function end

Worst Case and Average Case Analysis: Let W(n) be the number of comparisons required in the worst case to find all the occurrences of Pat in Text. Such worst case can be found when:

* The character at (base + 1) occurs at the last position in the pref infx part (i.e., important part) and

* The K_mismatches are found at the last K positions in the unimportant part (suffix part).

This causes one skip position and m comparisons in each iteration. So, W(n) = m*(n-m+1 + K_mismatches) comparisons. Thus, the number of shifts with the worst case is equal (n – m + K -Mismatches +1). Regarding the average case, let A(n) be the number of shifts performed in the average case. The shift distance in each iteration will be equal to either 1, 2, 3, … 2(2m/3)+1 = 4m/3 + 1. Assume the shift distance is equally likely to be in any particular iteration that equals to 11((4m13)+1). Then,

[A.sub.i],(n)=1/((4m13)+1)*[[4m/3+1].summation over (i=1)] (2m13+2}12 =(4m+6)13

Example: The same text and pattern used in the previous example are used here.

Text: PPEEPPPIIDPPIDPPIISE Pat: PPEESS

As in the previous example, “PPEE” in Pat is the important part and “SS” is the unimportant part. According to the given pattern, the values of the sign array are: sign[‘P’] = sign[‘E’] = sign[‘S’] = 1.

Otherwise, the initial value of the sign array is zero. At the beginning and in iteration number 1, MWA starts the comparison at the important part at the last character and then the first, followed by the rest from right to left (i.e., [Text.sub.3], [Text.sub.3], [Text.sub.0] and, [Text.sub.2] and, [Text.sub.1] respectively). Since there is no mismatch for the important part, the algorithm moves to the unimportant part, Text4 and Texts (unimportant part compared from left to right). The algorithm detects two mismatches (making mis = 2) and that is greater than the specified criterion (kmis = 1). This is the end of the checking step. Six character comparisons and one character access are needed. In order to complete the skipping step, ref must be calculated. pT = pos[‘P’] = 2, where ‘P’ at [Text.sub.4]. The value of ref = (base + 1) + (pref inf + 1) – pT = 4 + 4 – 2 = 6. Thus the algorithm moves forward five positions to align [Text.sub.6] with [Pat.sub.1]. The result is as follows:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

The checking step starts again. The existence of the base character at [Text.sub.8] in Pat is tested. According to the sign array, it occurs in Pat. There is a match between each character in Text with the corresponding characters in the important part of Pat. Moving to the unimportant part, the algorithm detects two mismatches, which is greater than kmis = 1. Six character comparisons and one character access are needed. The ref must be calculated. The character ‘D’ at [Text.sub.9] will be used as reference for calculating ref. PT = pos[‘D’] = 0. The value of ref = (base + 1) + (pref inf + 1) – pT = 9 + 4 – 0 = 13. Since Text13 doesn’t occur in Pat, the algorithm moves forward nine positions. The result is as follows:

Text: PPEEPPPEEDPPEDPPEESE Pat: PPEESS

The existence of the base character at [Text.sub.17] in Pat is tested. According to the sign array, it occurs in Pat. There is a match between each character in Text with the corresponding characters in the important part of Pat. Moving to the unimportant part, the algorithm detects one mismatch, but that number is less than or equal to kmis = 1, thus, there is an occurrence of Pat in Text. Six character comparisons and one character access are needed. Moving to the next position ends the search process since the number of characters left in the text is less than the Pat length. So, the number of shifts is three and the total number of character comparisons is 18 in addition to 3 character accesses and 4 number comparisons.

RESULTS AND DISCUSSION

In this experiment, the two algorithms WA and MWA were implemented and compared on English text with size exactly 1,005,077 characters. A program was designed and implemented in C++ to select randomly 3000 patterns divided into 10 groups. Each group consists of 300 patterns. All the information about the different groups can be seen in Table 1 (A). The cost of the search process is measured by finding the average number of shifts, the clock time and the average number of conditions with the different types including character access, character-comparisons and number comparisons to find all the occurrences of the patterns in each group in Text. The clock time measure includes the preprocessing of the patterns in the two algorithms. The results of the experiment can be seen in Table 1. In Table 1(A), the number of groups, the pattern lengths, the number of k-mismatches and the average number of occurrences are presented. The pattern length ranges from 11 to 92 characters. The number of k_mismatches ranges from 5 to 23. The average number of occurrences ranges from 1.63 (group 8) to 105.84 (group 1). The clock time required to find each group of patterns is recorded in Table 1 (B). By using WA, the clock time required ranges from 2.487 seconds (group 4) to 2.625 seconds (group 2). By using MWA, it ranges from 0.235 seconds (group 9) to 0.906 seconds (group 1). By using MWA, the clock time required by the algorithm WA is reduced by 65.29% (group 1) to 90.83% (group 9). The average number of comparisons required to find the occurrences of patterns in Text is recorded in Table 1 (C). From the table, it is clear that MWA uses three types of conditions, including character accesses, number comparisons and character comparisons. On the other hand, WA uses only two types of conditions, including number comparisons and character comparisons. The character accesses are of the following form:

if (!sign[ Text[ref]]);11 see MWA algorithm.

The second type of condition is number-comparisons of the following

form:

if(i0 <= n - 1 + kmis); 11 see both algorithms.

The third type of condition is character comparisons of the following

form:

if ( Text[k] == Pat[j]); 1I see both algorithms.

In order to test if there is any difference among these three types of conditions, in terms of clock time execution, each type was repeated [10.sup.7] times. It was found that the clock time needed to execute a condition of type character comparison is reduced to 2% of that used by number comparison and 41 % of that used to do a character access.

Looking at Table 1 (C), one can see that WA uses only number comparisons and character comparisons. The total number of comparisons for both conditions ranges from 3,025,097 (group 10) to 3,029,718 (group 1). While using MWA, it ranges from 118,433 (group 10) to 434,553 (group 1). The new algorithm MWA reduces the total average number of comparisons required by WA by 85.66% (group 1) to 96.09% (group 10).

The performance improvement of MWA comes from different factors, including:

* Shift distance: using MWA, the shift distance ranges from one position to 2 * important length. On the other hand, the shift distance in WA is one position only. Of course, increasing the shift distance reduces the number of shifts and in turn decreases the number of comparisons.

* Number of conditions: the average total number of conditions using WA is much larger than the average number of conditions using MWA (3,029,718 conditions vs. 4345,53 conditions). This result comes directly from the shift distance.

* Condition type: using MWA, about half of the average totals conditions of type character access, while all the conditions are of type character comparison and number comparison using WA (See Table 1 C). From the previous discussion, one can notice that the condition of type character access needs less clock time by 40% than the clock time required by either the condition of type character comparison or number comparison. This has a major effect on the performance of the two algorithms.

In order to test the effect of text size on the performance of the two algorithms, the text size has been increased from 1,005,077 characters into 1,907,938 characters. The experiment is repeated with the same specifications except the text size. Table 2 reflects the results.

In Table 2(A), the average number of occurrences ranges from 1.71 (group 8) to 142.64 (group 1). The clock time required to find each group of patterns is recorded in Table 1 (B). By using WA, the clock time required ranges from 4.562 seconds (group 10) to 4.703 seconds (group 2). By using NIWA, it ranges from 0.453 seconds (group 9 and group 10) to 1.703 seconds (group 1). By using MWA, the clock time required by the algorithm WA is reduced by 63.79% (group 1) to 90.11% (group 9). This percentage of reduction is very close to the percentage of reduction (65.29% to 90.83%) for the text size 1,005,077 characters. This led us not to include the other associated information for the size 1907,938 in this study to avoid redundancy.

CONCLUSION

A new approximate string searching algorithm, the Modified Weight Algorithm (MWA) has been developed and compared with a recent approximate string-searching algorithm called the Weight Algorithm (WA). The two algorithms preprocess the pattern only. An experiment was performed to evaluate the new algorithm MWA. The number of comparisons, the running time and the shift distance are the different criteria used to compare the different algorithms. In comparison between WA and the new algorithm MWA, the results of the experiment suggest that: (I) the average number of comparisons required by WA (3,025,097 to 3,029,718) has been improved by MWA (118,433 to 434,553). So, the improvement ranges from 85.66% to 96.09%, so the new algorithm requires only 4% to 15% as many comparisons. (II) The clock time range required by WA (2.61 to 2.653 sec) has been improved by MWA to become (0.906 to 0.235 sec). Thus, the percentage of improvement ranges from 65.29% to 90.83% and the new algorithm clock times range from 10% to 35% of the old ones. Furthermore, using the MWA, the shift distance ranges from 1 to 2* important_partlength + 1 positions. On the other hand, using WA, the shift distance is one position only. The MWA algorithm gains its performance from two main changes. One change is the use of character access conditions. All the conditions used by WA, whether number comparisons or character comparisons require high execution clock time. On the other hand, using MWA about half the conditions are of type character access, which need less clock time than the other types of conditions. This increases the performance of MWA in comparison with WA. In general, increasing the percentage of the required conditions of type character-access increases the performance of string searching algorithms.

The other direction is the shift distance (1 to 2* important_part_length + 1). This decreases the number of required conditions, which, in turn, decreases the clock time required to finish the task. However, in our experiments using MWA we considered the first two-thirds of the pattern as the important part. In practice, the important part might be any part of the pattern. The question arises, is it possible to determine the important part automatically. How does that affect the relevance of the occurrences? How does that affect the system performance? Such questions need to be investigated next.

ACKNOWLEDGMENTS

The authors would like to thank Professor Martha Evens from Illinois Institute of Technology for her helpful comments and suggestions that reflected many improvements in the presentation of this study.

REFERENCES

[1.] Fredriksson, K., 2001 Shift or string matching with super-alphabets, Information Processing Letters, 87: 201-204.

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[3.] Mhashi, M., 2001 A Fast String Matching Algorithm using Double-Length Skip Distances. Dirasat J., University of Jordan, Jordan, 30: 84-92.

[4.] Fenwick, M., 2001. Fast string matching for multiple searches. Software-Practice and Experience, 31: 815-833.

[5.] Lecroq, T., 1998. Experiments on string matching in memory structures, Software-Practice and Experience, 28: 561-568.

[6.] Dermouche, A., 1995. A fast algorithm for string matching with mismatches, Information Processing Letters, 55: 105-110.

[7.] Park, K. and Z. Galil, 1990. An improved algorithm for approximate string matching, SIAM J. Comput., 19: 989-999.

[8.] Wu, S. and U. Manber, 1995. Fast text searching with errors, Comm. ACM, 35: 83-91.

[9.] Galil, Z. and K. Park, 1998. An improved algorithm for approximate string matching, in ICALP, pp: 394-404.

[10.] Cole, R. and R. HariHaran, Approximate String Matching: A faster simpler algorithm, In Proc. 9h ACM-SIAM Symposium on Discrete Algorithms (SODA), pp: 463-472.

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[12.] Mhashi, M., 2001. Character-Weight based k_mismatches and Relevancy. Al-Manarah, 7: 95-114.

Mahmoud Mhashi, Adnan Rawashdeh and Awni Hammouri

Department of IT, Faculty of Science, Mu’tah University, Mu’tah, Karak, 61710, Jordan.

Table 1: A Comparison between MWA and WA for Each Group in Terms of 1)

the Clock Time Required to Find Each Group 2) the Average Number of

Comparisons 3) and 4) the Percentage of Improvements

(A) Pattern Information

Pattern Number

Group Length in of K- Ave. Num. Of

No. characters mismatches Occurrences

1 11 5 105.84

2 20 7 2.53

3 29 9 3.29

4 38 11 2.73

5 47 13 2.07

6 56 15 1.69

7 65 17 1.90

8 74 19 1.63

9 83 21 1.68

10 92 23 1.75

(B) Average Clock Time (in Seconds)

Average clock time in seconds by using the two algorithms

Group

No. Using WA Using MWA Improvements of MWA vs. WA

1 2.610 0.906 65.29%

2 2.625 0.562 78.59%

3 2.622 0.438 83.30%

4 2.487 0.375 84.92%

5 2.609 0.328 87.43%

6 2.594 0.281 89.17%

7 2.594 0.282 89.13%

8 2.562 0.265 89.66%

9 2.563 0.235 90.83%

10 2.531 0.250 90.12%

(C) Average Number of Comparisons Including Character Accesses,

Number Comparisons and Character Comparisons. Note that

WA Requires No

Character Accesses

Ave. of Comparisons using WA

Group Number- Character-

No. Comp. Comp. Total

1 1018109 2011609 3029718

2 1016034 2010830 3026864

3 1015969 2011000 3026969

4 1015307 2010802 3026109

5 1016192 2010863 3027055

6 1017226 2010912 3028138

7 1017504 2011000 3028504

8 1015662 2010871 3026533

9 1014461 2010786 3025247

10 1014292 2010805 3025097

Ave. of Comparisons using MWA

Group Char. Number Character Total Improvement

No. Access Comp. Comp. Total of MWA Versus WA

1 249582 102091 82880 434553 85.66%

2 136270 58628 66836 261734 91.35%

3 99948 44843 59585 204376 93.25%

4 82590 37988 55089 175667 94.19%

5 71089 33361 50824 155274 94.87%

6 65240 31206 49751 146197 95.17%

7 60107 29107 47311 136525 95.49%

8 53932 26023 42707 122662 95.95%

9 52187 25423 42968 120578 96.01%

10 50581 24876 42976 118433 96.09%

Table 2: A Comparison between WA and MWA with Text Size 1,907,938

Characters

(A) Pattern Information

Pattern Number

Group Length in of K- Ave. Num. of

No. characters mismatches Occurrences

1 11 5 142.64

2 20 7 2.90

3 29 9 3.83

4 38 11 3.35

5 47 13 2.33

6 56 15 1.94

7 65 17 1.97

8 74 19 1.71

9 83 21 1.72

10 92 23 1.89

(B) Average Clock Time (in Seconds)

Average clock time in seconds by using the two algorithms

Group

No. Using WA Using MWA Improvements of MWA vs. WA

1 4.703 1.703 63.79%

2 4.609 1.063 76.94%

3 4.609 0.828 82.04%

4 4.594 0.687 85.05%

5 4.656 0.610 86.90%

6 4.657 0.562 87.93%

7 4.672 0.516 88.96%

8 4.578 0.469 89.76%

9 4.578 0.453 90.11%

10 4.562 0.453 90.07%

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