Inclusive transformation consistency control algorithm in distributed system

Operational transformation (OT) is the most effective method for consistency and concurrency control in multi-user groupware applications. This study proposes a new string-based OT algorithm to address the challenge of swapping and transposing two deletions. It has removed the faults of previous existing algorithm swapDD (ABTS: a transformation-based consistency control algorithm for wide-area collaborative applications, collaborative computing: networking, applications and worksharing 1–10, 2009). Existing algorithm swapDD fails totally in transposing two deletions if the first operation region is included in the second operation region or the second operation string is covered by the first operation string. In addition, swapDD has not considered partial overlapping between two deletions in swapping and fails at boundary conditions. New proposed algorithm works well in all possible cases of transposing two deletions. It handles overlapping and splitting of operations.

Real-time groupware systems, such as multi-player game, and real-time computer conferencing in the area of computer-supported cooperative work have multiple users where the actions of all users must be propagated to all other users.

Groupware systems are multi-user systems that provide an interface to a multi-user shared environment, which require sharing of data, fine-granularity, concurrency control, and fast response times. Concurrency control protocols are needed to repair inconsistencies in the multi-user transactions and areas of computing systems, such as database systems, distributed systems, and groupware systems. Therefore, there are specific requirements (Sun et al. 1998): high local responsiveness, unconstrained interaction, real-time communication, and consistency.

Theorem 1

In a consistent shared environment which has replicated data after execution of all operations, all have the same data.

Traditional concurrency control methods, such as locking, transactions, single active participant, dependency detection, and reversible execution, may cause the loss of interaction results and were not suitable for distributed interactive applications that demand fast local response satisfying user intentions, intention consistency, and convergence.

Over the past decade, operational transformation (OT) has become an established acceptable method for consistency maintenance in group editors. Compared with alternative concurrency control methods, OT has been found uniquely promising in better way achieving convergence, causality, and intention preservation without killing responsiveness and concurrent work (Shao et al. 2009). OT allows users to edit any part of the shared data at any time. Local operations are always executed as soon as they are generated by the user. Remote operations are transformed before execution to repair inconsistencies. Most of the existing OT algorithms only support primitive character operations like insert and delete. Only a few OT algorithms support string primitive operations like insert and delete.

Operational transformation algorithms have been studied over the past 25 years. OT algorithms correctness cannot be formally proved due to informal condition called “intention preservation.” OT algorithms only consider two primitive character-based operations like insert and delete.

We have reviewed a number of major OT algorithms for consistency maintenance in real-time group editors, including the distributed operation transformation (dOPT) algorithm (Ellis and Gibbs 1989), the generic operational transformation (GOT) algorithm (Sun and 1998), GOT optimized (GOTO) algorithm (Sun et al. 1998), state difference transformation (SDT) algorithm (Li and Li 2006), SCOT2 (Suleiman et al. 1998), SCOT 3/4 algorithm (Vidot et al. 2000), adopted (adOPTed) algorithm (Ressel et al. 1996), admissibility-based transformation (ABT) algorithm (Li and Li 2010), ABT-undo (ABTU) algorithm (Shao et al. 2010), admissibility-based sequence transformation (ABST) (Sun and 1998), and admissibility-based transformation with strings (ABTS) algorithm (Shao et al. 2009).

On categorizing all existing OT algorithms on the basis of major existing algorithms, such as dOPT, adOPT, GOT, GOTO, SDT, SOCT2, SOCT3/4, ABT (Li and Li 2007), and then further classified on the basis of area of operation, such as undo, char, string, web, graph and so on, we get that only three algorithms support string handling—GOT, GOTO and ABTS. We conclude that ABTS supports for string handling and is better than GOT and GOTO, because it has less time complexity and space complexity. In addition, ABTS is based on ABT framework, which can be formally proved. We conclude that ABTS is the best string-based OT algorithm as has less time and space complexity than GOT and GOTO (see Fig. 1). This study is focused on string-based OT algorithm based on ABT framework and removed the faults of ABTS algorithm.

Flow of problem justification process

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In a multi-user system on starting of session, the shared data are replicated at all sharing sites. In OT, local operations are executed immediately without delay, and local operations are propagated to remote sites in the background, so local operations execution do not suffer. The shared data are like a linear string ‘s’ of atomic characters and positions ‘p’ in the string that starts from zero and consider two only primitive string operations, called, insert(p, s) and delete(p, s). Here, insert(p, s) insert ‘s’ at location ‘p’ in given string definition. In addition, delete(p, s) delete ‘s’ at location ‘p’ in given string definition. The operations o1 and o2 are contextually serialized, denoted by o1 → o2, if o2′s position is defined in the resulting state of applying o1 (but no other operation). The standard notations are summarized in Table 1, where a few standard notations are taken from (Shao et al. 2009).

Definition 1

o1 and o2 are contextually equivalent o1||o2, o1Uo2 and if input is o1 and output then output should be o2 → o1′.

Definition 2

If we have exec(o_i), then all exec(o_i−1) must be completed then only o_i satisfy causality.

Definition 3

If o1Uo2, then IT(o1,o2) satisfy admissibility. It does not have inconsistent order at shared environment.

The basic swap functions for swapping two primitive operations insert and delete exist in (Shao et al. 2009). Given two operations o1 and o2, where o1 → o2, function swap(o1, o2) transposes them into o1′ and o2′, such that o2′ → o1′. Depending on their types, insert (I) and delete (D), we call different swapping functions. The basic swap function for swapping primitive operations two deletions is swapDD (Shao et al. 2009). Here, swapDD and MGswapDD take two string operations o1 and o2 as parameters. Here, o1.type = o2.type = delete. Before swapping, we have o1 → o2 and after swapping, we get o1′ and o2′, so that we can have o2′ → o1′.

Algorithm swapDD

Algorithm swapDD (o1, o2) transposes two deletions o1 and o2. There are three cases considered by (Shao et al. 2009). First, if o2.pos ≥o1.pos, it means that o2 is to delete a substring on the right side of the substring o1.str deleted by o1. Hence, if we execute o2 before o1 instead, then o2.pos should consider o1.str, because it has not been deleted yet. Therefore, o2 position shifted right by length of o1.str.

Second, if o2.pos + |o2.str| ≤o1.pos, it means that o2.str is completely on the left side of o1.pos. Hence, if o2 get executed before o1 instead, o1.pos should be shifted to the left, because o2.str has already been deleted.

Third, as in lines 6–12, o1.str is completely covered by o2.str. Then, if o2 get executed before o1 instead, o2.str is divided into three parts, among which the middle overlapping part is to be deleted by o1. The remaining left and right parts, as divided by position o1.pos, are deleted by two suboperations o2L and o2R, respectively. At last, finally, o1.pos should be set to o2.pos due to the deletion of o2L.str.

Algorithm swapDD (o₁, o₂): (o₂′, o₁′)

1. 1.

   o₁′ ← o₁; o₂′ ← o₂;

2. 2.

   if o₂.pos > = o₁.pos then

3. 3.

   o₂′.pos ← o₂.pos + | o₁.str|

4. 4.

   else if o₂.pos + | o₂.str| ≤ o₁.pos then

5. 5.

   o₁′.pos ← o₁′.pos−| o₂.str|

6. 6.

   else

7. 7.

   o_2L ← o_2R ← o₂

8. 8.

   o_2L.str ← o₂.str[0: o₁.pos−o₂.pos]

9. 9.

   o_2R.pos ← o₁.pos + | o₁.str|

10. 10.

    o_2R.str ← o₂.str[o₁.pos−o₂.pos:]

11. 11.

    o₂′.sol ← [o_2L, o_2R]

12. 12.

    o₁′.pos ← o₂.pos

13. 13.

    endif

14. 14.

    return(o₂′, o₁′)

Failure of algorithm swapDD

Algorithm swapDD fails in most of cases in swapping two deletions. Failure of algorithm swapDD in various conditions is highlighted in the following cases:

Case 1: If o2.pos ≥o1.pos

In this case, swapDD fails at boundary condition means that if o₂.pos = o₁.pos, it fails totally (lines 2–3 of algorithm swapDD).

Case 2: If there exist partial overlapping between deletion operations o1 and o2 regions

Here, partial overlapping between o₁ and o₂ means region of o₁ and o₂ overlaps to each other. In addition, we can say that o₁.str partially overlaps by o₂.str. There can be either overlapping along the left border of o₁ with o₂ or overlapping along the right border of o₁ with o₂. In this case, lines 2–3 of algorithm swapDD execute for the right overlapping of o₁.str with o₂.str, and lines 6–13 of algorithm swapDD execute for the left overlapping of o₁.str with o₂.str and it totally fails. As per the details of this algorithm given in (Shao et al. 2009), it has not discussed partial overlapping between two deletion operations o₁ and o₂ but in algorithm not put required conditions to avoid partial overlapping of o₁ and o₂. Therefore, either it has not considered the partial overlapping of o₁ and o₂ in swapDD just by assumption or it totally fails in this case.

Case 3: If o _1. str completely overlaps by o ₂ .str

In this case, swapDD lines 6–13 get executed, and it gives total wrong output in all cases. Ideally as per algorithm swapDD theory specified in (Shao et al. 2009), it should divide o₂.str into three parts, among which the middle overlapping part is to be deleted by o1. However, it fails in splitting o₂.str in the remaining left and right parts which are to be deleted by two suboperations o_2L and o_2R, respectively.

Case 4: If o2.str completely overlaps by o1.str

This case is not discussed in theory of swapDD given in (Shao et al. 2009), but if we have this case, lines 2–3 of swapDD get executed and give total faulty result.

Therefore, it is concluded that swapDD fails totally in swapping two deletions if there exist partial or total overlapping of o₁.str by o₂.str. In addition, in a few cases, it fails totally at boundary conditions.

Algorithm MGswapDD

The new proposed algorithm MGswapDD has removed all faults of the existing algorithm swapDD and is working well in all possible cases of swapping two deletions. It works well at all boundary conditions. It has also considered the partial overlapping of operations o1.str and o2.str. Also if o1.str completely overlaps by o2.str or o2.str completely overlaps by o1.str, then also it works well totally. Thus, it considers well overlapping and splitting of operations. The MGswapDD is practically implemented in lab and works well on partial or total overlapping of operations. In addition, it works well on not overlapping operations and boundary conditions.

Algorithm MGswapDD is for swapping and transposing two deletions. The process of MgswapDD is explained in the following points. Here, if we have o ← null means, o is initialized to null and will not perform any operation:

1. 1.

   From Line 3 if(o2.pos > (o1.pos + |o1.str|)), means if o2 lies completely on the right side of o1 then, if we execute o2 before o1 instead, then o2.pos should consider o1.str, because it has not been deleted yet. Therefore, o2 position shifted right by length of o1.str.

2. 2.

   From Line 5 if ((o2.pos + |o2.str|) < o1.pos), means o2.str is completely on the left side of o1.pos. Hence, if o2 get executed before o1 instead, o1.pos should be shifted to the left, because o2.str has already been deleted. Therefore, o1 shift left equal to length of o2.str.

3. 3.

   From Line 7 if (o2.pos > o1.pos&&o2.pos ≤ (o1.pos + |o1.str|)&&(o2.pos + |o2.str|) > (o1.pos + |o1.str|)), means o1.str overlaps partially with o2.str along its right boundary, then o1.str and o1 position will remain unchanged and o2′ position will shift right by the length of overlapping region of o1.str and o2.str. In addition, o2′.str will be set to not overlapping part of o2 string. Here, the overlapped region gets deleted by o1′, and o2′ deletes the remaining not overlapping region of o2.str.

4. 4.

   From Line10 if(o2.pos < o1.pos&&(o2.pos + |o2.str|) ≥o1.pos && o1.pos + |o1.str| > (o2.pos + |o2.str|), means o1.str overlaps partially with o2.str along its left boundary then the overlapped region gets deleted by o1, and o2 deletes the remaining not overlapping region. Here, o2′ string will reduced to not overlapping part of o2 string by deducting the overlapped region from the existing o2 string. In addition, o1′ position is shifted right by length of o2′ string, since o2′ is already deleted since after swapping, we have o₂′ → o₁′.

5. 5.

   From lines 13–25 get executed if none of the above conditions are true. Line 14 check if o2.str completely covered by string o1.str. If o1 and o2 delete the same substring of given string sequence ‘s’ which lie at the same position, then also condition at line 14 is true. In this case, o2 initialized to null, and o1 deletes the o1.str from o1 position. Lines 16–25 are executed if o1.str is completely covered by o2.str. Then, if o2 get executed before o1 instead, o2.str is divided into three parts, among which the middle overlapping part is to be deleted by o1. The remaining left and right parts are deleted by two suboperations o2L and o2R of o2′, respectively. At last, o1′.pos should be set to o2.pos due to the deletion of o2L.str. Therefore, if o1 is totally overlapped by o2 string, then the overlapping region gets deleted by o1, and o2 deletes its remaining regions left and right called o2Lpart and o2Rpart, respectively, which are separated by o1 region.

In multi-user environment, practically, we have implemented ABTS and MGswapDD in lab using Qualnet and ASP.Net software.

Algorithm swapDD

Case 1: If o ₂ pos = o ₁ .pos

In this case, swapDD fails at boundary condition means that if o₂.pos = o₁.pos, it fails totally (lines 2–3 of algorithm swapDD).

For example, let s = “TheGodHelpAllEqually.” Here, suppose o₁ = delete(3, “GodHelp”) and o₂ = delete(3, “God”). Therefore, condition at line 2 if(o ₂ .pos ≥o ₁ .pos) is true, since o₂.pos = o₁.pos, so by line 3, we get o₂′.pos = o₂.pos + |o₁.str| so we get o₂′.pos = 3+7 = 10. Here, in given string definition s, we apply o₂′ = delete(10, “God”); the operation fails since at starting position ‘10’ substring “God” not found (see Fig. 2). Therefore, swapDD fails totally.

Practical implementation of swapDD in lab for inputs of case 1 (wrong output)

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If we implement new proposed MGswapDD for the same inputs like case 1, we get o1′ = o1 and o2′ = null, so get right input because if o1′ get executed, then no need to execute o2′ because o2 string get deleted by o1′ since o2 string is overlapped by o1 string (see Fig. 3).

Practical implementation of MGswapDD in lab (right output)

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Case 2: If there exist partial overlapping between deletion operations o1 and o2 regions

Here, partial overlapping between o1 and o2 means region of o1 and o2 overlaps to each other.

For example, let s = “TheBirdsAreFlyingInTheSky”

Let o₁.str = “BirdsAreFlying” and o₁.pos = 3,|o₁.str| = 14

o₂.str = “FlyingInTheSky” and o₂.pos = 11.

Here, o₁ overlaps with o₂ along its right boundary. And if we execute swapDD; condition at line 2 is correct that is (o₂.pos ≥o₁.pos), since 11 > 3, so enter in if block and execute the code at line 3 that are o₂′.pos = o₂.pos + |o₁.str|, so here, we get o₂′.pos = 11 + 14 = 25, so we get o₂′ = delete (25, “FlyingInTheSky”). The operation o₂′ fails since at starting position ‘25’ substring “FlyingInTheSky” not found. Even position ‘25’ not exist in given ‘s’. Thus, swapDD fails totally (see Fig. 4).

Practical implementation of swapDD in lab for inputs of case 2 (wrong output)

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When we implement MGswapDD in lab practically for inputs of case 2, we get o1′ = o1. o₂′.pos = 17 and o2′.str=‘‘InTheSky” which give right output, because there exist no overlapping in o1′ and o2′ and both lie at given position in string ‘s’ (see Fig. 5).

Practical implementation of MGswapDD in lab for inputs of case 2 (right output)

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Case 3: If o _1. str completely overlaps by o ₂ .str

In this case, swapDD lines 6–13 get executed and it gives total wrong output in all cases.

For example, s = “WorkNotO nlyHardButGoodAlso”.

Let o₁ = delete(7, “Only”); o₂ = delete(4, “NotOnlyHard”)

Here, on executing swapDD lines 6–13, it will get executed, and we get from line7: o_2L ← o₂ and o_2R ← o₂. From line 8: o_2L.str ← o₂.str [0:o₁.pos−o₂.pos], so we get o_2L.str ← o₂.str [0:7−4] = “Not”;

Also from line 9, we get o_2R.pos ← o₁.pos + |o₁.str|; so we get o_2R.pos ← 7 + 4 = 11. And from Line 10 we get o_2R.str ← o₂.str [o₁.pos−o₂.pos:]; so we get o_2R.str ← o₂.str[7 − 4:]; so we get o_2R.str ← “OnlyHard”. Therefore, we get o_2R = delete (11, “OnlyHard”) but in given ‘s’ at position 11 “OnlyHard” not exist so o₂′.sol ← [o_2L, o_2R] also fails totally (see Fig. 6).

Practical implementation of swapDD in lab for inputs of case 3 (wrong output)

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When we practically implemented MGswapDD in lab for inputs of case 3, we get o1′.str = o1.str and o1′.pos = 4. We get o2L’.str = “Not” and o2L’.pos = 4, o2R’.str = ‘‘Hard”, o2R’.pos = 11 which give desired output, because o1 overlapped completely by o2 and overlapped region of o2 get deleted by o1′ so o2′ split in o2L’ and o2R’ to get desired output (see Fig. 7).

Practical implementation of MGswapDD in lab for inputs of case 3 (right output)

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Case 4: If o2.str completely overlaps by o1.str

If o₂.str completely overlaps by o₁.str, then in this case, swapDD lines 2–3 get executed and it gives total wrong output in all cases.

For example:

Let s = “GodHelpThoseWhoHelpThemselves”

o₂ = delete (12, “Who”);

o₁ = delete (3, “HelpThoseWhoHelp”);

Here on executing swapDD lines 2–3 will get executed and we get wrong output.

o1’ = “HelpThoseWhoHelp” o1’ position = 3 o2’ = “Who” o2’ position = 28

Algorithm swapDD failed. Thus, o₂′.sol ← [o_2L, o_2R] fails totally (see Fig. 8).

Practical implementation of swapDD in lab for inputs of case 4 (wrong output)

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When we practical implement MGswapDD in lab for inputs of case 4, we get o1′.pos = “HelpThoseWhoHelp” and o1′.pos = 3. Also o2′ = null, because o2 overlapped by o1 and o1′ delete the overlapped region so no need to execute o2′ (see Fig. 9).

Practical implementation of MGswapDD in lab for inputs of case 4 (right output)

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Algorithm MGswapDD

Case 1: If o2.pos = o1.pos

Here, in this case on executing MGswapDD lines 13–25, it will get executed and will give right result.

For example, let s = “TheBirdsAreFlyingInTheSky”

o₁ = delete(3, “BirdsAreFlying”); o₂ = delete(3, “Birds”)

Condition at line 14 is correct so switch to line 15. Condition if (((o₁.pos + |o₁.str|) ≥(o₂.pos + |o₂.str|))&& (o₁.pos ≤ o₂.pos)). Here, we get if((3 + 14) ≥ (3 + 5)&&3 ≤3) returns true so code at line 15 that is o₂′ ← null get executed means o₂′ will not execute any operation both its string and position are null. And o₁′ ← o₁ from line 1, so we get desired output “TheInTheSky” after execution of o₁′ and o₂′ where o₂′ is null. It satisfies user intentions also (see Fig. 10).

Practical implementation of MGswapDD in lab for inputs of case 1 (right output)

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Case 2: If there exist partial overlapping between deletion operations o1 and o2 regions

Here, two cases are possible either o₁.str overlaps with o₂.str along its right border or left border.

First, we consider the case when o₁.str overlaps with o₂.str along its rightboundary. For example, let s = “GodPleaseHelpMeToTakeCareMyChild”.

o₁ = (3, “PleaseHelpMe”); |o₁.str| = 12 and |o₂.str| = 12; o₂ = delete(13, “MeToTakeCare”); o₁.pos = 3 and o₂.pos = 13.

Since on executing MGswapDD condition at line 7 is true that is if (o₂.pos > o₁.pos&&o₂.pos ≤(o₁.pos +|o₁.str|) && (o₂.pos + |o₂.str|) > (o₁.pos + |o₁.str|)) returns true, so lines 8 and 9 will get executed.

Step 8: o₂′.pos = o₂.pos + ((o₁.pos + |o₁.str|)−o₂.pos);

Step 9: o₂′.str = o₂.Substring ((o₁.pos + |o₁.str|)−o₂.pos);

From step 8, o₂′.pos = 13 + (3 + 12)−13; o₂′ = 15;

Step 9: o₂′.str = o₂.Substring (3 + 12−13) = o₂.Substring(2), so o₂′.str = “ToTakeCare”. We get o₂′ = delete (15, “ToTakeCare”) and it runs well since at position 15 “ToTakeCare” exist in given ‘s’. Therefore, the overlapped substring “Me” get deleted by o₁′ and o₂′ has deleted just unoverlapped part of o₂. Here, o₁′ ← o1 from line 1. So again, we get totally right output satisfying user intentions (see Fig. 11).

Practical implementation of MGswapDD in lab for inputs of case 2 (right output)

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Second, we consider the case when o₁.str overlaps with o₂.str along its left boundary.

For example, let s = “GodPleaseHelpMeToTakeCareMyChild”.

o₂ = (3, “PleaseHelpMe”); |o₂.str| = 12 and |o₁.str| = 12; o₁ = delete(13, “MeToTakeCare”); o₂.pos = 3 and o₁.pos = 13. Since on executing MGswapDD condition at line 10 is true that is so the given code will get executed. Therefore, condition at line 10 is as follows: if(o₂.pos < o₁.pos&&(o₂.pos + |o₂.str|) ≥ o₁.pos && o₁.pos + |o₁.str| > (o₂.pos + |o₂.str|)) returns true so from Step 11: o₂′.str = o₂.Substring (0, (o₁.pos−o₂.pos)); so we get o₂′.str = o₂.Substring (0,(13−3)) = “PleaseHelp”; here |o₂′.str| = 10; and from step Step 12: o₁′.pos = o₁.pos−|o₂′.str|; we get o₁′.pos = 13−10 = 3. Since after deletion by o₂′ the o₁′. pos should left by the length of o₂′.str as o₂′ lies left of o₁′ and is already deleted. Here, we get finally o₁′ = delete(3, “PleaseHelpMe”) and o₂′ = delete(15, “ToTakeCare”) and o₂′ → o₁′ work well after swapping of o₁ → o₂.

In this case, also we get right output.

Case 3: If o _1. str completely overlaps by o ₂ .str

In this case, all the above conditions before line 13 are false so enter in else block at line 13. Here, condition at Step 14: if(((o₁.pos + |o₁.str|) ≥ (o₂.pos + |o₂.str|))&& (o₁.pos ≤ o₂.pos)) is false so enter in its else part. So lines 16 to 25 get executed.

For example, let s = “The Sun give us Heat and Light”; o₁ = delete(13, “us”) and o₂ = delete(4, “Sun give us Heat”). From Step 17: o_2Lpart ← o₂; o_2Lpart.str = o₂.str; here, we have o_2Lpart.str = “Sun give us Heat”. From Step 18: o_2Lpart.pos = o₂.pos; Here we have o_2Lpart.pos = 4. From Step 19: o_2Rpart ← o₂; o_2Rpart.str = o₂.str; here, we have o_2Rpart.str = “Sun give us Heat”. From Step 20: o_2Rpart.pos = o₂.pos; here, we have o_2Rpart.pos = 4.

From Step 21: o_2Lpart.str = o₂.Substring (0, o₁.pos−o₂.pos); here, we have o_2Lpart.str = o₂.Substring (0,13−4) ≥ o_2Lpart.str = o₂.Substring (0,9) = “Sun give”. From Step 22: o_2Rpart.pos = o1.pos + |o1.str|; here, we have o_2Rpart.pos = 13 + 2=15. From Step 23: o_2Rpart.str = o₂.Substring (o₁.pos−o₂.pos + |o₁.str|); here, we have o_2Rpart.str = o₂.Substring (13−4 + 2) = o₂.Substring (11) = “Heat”. From Step 24: o₂′.sol ← [o_2Lpart, o_2Rpart]; so the left part “Sun give”. get deleted by operation o_2Lpart;right part “Heat” get deleted by o_2Rpart and the middle overlapping región “us” get deleted by o₁ and by Step 25: o₁′.pos = o₂.pos;so we get o₁′.pos = 4 since at first o₂′ get executed and since o_2Lpart is already executed the position of o₁′ shift left to o₂.pos. So o₂′ → o₁′ works correctly here (see Fig. 12).

Practical implementation of MGswapDD in lab for inputs of case 3 (right output)

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Case 4: If o _2. str completely overlaps by o ₁ .str

In this case, all the above conditions before line 13 are false so enter in else block at line 13. Here, condition at Step 14: if (((o₁.pos + |o₁.str|) ≥ (o₂.pos + |o₂.str|))&& (o₁.pos ≤ o₂.pos)) is true so enter in its if part. So step 15 will get executed.

For example, let s = “The God will help me always everywhere”. o₁ = delete (4, “God will help me”) and o₂ = delete (8, “will”). Here, condition at Step 14: if(((o₁.pos + |o₁.str|) ≥ (o₂.pos + |o₂.str|))&&(o₁.pos ≤o₂.pos)) that is if((4 + 15) ≥ (8 + 4)&&4 ≤8) is true so condition at line 15 get executed where o₂′ is set to null means will not perform any operation and o₁ will remain as it is. If o₁ will execute the region of o₂ which is covered by o₁ will automatically deleted giving right output. Here, o₂′ → o₁′ is equal to execution of o₁′ only, since o₂′ is null and also o₁′ ← o₁ from line 1 (see Fig. 13).

Practical implementation of MGswapDD in lab for inputs of case 4 (right output)

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Operational transformation is the most optimistic method for concurrency and consistency control in muti-user groupware systems.

ABTS is the best string handling OT algorithm. The swapDD function of ABTS is proposed to swap two deletions, but swapDD fails totally if there exist partial overlapping between two deletions. In addition, it fails if one deletion operation string is totally covered by other deletion operation string. In few other cases, also swapDD fails at boundary conditions.

We propose a new algorithm MGswapDD to swap two deletions. It is also based on ABT framework and support string handling. It considers and works well in splitting and overlapping of operations. It works well on all boundary conditions also. It is practically implemented in lab also covering all possible cases of swapping two deletions. It gives totally right result if either there exist partial overlapping between two deletions or if one deletion operation string is totally covered by other deletion operation string. Therefore, in brief, it has removed all faults of the existing swapDD and work well in all possible cases of swapping two deletions.

Still there is scope to extend the support to other composite operations of string handling and char handling. Also there is need to support better data structures. A lot of work is done to reduce time complexity and space complexity. Still there is a scope to reduce time complexity and space complexity.

SK made substantial contributions to conception and design, acquisition of data, and analysis and interpretation of data; has been involved in drafting the manuscript or revising it critically for important intellectual content; has given final approval of the version to be published, and agrees to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. AK provided full guidance and support in all of the above works. Both authors read and approved the final manuscript.

Acknowledgements

The paper is dedicated to my mother Meera Devi and daughter Harshita Kumawat.

Competing interests

The authors declare that they have no competing interests.

Affiliations

1. School of Engineering and Technology, Poornima University, IS-2027 To 2031 Ramchandrapura P.O. Vidhani Vatika Sitapura Extension, Jaipur, Rajasthan, 303905, India
   • Santosh Kumawat
2. Poornima College of Engineering, Jaipur, Rajasthan, India
   • Ajay Khunteta

Corresponding author

Correspondence to Santosh Kumawat.

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