Patent Publication Number: US-8112496-B2

Title: Efficient algorithm for finding candidate objects for remote differential compression

Description:
RELATED APPLICATIONS 
     This application is a continuation of prior U.S. application Ser. No. 10/948,980, filed on Sep. 24, 2004, now U.S. Pat. No. 7,613,787 which is hereby incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The proliferation of networks such as intranets, extranets, and the internet has lead to a large growth in the number of users that share information across wide networks. A maximum data transfer rate is associated with each physical network based on the bandwidth associated with the transmission medium as well as other infrastructure related limitations. As a result of limited network bandwidth, users can experience long delays in retrieving and transferring large amounts of data across the network. 
     Data compression techniques have become a popular way to transfer large amounts of data across a network with limited bandwidth. Data compression can be generally characterized as either lossless or lossy. Lossless compression involves the transformation of a data set such that an exact reproduction of the data set can be retrieved by applying a decompression transformation. Lossless compression is most often used to compact data, when an exact replica is required. 
     In the case where the recipient of a data object already has a previous, or older, version of that object, a lossless compression approach called Remote Differential Compression (RDC) may be used to determine and only transfer the differences between the new and the old versions of the object. Since an RDC transfer only involves communicating the observed differences between the new and old versions (for instance, in the case of files, file modification or last access dates, file attributes, or small changes to the file contents), the total amount of data transferred can be greatly reduced. RDC can be combined with another lossless compression algorithm to further reduce the network traffic. The benefits of RDC are most significant in the case where large objects need to be communicated frequently back and forth between computing devices and it is difficult or infeasible to maintain old copies of these objects, so that local differential algorithms cannot be used. 
     SUMMARY OF THE INVENTION 
     Briefly stated, the present invention is related to a method and system for finding candidate objects for remote differential compression. Objects are updated between two or more computing devices using remote differential compression (RDC) techniques such that required data transfers are minimized. In one aspect, an algorithm provides enhanced efficiencies by allowing the sender to communicate a small amount of meta-data to the receiver, and the receiver to use this meta-data to locate a set of objects that are similar to the object that needs to be transferred from the sender. Once this set of similar objects has been found, the receiver may reuse any parts of these objects as needed during the RDC algorithm. 
     A more complete appreciation of the present invention and its improvements can be obtained by reference to the accompanying drawings, which are briefly summarized below, to the following detailed description of illustrative embodiments of the invention, and to the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Non-limiting and non-exhaustive embodiments of the present invention are described with reference to the following drawings. 
         FIG. 1  is a diagram illustrating an operating environment; 
         FIG. 2  is a diagram illustrating an example computing device; 
         FIGS. 3A and 3B  are diagrams illustrating an example RDC procedure; 
         FIGS. 4A and 4B  are diagrams illustrating process flows for the interaction between a local device and a remote device during an example RDC procedure; 
         FIGS. 5A and 5B  are diagrams illustrating process flows for recursive remote differential compression of the signature and chunk length lists in an example interaction during an RDC procedure; 
         FIG. 6  is a diagram that graphically illustrates an example of recursive compression in an example RDC sequence; 
         FIG. 7  is a diagram illustrating the interaction of a client and server application using an example RDC procedure; 
         FIG. 8  is a diagram illustrating a process flow for an example chunking procedure; 
         FIG. 9  is a diagram of example instruction code for an example chunking procedure; 
         FIGS. 10 and 11  are diagrams of example instruction code for another example chunking procedure; 
         FIG. 12  illustrates an RDC algorithm modified to find and use candidate objects; 
         FIGS. 13 and 14  show a process and an example of a trait computation; 
         FIGS. 15 and 16  may be used when selecting the parameters for b and t; 
         FIG. 17  illustrates data structures that make up a compact representation of: an Object Map and a set of Trait Tables; and 
         FIG. 18  illustrates a process for computing similar traits, in accordance with aspects of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Various embodiments of the present invention will be described in detail with reference to the drawings, where like reference numerals represent like parts and assemblies throughout the several views. Reference to various embodiments does not limit the scope of the invention, which is limited only by the scope of the claims attached hereto. Additionally, any examples set forth in this specification are not intended to be limiting and merely set forth some of the many possible embodiments for the claimed invention. 
     The present invention is described in the context of local and remote computing devices (or “devices”, for short) that have one or more commonly associated objects stored thereon. The terms “local” and “remote” refer to one instance of the method. However, the same device may play both a “local” and a “remote” role in different instances. Remote Differential Compression (RDC) methods are used to efficiently update the commonly associated objects over a network with limited-bandwidth. When a device having a new copy of an object needs to update a device having an older copy of the same object, or of a similar object, the RDC method is employed to only transmit the differences between the objects over the network. An example described RDC method uses (1) a recursive approach for the transmission of the RDC metadata, to reduce the amount of metadata transferred for large objects, and (2) a local maximum-based chunking method to increase the precision associated with the object differencing such that bandwidth utilization is minimized. Some example applications that benefit from the described RDC methods include: peer-to-peer replication services, file-transfer protocols such as SMB, virtual servers that transfer large images, email servers, cellular phone and PDA synchronization, database server replication, to name just a few. 
     Operating Environment 
       FIG. 1  is a diagram illustrating an example operating environment for the present invention. As illustrated in the figure, devices are arranged to communicate over a network. These devices may be general purpose computing device, special purpose computing devices, or any other appropriate devices that are connected to a network. The network  102  may correspond to any connectivity topology including, but not limited to: a direct wired connection (e.g., parallel port, serial port, USB, IEEE 1394, etc), a wireless connection (e.g., IR port, Bluetooth port, etc.), a wired network, a wireless network, a local area network, a wide area network, an ultra-wide area network, an internet, an intranet, and an extranet. 
     In an example interaction between device A ( 100 ) and device B ( 101 ), different versions of an object are locally stored on the two devices: object O A on  100  and object O B  on  101 . At some point, device A ( 100 ) decides to update its copy of object O A  with the copy (object O B ) stored on device B ( 101 ), and sends a request to device B ( 101 ) to initiate the RDC method. In an alternate embodiment, the RDC method could be initiated by device B ( 101 ). 
     Device A ( 100 ) and device B ( 101 ) both process their locally stored object and divide the associated data into a variable number of chunks in a data-dependent fashion (e.g., chunks  1 -n for object O B , and chunks  1 -k for object O A , respectively). A set of signatures such as strong hashes (SHA) for the chunks are computed locally by both the devices. The devices both compile separate lists of the signatures. During the next step of the RDC method, device B ( 101 ) transmits its computed list of signatures and chunk lengths  1 -n to device A ( 100 ) over the network  102 . Device A ( 100 ) evaluates this list of signatures by comparing each received signature to its own generated signature list  1 -k. Mismatches in the signature lists indicate one or more differences in the objects that require correction. Device A ( 100 ) transmits a request for device B ( 101 ) to send the chunks that have been identified by the mismatches in the signature lists. Device B ( 101 ) subsequently compresses and transmits the requested chunks, which are then reassembled by device A ( 100 ) after reception and decompression are accomplished. Device A ( 100 ) reassembles the received chunks together with its own matching chunks to obtain a local copy of object O B . 
     Example Computing Device 
       FIG. 2  is a block diagram of an example computing device that is arranged in accordance with the present invention. In a basic configuration, computing device  200  typically includes at least one processing unit ( 202 ) and system memory ( 204 ). Depending on the exact configuration and type of computing device, system memory  204  may be volatile (such as RAM), non-volatile (such as ROM, flash memory, etc.) or some combination of the two. System memory  204  typically includes an operating system ( 205 ); one or more program modules ( 206 ); and may include program data ( 207 ). This basic configuration is illustrated in  FIG. 2  by those components within dashed line  208 . 
     Computing device  200  may also have additional features or functionality. For example, computing device  200  may also include additional data storage devices (removable and/or non-removable) such as, for example, magnetic disks, optical disks, or tape. Such additional storage is illustrated in  FIG. 2  by removable storage  209  and non-removable storage  210 . Computer storage media may include volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules or other data. System memory  204 , removable storage  209  and non-removable storage  210  are all examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computing device  200 . Any such computer storage media may be part of device  200 . Computing device  200  may also have input device(s)  212  such as keyboard, mouse, pen, voice input device, touch input device, etc. Output device(s)  214  such as a display, speakers, printer, etc. may also be included. All these devices are known in the art and need not be discussed at length here. 
     Computing device  200  also contains communications connection(s)  216  that allow the device to communicate with other computing devices  218 , such as over a network. Communications connection(s)  216  is an example of communication media. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means 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 media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, microwave, satellite, infrared and other wireless media. The term computer readable media as used herein includes both storage media and communication media. 
     Various procedures and interfaces may be implemented in one or more application programs that reside in system memory  204 . In one example, the application program is a remote differential compression algorithm that schedules file synchronization between the computing device (e.g., a client) and another remotely located computing device (e.g., a server). In another example, the application program is a compression/decompression procedure that is provided in system memory  204  for compression and decompressing data. In still another example, the application program is a decryption procedure that is provided in system memory  204  of a client device. 
     Remote Differential Compression (RDC) 
       FIGS. 3A and 3B  are diagrams illustrating an example RDC procedure according to at least one aspect of the present invention. The number of chunks in particular can vary for each instance depending on the actual objects O A  and O B . 
     Referring to  FIG. 3A , the basic RDC protocol is negotiated between two computing devices (device A and device B). The RDC protocol assumes implicitly that the devices A and B have two different instances (or versions) of the same object or resource, which are identified by object instances (or versions) O A  and O B , respectively. For the example illustrated in this figure, device A has an old version of the resource O A , while device B has a version O B  with a slight (or incremental) difference in the content (or data) associated with the resource. 
     The protocol for transferring the updated object O B  from device B to device A is described below. A similar protocol may be used to transfer an object from device A to device B, and that the transfer can be initiated at the behest of either device A or device B without significantly changing the protocol described below.
         1. Device A sends device B a request to transfer Object O B  using the RDC protocol. In an alternate embodiment, device B initiates the transfer; in this case, the protocol skips step 1 and starts at step 2 below.   2. Device A partitions Object O A  into chunks  1 -k, and computes a signature Sig Ai  and a length (or size in bytes) Len Ai  for each chunk 1 . . . k of Object O A . The partitioning into chunks will be described in detail below. Device A stores the list of signatures and chunk lengths ((Sig A1 , Len A1 ) (Sig Ak , Len Ak )).   3. Device B partitions Object O B  into chunks  1 -n, and computes a signature Sig Bi  and a length Len Bi  for each chunk  1  . . . n of Object O B . The partitioning algorithm used in step 3 must match the one in step 2 above.   4. Device B sends a list of its computed chunk signatures and chunk lengths ((Sig B1 , Len B1 ) . . . (Sig Bn , Len Bn )) that are associated with Object O B  to device A. The chunk length information may be subsequently used by device A to request a particular set of chunks by identifying them with their start offset and their length. Because of the sequential nature of the list, it is possible to compute the starting offset in bytes of each chunk Bi by adding up the lengths of all preceding chunks in the list.
           In another embodiment, the list of chunk signatures and chunk lengths is compactly encoded and further compressed using a lossless compression algorithm before being sent to device A.   
           5. Upon receipt of this data, device A compares the received signature list against the signatures Sig A1  . . . Sig Ak  that it computed for Object O A  in step 2, which is associated with the old version of the content.   6. Device A sends a request to device B for all the chunks whose signatures received in step 4 from device B failed to match any of the signatures computed by device A in step 2. For each requested chunk Bi, the request comprises the chunk start offset computed by device A in step 4 and the chunk length.   7. Device B sends the content associated with all the requested chunks to device A. The content sent by device B may be further compressed using a lossless compression algorithm before being sent to device A.   8. Device A reconstructs a local copy of Object O B  by using the chunks received in step 7 from device B, as well as its own chunks of Object O A  that matched signatures sent by device B in step 4. The order in which the local and remote chunks are rearranged on device A is determined by the list of chunk signatures received by device A in step 4.       

     The partitioning steps 2 and 3 may occur in a data-dependent fashion that uses a fingerprinting function that is computed at every byte position in the associated object (O A  and O B , respectively). For a given position, the fingerprinting function is computed using a small data window surrounding that position in the object; the value of the fingerprinting function depends on all the bytes of the object included in that window. The fingerprinting function can be any appropriate function, such as, for example, a hash function or a Rabin polynomial. 
     Chunk boundaries are determined at positions in the Object for which the fingerprinting function computes to a value that satisfies a chosen condition. The chunk signatures may be computed using a cryptographically secure hash function (SHA), or some other hash function such as a collision-resistant hash function. 
     The signature and chunk length list sent in step 4 provides a basis for reconstructing the object using both the original chunks and the identified updated or new chunks. The chunks that are requested in step 6 are identified by their offset and lengths. The object is reconstructed on device A by using local and remote chunks whose signatures match the ones received by device A in step 4, in the same order. 
     After the reconstruction step is completed by device A, Object O A  can be deleted and replaced by the copy of Object O B  that was reconstructed on device A. In other embodiments, device A may keep Object O A  around for potential “reuse” of chunks during future RDC transfers. 
     For large objects, the basic RDC protocol instance illustrated in  FIG. 3A  incurs a significant fixed overhead in Step 4, even if Object O A  and Object O B  are very close, or identical. Given an average chunk size C, the amount of information transmitted over the network in Step 4 is proportional to the size of Object O B , specifically it is proportional to the size of Object O B  divided by C, which is the number of chunks of Object B, and thus of (chunk signature, chunk length) pairs transmitted in step 4. 
     For example, referring to  FIG. 6 , a large image (e.g., a virtual hard disk image used by a virtual machine monitor such as, for example, Microsoft Virtual Server) may result in an Object (O B ) with a size of 9.1 GB. For an average chunk size C equal to 3 KB, the 9 GB object may result in 3 million chunks being generated for Object O B , with 42 MB of associated signature and chunk length information that needs to be sent over the network in Step 4. Since the 42 MB of signature information must be sent over the network even when the differences between Object O A  and Object O B  (and thus the amount of data that needs to be sent in Step 7) are very small, the fixed overhead cost of the protocol is excessively high. 
     This fixed overhead cost can be significantly reduced by using a recursive application of the RDC protocol instead of the signature information transfer in step 4. Referring to  FIG. 3B , additional steps 4.2-4.8 are described as follows below that replace step 4 of the basic RDC algorithm. Steps 4.2-4.8 correspond to a recursive application of steps 2-8 of the basic RDC protocol described above. The recursive application can be further applied to step 4.4 below, and so on, up to any desired recursion depth.
         4.2. Device A performs a recursive chunking of its signature and chunk length list ((Sig A1 , Len A1 ) . . . (Sig Ak , Len Ak )) into recursive signature chunks, obtaining another list of recursive signatures and recursive chunk lengths ((RSig A1 , RLen A1 ) . . . (RSig As , RLen As )), where s&lt;&lt;k.   4.3. Device B recursively chunks up the list of signatures and chunk lengths ((Sig B1 , Len B1 ) . . . (Sig Bn , Len Bn )) to produce a list of recursive signatures and recursive chunk lengths ((RSig B1 , RLen B1 ) . . . (RSig Br , RLen Br )), where r&lt;&lt;n.   4.4. Device B sends an ordered list of recursive signatures and recursive chunk lengths ((RSig B1 , RLen B1 ) . . . (RSig Br , RLen Br )) to device A. The list of recursive chunk signatures and recursive chunk lengths is compactly encoded and may be further compressed using a lossless compression algorithm before being sent to device A.   4.5. Device A compares the recursive signatures received from device B with its own list of recursive signatures computed in Step 4.2.       

     4.6. Device A sends a request to device B for every distinct recursive signature chunk (with recursive signature RSig Bk ) for which device A does not have a matching recursive signature in its set (RSig As  . . . RSig As ).
         4.7. Device B sends device A the requested recursive signature chunks. The requested recursive signature chunks may be further compressed using a lossless compression algorithm before being sent to device A.   4.8. Device A reconstructs the list of signatures and chunk information ((Sig B1 , Len B1 ) . . . (Sig Bn , Len Bn )) using the locally matching recursive signature chunks, and the recursive chunks received from device B in Step 4.7.       

     After step 4.8 above is completed, execution continues at step 5 of the basic RDC protocol described above, which is illustrated in  FIG. 3A . 
     As a result of the recursive chunking operations, the number of recursive signatures associated with the objects is reduced by a factor equal to the average chunk size C, yielding a significantly smaller number of recursive signatures (r&lt;&lt;n for object O A  and s&lt;&lt;k for object O B , respectively). In one embodiment, the same chunking parameters could be used for chunking the signatures as for chunking the original objects O A  and O B . In an alternate embodiment, other chunking parameters may be used for the recursive steps. 
     For very large objects the above recursive steps can be applied k times, where k≧1. For an average chunk size of C, recursive chunking may reduce the size of the signature traffic over the network (steps 4.2 through 4.8) by a factor approximately corresponding to C k . Since C is relatively large, a recursion depth of greater than one may only be necessary for very large objects. 
     In one embodiment, the number of recursive steps may be dynamically determined by considering parameters that include one or more of the following: the expected average chunk size, the size of the objects O A  and/or O B , the data format of the objects O A  and/or O B , the latency and bandwidth characteristics of the network connecting device A and device B. 
     The fingerprinting function used in step 2 is matched to the fingerprinting function that is used in step 3. Similarly, the fingerprinting function used in step 4.2 is matched to the fingerprinting function that is used in step 4.3. The fingerprinting function from steps 2-3 can optionally be matched to the fingerprinting function from steps 4.2-4.3. 
     As described previously, each fingerprinting function uses a small data window that surrounds a position in the object; where the value associated with the fingerprinting function depends on all the bytes of the object that are included inside the data window. The size of the data window can be dynamically adjusted based on one or more criteria. Furthermore, the chunking procedure uses the value of the fingerprinting function and one or more additional chunking parameters to determine the chunk boundaries in steps 2-3 and 4.2-4.3 above. 
     By dynamically changing the window size and the chunking parameters, the chunk boundaries are adjusted such that any necessary data transfers are accomplished with minimal consumption of the available bandwidth. 
     Example criteria for adjusting the window size and the chunking parameters include: a data type associated with the object, environmental constraints, a usage model, the latency and bandwidth characteristics of the network connecting device A and device B, and any other appropriate model for determining average data transfer block sizes. Example data types include word processing files, database images, spreadsheets, presentation slide shows, and graphic images. An example usage model may be where the average number of bytes required in a typical data transfer is monitored. 
     Changes to a single element within an application program can result in a number of changes to the associated datum and/or file. Since most application programs have an associated file type, the file type is one possible criteria that is worthy of consideration in adjusting the window size and the chunking parameters. In one example, the modification of a single character in a word processing document results in approximately 100 bytes being changed in the associated file. In another example, the modification of a single element in a database application results in 1000 bytes being changed in the database index file. For each example, the appropriate window size and chunking parameters may be different such that the chunking procedure has an appropriate granularity that is optimized based on the particular application. 
     Example Process Flow 
       FIGS. 4A and 4B  are diagrams illustrating process flows for the interaction between a local device (e.g., device A) and a remote device (e.g., device B) during an example RDC procedure that is arranged in accordance with at least one aspect of the present invention. The left hand side of  FIG. 4A  illustrates steps  400 - 413  that are operated on the local device A, while the right hand side of  FIG. 4A  illustrates steps  450 - 456  that are operated on the remote device B. 
     As illustrated in  FIG. 4A , the interaction starts by device A requesting an RDC transfer of object O B  in step  400 , and device B receiving this request in step  450 . Following this, both the local device A and remote device B independently compute fingerprints in steps  401  and  451 , divide their respective objects into chunks in steps  402  and  452 , and compute signatures (e.g., SHA) for each chunk in steps  403  and  453 , respectively. 
     In step  454 , device B sends the signature and chunk length list computed in steps  452  and  453  to device A, which receives this information in step  404 . 
     In step  405 , the local device A initializes the list of requested chunks to the empty list, and initializes the tracking offset for the remote chunks to 0. In step  406 , the next (signature, chunk length) pair (Sig Bi , Len Bi ) is selected for consideration from the list received in step  404 . In step  407 , device A checks whether the signature Sig Bi  selected in step  406  matches any of the signatures it computed during step  403 . If it matches, execution continues at step  409 . If it doesn&#39;t match, the tracking remote chunk offset and the length in bytes Len Bi  are added to the request list in step  408 . At step  409 , the tracking offset is incremented by the length of the current chunk Len Bi . 
     In step  410 , the local device A tests whether all (signature, chunk length) pairs received in step  404  have been processed. If not, execution continues at step  406 . Otherwise, the chunk request list is suitably encoded in a compact fashion, compressed, and sent to the remote device B at step  411 . 
     The remote device B receives the compressed list of chunks at step  455 , decompresses it, then compresses and sends back the chunk data at step  456 . 
     The local device receives and decompresses the requested chunk data at step  412 . Using the local copy of the object O A  and the received chunk data, the local devices reassembles a local copy of O B  at step  413 . 
       FIG. 4B  illustrates a detailed example for step  413  from  FIG. 4A . Processing continues at step  414 , where the local device A initializes the reconstructed object to empty. 
     In step  415 , the next (signature, chunk length) pair (Sig Bi , Len Bi ) is selected for consideration from the list received in step  404 . In step  416 , device A checks whether the signature Sig Bi  selected in step  417  matches any of the signatures it computed during step  403 . 
     If it matches, execution continues at step  417 , where the corresponding local chunk is appended to the reconstructed object. If it doesn&#39;t match, the received and decompressed remote chunk is appended to the reconstructed object in step  418 . 
     In step  419 , the local device A tests whether all (signature, chunk length) pairs received in step  404  have been processed. If not, execution continues at step  415 . Otherwise, the reconstructed object is used to replace the old copy of the object O A  on device A in step  420 . 
     Example Recursive Signature Transfer Process Flow 
       FIGS. 5A and 5B  are diagrams illustrating process flows for recursive transfer of the signature and chunk length list in an example RDC procedure that is arranged according to at least one aspect of the present invention. The below described procedure may be applied to both the local and remote devices that are attempting to update commonly associated objects. 
     The left hand side of  FIG. 5A  illustrates steps  501 - 513  that are operated on the local device A, while the right hand side of  FIG. 5A  illustrates steps  551 - 556  that are operated on the remote device B. Steps  501 - 513  replace step  404  in  FIG. 4A  while steps  551 - 556  replace step  454  in  FIG. 4A . 
     In steps  501  and  551 , both the local device A and remote device B independently compute recursive fingerprints of their signature and chunk length lists ((Sig A1 ,Len A1 ), . . . (Sig Ak ,Len Ak )) and ((Sig B1 ,Len B1 ), (Sig Bn ,Len Bn )), respectively, that had been computed in steps  402 / 403  and  452 / 453 , respectively. In steps  502  and  552  the devices divide their respective signature and chunk length lists into recursive chunks, and in steps  503  and  553  compute recursive signatures (e.g., SHA) for each recursive chunk, respectively. 
     In step  554 , device B sends the recursive signature and chunk length list computed in steps  552  and  553  to device A, which receives this information in step  504 . 
     In step  505 , the local device A initializes the list of requested recursive chunks to the empty list, and initializes the tracking remote recursive offset for the remote recursive chunks to 0. In step  506 , the next (recursive signature, recursive chunk length) pair (RSig Bi , RLen Bi ) is selected for consideration from the list received in step  504 . In step  507 , device A checks whether the recursive signature RSig Bi  selected in step  506  matches any of the recursive signatures it computed during step  503 . If it matches, execution continues at step  509 . If it doesn&#39;t match, the tracking remote recursive chunk offset and the length in bytes RLen Bi  are added to the request list in step  508 . At step  509 , the tracking remote recursive offset is incremented by the length of the current recursive chunk RLen Bi . 
     In step  510 , the local device A tests whether all (recursive signature, recursive chunk length) pairs received in step  504  have been processed. If not, execution continues at step  506 . Otherwise, the recursive chunk request list is compactly encoded, compressed, and sent to the remote device B at step  511 . 
     The remote device B receives the compressed list of recursive chunks at step  555 , uncompressed the list, then compresses and sends back the recursive chunk data at step  556 . 
     The local device receives and decompresses the requested recursive chunk data at step  512 . Using the local copy of the signature and chunk length list ((Sig A1 ,Len A1 ), (Sig Ak ,Len Ak )) and the received recursive chunk data, the local devices reassembles a local copy of the signature and chunk length list ((Sig B1 ,Len B1 ), . . . (Sig Bk ,Len Bn )) at step  513 . Execution then continues at step  405  in  FIG. 4A . 
       FIG. 5B  illustrates a detailed example for step  513  from  FIG. 5A . Processing continues at step  514 , where the local device A initializes the list of remote signatures and chunk lengths, SIGCL, to the empty list. 
     In step  515 , the next (recursive signature, recursive chunk length) pair (RSig Bi , RLen Bi ) is selected for consideration from the list received in step  504 . In step  516 , device A checks whether the recursive signature RSig Bi  selected in step  515  matches any of the recursive signatures it computed during step  503 . 
     If it matches, execution continues at step  517 , where device A appends the corresponding local recursive chunk to SIGCL. If it doesn&#39;t match, the remote received recursive chunk is appended to SIGCL at step  518 . 
     In step  519 , the local device A tests whether all (recursive signature, recursive chunk length) pairs received in step  504  have been processed. If not, execution continues at step  515 . Otherwise, the local copy of the signature and chunk length list ((Sig B1 ,Len B1 ), . . . (Sig Bk ,Len Bn )) is set to the value of SIGCL in step  520 . Execution then continues back to step  405  in  FIG. 4A . 
     The recursive signature and chunk length list may optionally be evaluated to determine if additional recursive remote differential compression is necessary to minimize bandwidth utilization as previously described. The recursive signature and chunk length list can be recursively compressed using the described chunking procedure by replacing steps  504  and  554  with another instance of the RDC procedure, and so on, until the desired compression level is achieved. After the recursive signature list is sufficiently compressed, the recursive signature list is returned for transmission between the remote and local devices as previously described. 
       FIG. 6  is a diagram that graphically illustrates an example of recursive compression in an example RDC sequence that is arranged in accordance with an example embodiment. For the example illustrated in  FIG. 6 , the original object is 9.1 GB of data. A signature and chunk length list is compiled using a chunking procedure, where the signature and chunk length list results in 3 million chunks (or a size of 42 MB). After a first recursive step, the signature list is divided into 33 thousand chunks and reduced to a recursive signature and recursive chunk length list with size 33 KB. By recursively compressing the signature list, bandwidth utilization for transferring the signature list is thus dramatically reduced, from 42 MB to about 395 KB. 
     Example Object Updating 
       FIG. 7  is a diagram illustrating the interaction of a client and server application using an example RDC procedure that is arranged according to at least one aspect of the present invention. The original file on both the server and the client contained text “The quick fox jumped over the lazy brown dog. The dog was so lazy that he didn&#39;t notice the fox jumping over him.” 
     At a subsequent time, the file on the server is updated to: “The quick fox jumped over the lazy brown dog. The brown dog was so lazy that he didn&#39;t notice the fox jumping over him.” 
     As described previously, the client periodically requests the file to be updated. The client and server both chunk the object (the text) into chunks as illustrated. On the client, the chunks are: “The quick fox jumped”, “over the lazy brown dog.”, “The dog was so lazy that he didn&#39;t notice”, and “the fox jumping over him.”; the client signature list is generated as: SHA 11 , SHA 12 , SHA 11 , and SHA 14 . On the server, the chunks are: “The quick fox jumped”, “over the lazy brown dog.”, “The brown dog was”, “so lazy that he didn&#39;t notice”, and “the fox jumping over him.”; the server signature list is generated as: SHA 21 , SHA 22 , SHA 23 , SHA 24 , and SHA 25 . 
     The server transmits the signature list (SHA 21 -SHA 25 ) using a recursive signature compression technique as previously described. The client recognizes that the locally stored signature list (SHA 11 -SHA 14 ) does not match the received signature list (SHA 21 -SHA 25 ), and requests the missing chunks  3  and  4  from the server. The server compresses and transmits chunks  3  and  4  (“The brown dog was”, and “so lazy that he didn&#39;t notice”). The client receives the compressed chunks, decompresses them, and updates the file as illustrated in  FIG. 7 . 
     Chunking Analysis 
     The effectiveness of the basic RDC procedure described above may be increased by optimizing the chunking procedures that are used to chunk the object data and/or chunk the signature and chunk length lists. 
     The basic RDC procedure has a network communication overhead cost that is identified by the sum of: 
     (S1) |Signatures and chunk lengths from B|=|O B |*|SigLen|/C, where |O B | is the size in bytes of Object O B , SigLen is the size in bytes of a (signature, chunk length) pair, and C is the expected average chunk size in bytes; and 
     (S2) Σchunk_length, where (signature, chunk_length)∉Signatures from B, 
     and signature ∈ Signatures from A 
     The communication cost thus benefits from a large average chunk size and a large intersection between the remote and local chunks. The choice of how objects are cut into chunks determines the quality of the protocol. The local and remote device must agree, without prior communication, on where to cut an object. The following describes and analyzes various methods for finding cuts. 
     The following characteristics are assumed to be known for the cutting algorithm: 
     1. Slack: The number of bytes required for chunks to reconcile between file differences. Consider sequences s 1 , s 2 , and s 3 , and form the two sequences s 1 s 3 , s 2 s 3  by concatenation. Generate the chunks for those two sequences Chunks 1 , and Chunks 2 . If Chunks 1 ′ and Chunks 2 ′ are the sums of the chunk lengths from Chunks 1  and Chunks 2 , respectively, until the first common suffix is reached, the slack in bytes is given by the following formula:
 
slack=Chunks 1   ′−|s   1 |=Chunks 2   ′−|s   2 |
 
     2. Average Chunk Size C: 
     When Objects O A  and O B  have S segments in common with average size K, the number of chunks that can be obtained locally on the client is given by:
 
S*└(K−slack)/C┘
 
and (S2) above rewrites to:
 
|O A |−S*└(K−slack)/C┘
 
     Thus, a chunking algorithm that minimizes slack will minimize the number of bytes sent over the wire. It is therefore advantageous to use chunking algorithms that minimize the expected slack. 
     Fingerprinting Functions 
     All chunking algorithms use a fingerprinting function, or hash, that depends on a small window, that is, a limited sequence of bytes. The execution time of the hash algorithms used for chunking is independent of the hash window size when those algorithms are amenable to finite differencing (strength reduction) optimizations. Thus, for a hash window of size k it is should be easy (require only a constant number of steps) to compute the hash #[b 1 , . . . , b k−1 ,b k ] using b 0 , b k , and #[b 0 , b 1 , . . . , b k−1 ] only. Various hashing functions can be employed such as hash functions using Rabin polynomials, as well as other hash functions that appear computationally more efficient based on tables of pre-computed random numbers. 
     In one example, a 32 bit Adler hash based on the rolling checksum can be used as the hashing function for fingerprinting. This procedure provides a reasonably good random hash function by using a fixed table with 256 entries, each a precomputed 16 bit random number. The table is used to convert fingerprinted bytes into a random 16 bit number. The 32 bit hash is split into two 16 bit numbers sum1 and sum2, which are updated given the procedure:
 
sum1+=table[ b   k ]−table[ b   0 ]
 
sum2+=sum1− k *table[ b   0 ]
 
     In another example, a 64 bit random hash with cyclic shifting may be used as the hashing function for fingerprinting. The period of a cyclic shift is bounded by the size of the hash value. Thus, using a 64 bit hash value sets the period of the hash to 64. The procedure for updating the hash is given as:
 
hash=hash^((table[ b   0 ]&lt;&lt;1)|(table[ b   0   ]&gt;&gt;u ))^table[ b   k ];
 
hash=(hash&lt;&lt;1)|(hash&gt;&gt;63);
 
where 1 =k % 64 and  u= 64−1
 
     In still another example, other shifting methods may be employed to provide fingerprinting. Straight forward cyclic shifting produces a period of limited length, and is bounded by the size of the hash value. Other permutations have longer periods. For instance, the permutation given by the cycles (1 2 3 0) (5 6 7 8 9 10 11 12 13 14 4) (16 17 18 19 20 21 15) (23 24 25 26 22) (28 29 27) (31 30) has a period of length 4*3*5*7*11=4620. The single application of this example permutation can be computed using a right shift followed by operations that patch up the positions at the beginning of each interval. 
     Analysis of Previous Art for Chunking at Pre-Determined Patterns 
     Previous chunking methods are determined by computing a fingerprinting hash with a pre-determined window size k (=48), and identifying cut points based on whether a subset of the hash bits match a pre-determined pattern. With random hash values, this pattern may as well be 0, and the relevant subset may as well be a prefix of the hash. In basic instructions, this translates to a predicate of the form:
 
CutPoint(hash)≡0==(hash &amp; ((1&lt;&lt; c )−1)),
 
     where c is the number of bits that are to be matched against. 
     Since the probability for a match given a random hash function is 2 −c , an average chunk size C=2 c  results. However, neither the minimal, nor the maximal chunk size is determined by this procedure. If a minimal chunk length of m is imposed, then the average chunk size is:
 
 C=m+ 2 c  
 
     A rough estimate of the expected slack is obtained by considering streams s 1 s 3  and s 2 s 3 . Cut points in s 1  and s 2  may appear at arbitrary places. Since the average chunk length is C=m+2 c , about (2 c /C) 2  of the last cut-points in s 1  and s 2  will be beyond distance m. They will contribute to slack at around 2 c . The remaining 1−(2 c /C) 2  contribute with slack of length about C. The expected slack will then be around (2 c /C) 3 +(1−(2 c /C) 2 )*(C/C)=(2 c /C) 3 +1−(2 c /C) 2 , which has global minimum for m=2 c−1 , with a value of about 23/27=0.85. A more precise analysis gives a somewhat lower estimate for the remaining 1−(2 c /C) 2  fraction, but will also need to compensate for cuts within distance m inside s 3 , which contributes to a higher estimate. 
     Thus, the expected slack for the prior art is approximately 0.85*C. 
     Chunking at Filters (New Art) 
     Chunking at filters is based on fixing a filter, which is a sequence of patterns of length m, and matching the sequence of fingerprinting hashes against the filter. When the filter does not allow a sequence of hashes to match both a prefix and a suffix of the filter it can be inferred that the minimal distance between any two matches must be at least m. An example filter may be obtained from the CutPoint predicate used in the previous art, by setting the first m−1 patterns to
 
0!=(hash &amp; ((1 &lt;&lt;c )−1))
 
and the last pattern to:
 
0==(hash &amp; ((1 &lt;&lt;c )−1)).
 
     The probability for matching this filter is given by (1−p) m−1  p where p is 2 −c . One may compute that the expected chunk length is given by the inverse of the probability for matching a filter (it is required that the filter not allow a sequence to match both a prefix and suffix), thus the expected length of the example filter is (1−p) −m+1 p −1 . This length is minimized when setting p:=1/m, and it turns out to be around (e*m). The average slack hovers around 0.8, as can be verified by those skilled in the art. An alternative embodiment of this method uses a pattern that works directly with the raw input and does not use rolling hashes. 
     Chunking at Local Maxima (New Art) 
     Chunking at Local Maxima is based on choosing as cut points positions that are maximal within a bounded horizon. In the following, we shall use h for the value of the horizon. We say that the hash at position offset is an h-local maximum if the hash values at offsets offset−h, . . . , offset−1, as well as offset+1, offset+h are all smaller than the hash value at offset. In other words, all positions h steps to the left and h steps to the right have lesser hash values. Those skilled in the art will recognize that local maxima may be replaced by local minima or any other metric based comparison (such as “closest to the median hash value”). 
     The set of local maxima for an object of size n may be computed in time bounded by 2·n operations such that the cost of computing the set of local maxima is close to or the same as the cost of computing the cut-points based on independent chunking Chunks generated using local maxima always have a minimal size corresponding to h, with an average size of approximately 2h+1. A CutPoint procedure is illustrated in  FIGS. 8 and 9 , and is described as follows below:
         1. Allocate an array M of length h whose entries are initialized with the record {is Max=false, hash=0, offset=0}. The first entry in each field (isMax) indicates whether a candidate can be a local maximum. The second field entry (hash) indicates the hash value associated with that entry, and is initialized to 0 (or alternatively, to a maximal possible hash value). The last field (offset) in the entry indicates the absolute offset in bytes to the candidate into the fingerprinted object.   2. Initialize offsets min and max into the array M to 0. These variables point to the first and last elements of the array that are currently being used.   3. CutPoint(hash, offset) starts at step  800  in  FIG. 8  and is invoked at each offset of the object to update M and return a result indicating whether a particular offset is a cutpoint.
           The procedure starts by setting result=false at step  801 . At step  803 , the procedure checks whether M[max].offset+h+1=offset. If this condition is true, execution continues at step  804  where the following assignments are performed: result is set to M[max].is Max, and max is set to max−1% h. Execution then continues at step  805 . If the condition at step  803  is false, execution continues at step  805 . At step  805 , the procedure checks whether M[min].hash&gt;hash. If the condition is true, execution continues at step  806 , where min is set to (min−1) % h. Execution the continues at step  807  where M[min] is set to {isMax=false, hash=hash, offset=offset}, and to step  811 , where the computed result is returned.   If the condition at step  805  is false, execution continues to step  808 , where the procedure checks for whether M[min].hash=hash. If this condition is true, execution continues at step  807 .   If the condition at step  808  is false, execution continues at step  809 , where the procedure checks whether min=max. If this condition is true, execution continues at step  810 , where M[min] is set to {isMax=true, hash=hash, offset=offset}. Execution then continues at step  811 , where the computed result is returned.   If the condition at step  809  is false, execution continues at step  811 , where min is set to (min+1) % h. Execution then continues back at step  805 .   
           4. When CutPoint(hash, offset) returns true, it will be the case that the offset at position offset−h−1 is a new cut-point.
 
Analysis of Local Maximum Procedure
       

     An object with n bytes is processed by calling CutPoint n times such that at most n entries are inserted for a given object. One entry is removed each time the loop starting at step  805  is repeated such that there are no more than n entries to delete. Thus, the processing loop may be entered once for every entry and the combined number of repetitions may be at most n. This implies that the average number of steps within the loop at each call to CutPoint is slightly less than 2, and the number of steps to compute cut points is independent of h. 
     Since the hash values from the elements form a descending chain between min and max, we will see that the average distance between min and max (|min−max|% h) is given by the natural logarithm of h. Offsets not included between two adjacent entries in M have hash values that are less than or equal to the two entries. The average length of such chains is given by the recurrence equation f(n)=1+1/n*Σ k&lt;n  f(k). The average length of the longest descending chain on an interval of length n is 1 greater than the average length of the longest descending chain starting from the position of the largest element, where the largest element may be found at arbitrary positions with a probability of 1/n. The recurrence relation has as solution corresponding to the harmonic number H n =1+½+⅓+¼+ . . . +1/n, which can be validated by substituting H n  into the equation and performing induction on n. H n  is proportional to the natural logarithm of n. Thus, although array M is allocated with size h, only a small fraction of size ln(h) is ever used at any one time. 
     Computing min and max with modulus h permits arbitrary growth of the used intervals of M as long as the distance between the numbers remain within h. 
     The choice of initial values for M implies that cut-points may be generated within the first h offsets. The algorithm can be adapted to avoid cut-points at these first h offsets. 
     The expected size of the chunks generated by this procedure is around 2h+1. We obtain this number from the probability that a given position is a cut-point. Suppose the hash has m different possible values. Then the probability is determined by:
 
Σ 0≦k&lt;m 1 /m ( k/m ) 2h .
 
     Approximating using integration ∫ 0≦x&lt;m  1/m(x/m) 2h dx=1/(2h+1) indicates the probability when m is sufficiently large. 
     The probability can be computed more precisely by first simplifying the sum to:
 
(1/m) 2h+1 Σ 0≦k&lt;m  k 2h ,
 
which using Bernoulli numbers B k  expands to:
 
(1/m) 2h+1 1/(2h+1)Σ 0≦k&lt;2h (2h+1)!/k!(2h+1−k)!B k m 2h+1−k  
 
     The only odd Bernoulli number that is non-zero is B 1 , which has a corresponding value of −½. The even Bernoulli numbers satisfy the equation:
 
 H   ∞   (2n) =(−1) n−1 2 2n−1 π 2n   B   2n /( 2   n )!
 
     The left hand side represents the infinite sum 1+(½)2n+(⅓)2n+ . . . , which for even moderate values of n is very close to 1. 
     When m is much larger than h, all of the terms, except for the first can be ignored, as we saw by integration. They are given by a constant between 0 and 1 multiplied by a term proportional to h k−1 /m k . The first term (where B 0 =1) simplifies to 1/(2h+1). (the second term is −1/(2m), the third is h/(6 m 2 )). 
     For a rough estimate of the expected slack consider streams s 1 s 3  and s 2 s 3 . The last cut points inside s i  and s 2  may appear at arbitrary places. Since the average chunk length is about 2h+1 about ¼′th of the last cut-points will be within distance h in both s 1  and S 2 . They will contribute to cut-points at around ⅞ h. In another ½ of the cases, one cut-point will be within distance h the other beyond distance h. These contribute with cut-points around ¾h. The remaining ¼′th of the last cut-points in s 1  and s 2  will be in distance larger than h. The expected slack will therefore be around ¼*⅞+*½*¾+¼*¼=0.66. 
     Thus, the expected slack for our independent chunking approach is 0.66*C, which is an improvement over the prior art (0.85*C). 
     There is an alternate way of identifying cut-points that require executing in average fewer instructions while using space at most proportional to h, or in average ln h. The procedure above inserts entries for every position 0 . . . n−1 in a stream of length n. The basic idea in the alternate procedure is to only update when encountering elements of an ascending chain within intervals of length h. We observed that there will in average only be ln h such updates per interval. Furthermore, by comparing the local maxima in two consecutive intervals of length h one can determine whether each of the two local maxima may also be an h local maximum. There is one peculiarity with the alternate procedure; it requires computing the ascending chains by traversing the stream in blocks of size h, each block gets traversed in reverse direction. 
     In the alternate procedure (see  FIGS. 10 and 11 ), we assume for simplicity that a stream of hashes is given as a sequence. The subroutine CutPoint gets called for each subsequence of length h (expanded to “horizon” in the Figures). It returns zero or one offsets which are determined to be cut-points. Only ln(h) of the calls to Insert will pass the first test. 
     Insertion into A is achieved by testing the hash value at the offset against the largest entry in A so far. 
     The loop that updates both  A [k] and  B [k].is M ax can be optimized such that in average only one test is performed in the loop body. The case  B [┐].hash&lt;= A [k].hash and  B [┐].is M ax is handled in two loops, the first checks the hash value against  B [ℏ].hash until it is not less, the second updates  A [k]. The other case can be handled using a loop that only updates  A [k] followed by an update to  B [┐].is M ax. 
     Each call to CutPoint requires in average ln h memory writes to A, and with loop hoisting h+ln h comparisons related to finding maxima. The last update to  A [k].is M ax may be performed by binary search or by traversing B starting from index 0 in at average at most log ln h steps. Each call to CutPoint also requires re-computing the rolling hash at the last position in the window being updated. This takes as many steps as the size of the rolling hash window. 
     Observed Benefits of the Improved Chunking Algorithms 
     The minimal chunk size is built into both the local maxima and the filter methods described above. The conventional implementations require that the minimal chunk size is supplied separately with an extra parameter. 
     The local max (or mathematical) based methods produce measurable better slack estimate, which translates to further compression over the network. The filter method also produces better slack performance than the conventional methods. 
     Both of the new methods have a locality property of cut points. All cut points inside s 3  that are beyond horizon will be cut points for both streams s 1 s 3  and s 2 s 3 . (in other words, consider stream s 1 s 3 , if p is a position≧|s 1 |+horizon and p is a cut point in s 1 s 3 , then it is also a cut point in s 2 s 3 . The same property holds the other direction (symmetrically), if p is a cut point in s 2 s 3 , then it is also a cut point in s 1 s 3 ). This is not the case for the conventional methods, where the requirement that cuts be beyond some minimal chunk size may interfere adversely. 
     Alternative Mathematical Functions 
     Although the above-described chunking procedures describe a means for locating cut-points using a local maxima calculation, the present invention is not so limited. Any mathematical function can be arranged to examine potential cut-points. Each potential cut-point is evaluated by evaluating hash values that are located within the horizon window about a considered cut-point. The evaluation of the hash values is accomplished by the mathematical function, which may include at least one of locating a maximum value within the horizon, locating a minimum values within the horizon, evaluating a difference between hash values, evaluating a difference of hash values and comparing the result against an arbitrary constant, as well as some other mathematical or statistical function. 
     The particular mathematical function described previously for local maxima is a binary predicate “_&gt;_”. For the case where p is an offset in the object, p is chosen as a cut-point if hash p &gt;hash k , for all k, where p-horizon≦k&lt;p, or p&lt;k≦p+horizon. However, the binary predicate&gt;can be replaced with any other mathematical function without deviating from the spirit of the invention. 
     Finding Candidate Objects for Remote Differential Compression 
     The effectiveness of the basic RDC procedure described above may be increased by finding candidate objects on the receiver, for signature and chunk reuse during steps 4 and 8 of the RDC algorithm, respectively. The algorithm helps Device A identify a small subset of objects denoted by: O A1 , O A2 , . . . , O An  that are similar to the object O B  that needs to be transferred from Device B using the RDC algorithm. O A1 , O A2 , . . . , O An  are part of the objects that are already stored on Device A. 
     The similarity between two objects O B  and O A  is measured in terms of the number of distinct chunks that the two objects share divided by the total number of distinct chunks in the first object. Thus if Chunks(O B ) and Chunks(O A ) are the sets of chunks computed for O B  and O A  of the RDC algorithm, respectively, then, using the notation |X| to denote the cardinality, or number of elements, of set X: 
     
       
         
           
             
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     As a proxy for chunk equality, the equality on the signatures of the chunks is used. This is highly accurate if the signatures are computed using a cryptographically secure hash function (such as SHA-1 or MD5), given that the probability of a hash collision is extremely low. Thus, if Signatures(O B ) and Signatures(O A ) are the sets of chunk signatures computed for O B  and O A  in the chunking portion of the RDC algorithm, then: 
     
       
         
           
             
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     Given an object O B  and the set of objects Objects A  that are stored on Device A, the members of Objects A  that have a degree of similarity with O B  which exceeds a given threshold s are identified. A typical value for s may be s=0.5, (50% similarity) i.e. we are interested in objects that have at least half of their chunks in common with O B . The value for s, however, may be set at any value that makes sense for the application. For example, s could be set between 0.01 and 1.0 (1% similar to 100% similar). This set of objects is defined as:
 
Similar( O   B ,Objects A   ,s )={ O   A   |O   A ∈Objects A ^Similarity( O   B   ,O   A )≧ s} 
 
     The set of objects O A1 , O A2 , O An  is computed as a subset of Similar(O B , Objects A , s) by taking the best n matches. 
     The basic RDC algorithm described above is modified as follows to identify and use the set of similar objects O A1 , O A2 , O An . 
       FIG. 12  illustrates an RDC algorithm modified to find and use candidate objects, in accordance with aspects of the invention. The protocol for finding and using candidate objects on Device A and the transferring the updated object O B  from device B to device A is described. A similar protocol may be used to transfer an object from device A to device B, and the transfer can be initiated at the behest of either device A or device B without significantly changing the protocol described below.
         1. Device A sends device B a request to transfer Object O B  using the RDC protocol.   1.5 Device B sends Device A a set of traits of Object O B , Traits(O B ). Generally, the traits are a compact representation of the characteristics relating to object O B . As will be described later, Device B may cache the traits for O B  so that it does not need to recompute them prior to sending them to Device A.   1.6. Device A uses Traits(O B ) to identify O A1 , O A2 , . . . , O An , a subset of the objects that it already stores, that are similar to Object O B . This determination is made in a probabilistic manner.   2. Device A partitions the identified Objects O A1 , O A2 , . . . , O An  into chunks. The partitioning occurs in a data-dependent fashion, by using a fingerprinting function that is computed at every byte position of the objects. A chunk boundary is determined at positions for which the fingerprinting function satisfies a given condition. Following the partitioning into chunks, Device A computes a signature Sig Aik  for each chunk k of each Object O Ai .   3. Using a similar approach as in step 2, Device B partitions Object O B  into chunks, and computes the signatures Sig Bj  for each of the chunks. The partitioning algorithm used in step 3 must match the one in step 2 above.   4. Device B sends list of chunk signatures (Sig B1  . . . Sig Bn ) to Device A. This list provides the basis for Device A being able to reconstruct Object O B . In addition to the chunk signatures Sig Bi , information will be sent about the offset and length of each chunk in Object O B .   5. As Device A receives the chunk signatures from Device B, it compares the received signatures against the set of signatures (Sig A11 , Sig A1m , . . . , Sig An1 , . . . Sig An1 ) that it has computed in step 2. As part of this comparison, Device A records every distinct signature value it received from Device B that does not match one of its own signatures Sig Aik  computed on the chunks of Objects O A1 , O A2 , . . . , O An .   6. Device A sends a request to Device B for all the chunks whose signatures were received in the previous step from Device B, but which did not have a matching signature on Device A. The chunks are requested by offset and length in Object O B , based on corresponding information that was sent in Step 4.   7. Device B sends the content associated with all the requested chunks to device A.   8. Device A reconstructs Object O B  by using the chunks received in step 6 from Device B, as well as its own chunks of objects O A1 , O A2 , . . . , O An  that matched signatures sent by Device B in step 4. After this reconstruction step is complete, Device A may now add the reconstructed copy of Object O B  to its already stored objects.       

     To minimize network traffic and CPU overhead, Traits(O B ) should be very small and the determination of the set of similar objects O A1 , O A2 , . . . , O An  be performed with very few operations on Device A. 
     Computing the Set of Traits for an Object 
     The set of traits for a object O, Traits(O), is computed based on the chunk signatures computed for O, as described for steps 2 or 3 of the RDC algorithm, respectively. 
       FIGS. 13 and 14  show a process and an example of a trait computation, in accordance with aspects of the invention. 
     The algorithm for identifying similar objects has four main parameters (q, b, t, x) that are summarized below. 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 q 
                 Shingle size 
               
               
                   
                 b 
                 Number of bits per trait 
               
               
                   
                 t 
                 Number of traits per object 
               
               
                   
                 x 
                 Minimum number of matching traits 
               
               
                   
                   
               
            
           
         
       
     
     The following steps are used to compute the traits for object O, Traits(O).
         1. At block  1310 , the chunk signatures of O, Sig 1  . . . Sig n  are grouped together into overlapping shingles of size q, where every shingle comprises q chunk signatures, with the exception of the last q−1 shingles, which will contain fewer than q signatures. Other groupings (discontiguous subsets, disjoint subsets, etc.) are possible, but it is practically useful that inserting an extra signature causes all of the previously considered subsets to still be considered.   2. At block  1320 , for each shingle  1  . . . n, a shingle signature Shingle 1  . . . Shingl n  is computed by concatenating the q chunk signatures forming the shingle. For the case where q=1, Shingle 1 =Sig 1 , . . . , Shingle n =Sig n .   3. At block  1330 , the shingle set {Shingle 1  . . . . Shingle n } is mapped into t image sets through the application of t hash functions H 1  . . . H t . This generates t image sets, each containing n elements:
 
 IS   1   ={H   1 (Shingle 1 ),  H   1 (Shingle 2 ), . . . ,  H   1 (Shingle n )}
 
. . .
 
 IS   t   ={H   t (Shingle 1 ),  H   t (Shingle 2 ), . . . ,  H   t (Shingle n }
   4. At block  1340 , the pre-traits PT 1  . . . PT t  are computed by taking the minimum element of each image set:
 
PT 1 =min( IS   1 )
 
. . .
 
PT t =min( IS   t ).
           Other deterministic mathematical functions may also be used to compute the pre-traits. For example, the pre-traits PT 1  . . . PT t  are computed by taking the maximum element of each image set:
 
PT 1 =max( IS   1 )
 
. . .
 
PT t =max( IS   t ).
   Mathematically, any mapping carrying values into a well-ordered set will suffice, max and min on bounded integers being two simple realizations.   
           5. At block  1350 , the traits T 1  . . . T t  are computed by selecting b bits out of each pre-trait PT 1  . . . PT t . To preserve independence of the samples, it is better to choose non-overlapping slices of bits, 0 . . . b−1 for the first, b . . . 2b−1 for the second, etc, if the pre-traits are sufficiently long:
 
 T   1 =select 0 . . . b−1 (PT 1 ).
 
. . .
 
 T   t =select (t−1)b . . . tb− 1(PT t )
           Any deterministic function may be used to create traits that are smaller in size than the pre-traits. For instance, a hash function could be applied to each of the pre-traits so long as the size of the result is smaller than the pre-trait; if the total number of bits needed (tb) exceeds the size of a pre-trait, some hash functions should be used to expand the number of bits before selecting subsets.   
               

     The number of traits t and the trait size b are chosen so that only a small total number of bits (t*b) is needed to represent the traits for an object. This is advantageous if the traits are precomputed and cached by Device A, as will be described below. According to one embodiment, some typical combinations of (b,t) parameters that have been found to work well are e.g. (4,24) and (6,16), for a total of 96 bits per object. Any other combinations may also be used. For purposes of explanation, the i th  trait of object A will be denoted by T i (A). 
     Efficiently Selecting the Pre-traits 
     To efficiently select the pre-traits PT 1  . . . PT t , the following approach is used, allowing partial evaluation of the shingles, and thus reducing the computational requirements for selecting the pre-traits. Logically, each H i  is divided into two parts, High i  and Low i . Since only the minimum element of each image set is selected, the High i  is computed for every chunk signature and the Low i  is computed only for those chunk signatures which achieve the minimum value ever achieved for High i . If the High values are drawn from a smaller space, this may save computation. If, further, several High values are bundled together, significant computation may be saved. Suppose, for instance, that each High value is 8 bits long. Eight of these can be packed into a long integer; at the cost of computing a single 8-byte hash from a signature, that value can be chopped into eight independent one byte-slices. If only the High value were needed, this would reduce computational costs by a factor of eight. However, on average one time in 256 a corresponding Low value needs to be computed and compared to other Low values corresponding to equal High values. 
     Finding Similar Objects Using the Sets of Traits 
     The algorithm approximates the set of objects similar to a given object O B  by computing the set of objects having similar traits to O B :
 
Traitsimilarity( O   B   ,O   A )=|{ i|T   i ( A )= T   i ( B )}|
 
SimilarTraits( O   B ,Objects A   ,x )={ O   A   |O   A ∈Objects A {circumflex over ( 0 )}TraitSimilarity( O   B   ,O   A )≧ t} 
 
     Other computations from which these values might be derived would work just as well. 
     To select the n most similar objects to a given object O B , SimilarTraits(O B , Objects A , x) is computed and the n best matching objects out of that set are taken. If the size of SimilarTraits(O B , Objects A , x) is smaller than n, the entire set is taken. The resulting set of objects forms a potential set of objects O A1 , O A2 , . . . , O An  identified in step 1.6 of the modified RDC algorithm illustrated in  FIG. 12 . 
     According to the embodiments, objects may be chosen guided by similarity, but trying also to increase diversity in the set of objects by choosing objects similar to the target, but dissimilar from one another, or by making other choices from the set of objects with similar traits. 
     According to one embodiment, the following combinations of parameters (q,b,t,x) may be used: (q=1,b=4,t=24,x=9) and (q=1,b=6,t=16,x=5). 
       FIGS. 15 and 16  may be used when selecting the parameters for b and t, in accordance with aspects of the present invention. The curves for the probability of detecting matches and for false positives first for (b=4, t=24) is shown in  FIG. 15 , and then for (b=6, t=16) is shown in  FIG. 16 . Both sets of similarity curves ( 1510  and  1610 ) allow the probabilistic detection similar objects with true similarity in the range of 0-100%. According to one embodiment, the false positive rate illustrated in displays  1520  and  1620  drops to an acceptable level at roughly 10 of 24 (providing 40 bits of true match), and at 6 of 16 (36 bits of match); the difference in the required number of bits is primarily due to the reduced number of combinations drawing from a smaller set. The advantage of the larger set is increased recall: fewer useful matches will escape attention; the cost is the increased rate of falsely detected matches. To improve both precision and recall, the total number of bits may be increased. Switching to (b=5, t=24), for instance would dramatically improve precision, at the cost of increasing memory consumption for object traits. 
     A Compact Representation for the Sets of Traits 
     It is advantageous for both Device A and Device B to cache the sets of traits for all of their stored objects so that they don&#39;t have to recompute their traits every time they execute steps 1.6 and 1.5, respectively, of the modified RDC algorithm (See  FIG. 12  and related discusssion). To speed up the RDC computation, the trait information may be stored in Device A&#39;s and Device B&#39;s memory, respectively. 
     The representation described below uses on the order of t+p memory bytes per object, where t is the number of traits and p is the number of bytes required to store a reference or a pointer to the object. Examples of references are file paths, file identifiers, or object identifiers. For typical values oft and p, this approach can support one million objects using less than 50 MB of main memory. If a device stores more objects, it may use a heuristic to prune the number of objects that are involved in the similarity computation. For instance, very small objects may be eliminated a priori because they cannot contribute too many chunks in steps 4 and 8 of the RDC algorithm illustrated in  FIG. 12 . 
       FIG. 17  illustrates data structures that make up a compact representation of: an ObjectMap and a set of t Trait Tables, in accordance with aspects of the invention. 
     Initially, short identifiers, or object IDs, are assigned to all of the objects. According to one embodiment, these identifiers are consecutive non-negative 4-byte integers, thus allowing the representation of up to 4 Billion objects. 
     A data structure (ObjectMap) maintains the mapping from object IDs to object references. It does not matter in which order objects stored on a device get assigned object IDs. Initially, this assignment can be done by simply scanning through the device&#39;s list of stored objects. If an object gets deleted, its corresponding entry in ObjectMap is marked as a dead entry (by using a reserved value for the object reference). If an object is modified, it corresponding entry in ObjectMap is marked as a dead entry, and the object gets assigned the next higher unused object ID. 
     When the ObjectMap becomes too sparse (something that can be easily determined by keeping track of the total size and the number of dead entries), both the ObjectMap and the Trait Tables are discarded and rebuilt from scratch. 
     The Trait Tables form a two-level index that maps from a trait number (1 to t) and a trait value (0 to 2 b −1) to a TraitSet, the set of object IDs for the objects having that particular trait. A TraitSet is represented as an array with some unused entries at the end for storing new objects. An index IX i,k  keeps track of the first unused entry in each TraitSet array to allow for appends. 
     Within a TraitSet, a particular set of objects is stored in ascending order of object IDs. Because the space of object IDs is kept dense, consecutive entries in the TraitSets can be expected to be “close” to each other in the object ID space on average, two consecutive entries should differ by about t*2 b  (but by at least 1). If the values oft and b are chosen so that t*2 b &lt;&lt;255, then consecutive entries can be encoded using on average only one unsigned byte representing the difference between the two object ID, as shown in  FIG. 17 . An escape mechanism is provided by using the 0x00 byte to indicate that a full 4-byte object ID follows next, for the rare cases where the two consecutive object IDs differ by more than 255. 
     According to a different embodiment, if an object ID difference is smaller than 256 then it can be represented as a single byte, otherwise the value zero is reserved to indicate that subsequent bytes represent the delta minus 256, say, by using a 7 in 8 representation. Then, for b=6, 98% of deltas will fit in one byte, 99.7% fit in two bytes, and all but twice in a billion into three bytes. It has been found that this scheme uses on average 1.02 bytes per object, compared to 1.08 bytes per object for the scheme shown in  FIG. 17 . 
     Entries in the Trait Tables corresponding to dead object IDs can be left in the Trait Tables. New entries are appended at the end (using indices IX 1,0  . . . IX t,2     b     −1 ). 
     Finding Similar Objects using the Compact Representation 
       FIG. 18  illustrates a process for finding objects with similar traits, in accordance with aspects of the invention. According to one embodiment, to compute SimilarTraits(O B , Objects A , x), the steps are are similar to a merge sort algorithm. The algorithm uses (t−x+1) object buckets, OB x OB t , that are used to store objects belonging to Objects A  that match at least x and up to and including t traits of O B , respectively.
         1. At block  1810 , select the t TraitSets corresponding to the t traits of O B : TS 1  . . . TS t . Initialize OB x OB t  to empty. Initialize indices Pi . . . P t  to point to the first element of TS 1  . . . TS t , respectively. TS k [P k ] is the notation for the object ID pointed to by P k .   2. At decision block  1820 , if all of P 1  . . . P t  point past the last element of their TraitSet arrays TS 1  . . . TS t , respectively, then go to step 6 (block  1860 ).   3. At block  1830 , the MinP set is selected which is the set of indices pointing to the minimum object ID, as follows:
 
Min P={P   k   |Λj∉[ 1 ,t]·TS   j   └P   j   ┘≧TS   k   └P   k ┘}
           Let MinID be the minimum object ID pointed to by all the indices in MinP.   
           4. At block  1840 , Let k=|MinP|, which corresponds to the number of matching traits. If k&gt;x and if ObjectMap(MinP) is not a dead entry, then append MinID to OB k .   5. Advance every index P k  in MinP to the next object ID in its respective TraitSet array TS k . Go to step 2 (block  1820 ).   6. At block  1860 , select the similar objects by first selecting objects from OB t , then from OB t−1 , etc., until the desired number of similar objects has been selected or no more objects are left in OB x . The object IDs produced by the above steps can be easily mapped into object references by using the ObjectMap.       

     The above specification, examples and data provide a complete description of the manufacture and use of the composition of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended.