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311 lines
14 KiB
TeX
311 lines
14 KiB
TeX
\documentclass[a4paper]{article}
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\begin{document}
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\title{The rsync algorithm}
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\author{Andrew Tridgell \quad\quad Paul Mackerras\\
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Department of Computer Science \\
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Australian National University \\
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Canberra, ACT 0200, Australia}
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\maketitle
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\begin{abstract}
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This report presents an algorithm for updating a file on one machine
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to be identical to a file on another machine. We assume that the
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two machines are connected by a low-bandwidth high-latency
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bi-directional communications link. The algorithm identifies parts
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of the source file which are identical to some part of the
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destination file, and only sends those parts which cannot be matched
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in this way. Effectively, the algorithm computes a set of
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differences without having both files on the same machine. The
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algorithm works best when the files are similar, but will also
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function correctly and reasonably efficiently when the files are
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quite different.
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\end{abstract}
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\section{The problem}
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Imagine you have two files, $A$ and $B$, and you wish to update $B$ to be
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the same as $A$. The obvious method is to copy $A$ onto $B$.
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Now imagine that the two files are on machines connected by a slow
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communications link, for example a dialup IP link. If $A$ is large,
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copying $A$ onto $B$ will be slow. To make it faster you could
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compress $A$ before sending it, but that will usually only gain a
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factor of 2 to 4.
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Now assume that $A$ and $B$ are quite similar, perhaps both derived
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from the same original file. To really speed things up you would need
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to take advantage of this similarity. A common method is to send just
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the differences between $A$ and $B$ down the link and then use this
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list of differences to reconstruct the file.
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The problem is that the normal methods for creating a set of
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differences between two files rely on being able to read both files.
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Thus they require that both files are available beforehand at one end
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of the link. If they are not both available on the same machine,
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these algorithms cannot be used (once you had copied the file over,
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you wouldn't need the differences). This is the problem that rsync
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addresses.
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The rsync algorithm efficiently computes which parts of a source file
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match some part of an existing destination file. These parts need not
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be sent across the link; all that is needed is a reference to the part
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of the destination file. Only parts of the source file which are not
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matched in this way need to be sent verbatim. The receiver can then
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construct a copy of the source file using the references to parts of
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the existing destination file and the verbatim material.
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Trivially, the data sent to the receiver can be compressed using any
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of a range of common compression algorithms, for further speed
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improvements.
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\section{The rsync algorithm}
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Suppose we have two general purpose computers $\alpha$ and $\beta$.
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Computer $\alpha$ has access to a file $A$ and $\beta$ has access to
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file $B$, where $A$ and $B$ are ``similar''. There is a slow
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communications link between $\alpha$ and $\beta$.
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The rsync algorithm consists of the following steps:
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\begin{enumerate}
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\item $\beta$ splits the file $B$ into a series of non-overlapping
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fixed-sized blocks of size S bytes\footnote{We have found that
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values of S between 500 and 1000 are quite good for most purposes}.
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The last block may be shorter than $S$ bytes.
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\item For each of these blocks $\beta$ calculates two checksums:
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a weak ``rolling'' 32-bit checksum (described below) and a strong
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128-bit MD4 checksum.
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\item $\beta$ sends these checksums to $\alpha$.
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\item $\alpha$ searches through $A$ to find all blocks of length $S$
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bytes (at any offset, not just multiples of $S$) that have the same
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weak and strong checksum as one of the blocks of $B$. This can be
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done in a single pass very quickly using a special property of the
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rolling checksum described below.
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\item $\alpha$ sends $\beta$ a sequence of instructions for
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constructing a copy of $A$. Each instruction is either a reference
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to a block of $B$, or literal data. Literal data is sent only for
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those sections of $A$ which did not match any of the blocks of $B$.
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\end{enumerate}
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The end result is that $\beta$ gets a copy of $A$, but only the pieces
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of $A$ that are not found in $B$ (plus a small amount of data for
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checksums and block indexes) are sent over the link. The algorithm
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also only requires one round trip, which minimises the impact of the
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link latency.
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The most important details of the algorithm are the rolling checksum
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and the associated multi-alternate search mechanism which allows the
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all-offsets checksum search to proceed very quickly. These will be
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discussed in greater detail below.
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\section{Rolling checksum}
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The weak rolling checksum used in the rsync algorithm needs to have
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the property that it is very cheap to calculate the checksum of a
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buffer $X_2 .. X_{n+1}$ given the checksum of buffer $X_1 .. X_n$ and
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the values of the bytes $X_1$ and $X_{n+1}$.
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The weak checksum algorithm we used in our implementation was inspired
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by Mark Adler's adler-32 checksum. Our checksum is defined by
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$$ a(k,l) = (\sum_{i=k}^l X_i) \bmod M $$
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$$ b(k,l) = (\sum_{i=k}^l (l-i+1)X_i) \bmod M $$
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$$ s(k,l) = a(k,l) + 2^{16} b(k,l) $$
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where $s(k,l)$ is the rolling checksum of the bytes $X_k \ldots X_l$.
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For simplicity and speed, we use $M = 2^{16}$.
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The important property of this checksum is that successive values can
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be computed very efficiently using the recurrence relations
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$$ a(k+1,l+1) = (a(k,l) - X_k + X_{l+1}) \bmod M $$
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$$ b(k+1,l+1) = (b(k,l) - (l-k+1) X_k + a(k+1,l+1)) \bmod M $$
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Thus the checksum can be calculated for blocks of length S at all
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possible offsets within a file in a ``rolling'' fashion, with very
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little computation at each point.
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Despite its simplicity, this checksum was found to be quite adequate as
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a first-level check for a match of two file blocks. We have found in
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practice that the probability of this checksum matching when the
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blocks are not equal is quite low. This is important because the much
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more expensive strong checksum must be calculated for each block where
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the weak checksum matches.
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\section{Checksum searching}
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Once $\alpha$ has received the list of checksums of the blocks of $B$,
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it must search $A$ for any blocks at any offset that match the
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checksum of some block of $B$. The basic strategy is to compute the
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32-bit rolling checksum for a block of length $S$ starting at each
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byte of $A$ in turn, and for each checksum, search the list for a
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match. To do this our implementation uses a
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simple 3 level searching scheme.
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The first level uses a 16-bit hash of the 32-bit rolling checksum and
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a $2^{16}$ entry hash table. The list of checksum values (i.e., the
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checksums from the blocks of $B$) is sorted according to the 16-bit
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hash of the 32-bit rolling checksum. Each entry in the hash table
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points to the first element of the list for that hash value, or
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contains a null value if no element of the list has that hash value.
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At each offset in the file the 32-bit rolling checksum and its 16-bit
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hash are calculated. If the hash table entry for that hash value is
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not a null value, the second-level check is invoked.
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The second-level check involves scanning the sorted checksum list
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starting with the entry pointed to by the hash table entry, looking
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for an entry whose 32-bit rolling checksum matches the current value.
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The scan terminates when it reaches an entry whose 16-bit hash
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differs. If this search finds a match, the third-level check is
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invoked.
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The third-level check involves calculating the strong checksum for the
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current offset in the file and comparing it with the strong checksum
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value in the current list entry. If the two strong checksums match,
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we assume that we have found a block of $A$ which matches a block of
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$B$. In fact the blocks could be different, but the probability of
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this is microscopic, and in practice this is a reasonable assumption.
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When a match is found, $\alpha$ sends $\beta$ the data in $A$ between
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the current file offset and the end of the previous match, followed by
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the index of the block in $B$ that matched. This data is sent
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immediately a match is found, which allows us to overlap the
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communication with further computation.
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If no match is found at a given offset in the file, the rolling
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checksum is updated to the next offset and the search proceeds. If a
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match is found, the search is restarted at the end of the matched
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block. This strategy saves a considerable amount of computation for
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the common case where the two files are nearly identical. In
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addition, it would be a simple matter to encode the block indexes as
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runs, for the common case where a portion of $A$ matches a series of
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blocks of $B$ in order.
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\section{Pipelining}
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The above sections describe the process for constructing a copy of one
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file on a remote system. If we have a several files to copy, we can
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gain a considerable latency advantage by pipelining the process.
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This involves $\beta$ initiating two independent processes. One of the
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processes generates and sends the checksums to $\alpha$ while the
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other receives the difference information from $\alpha$ and
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reconstructs the files.
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If the communications link is buffered then these two processes can
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proceed independently and the link should be kept fully utilised in
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both directions for most of the time.
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\section{Results}
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To test the algorithm, tar files were created of the Linux kernel
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sources for two versions of the kernel. The two kernel versions were
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1.99.10 and 2.0.0. These tar files are approximately 24MB in size and
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are separated by 5 released patch levels.
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Out of the 2441 files in the 1.99.10 release, 291 files had changed in
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the 2.0.0 release, 19 files had been removed and 25 files had been
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added.
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A ``diff'' of the two tar files using the standard GNU diff utility
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produced over 32 thousand lines of output totalling 2.1 MB.
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The following table shows the results for rsync between the two files
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with a varying block size.\footnote{All the tests in this section were
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carried out using rsync version 0.5}
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\vspace*{5mm}
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\begin{tabular}{|l|l|l|l|l|l|l|} \hline
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{\bf block} & {\bf matches} & {\bf tag} & {\bf false} & {\bf data} & {\bf written} & {\bf read} \\
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{\bf size} & & {\bf hits} & {\bf alarms} & & & \\ \hline \hline
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300 & 64247 & 3817434 & 948 & 5312200 & 5629158 & 1632284 \\ \hline
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500 & 46989 & 620013 & 64 & 1091900 & 1283906 & 979384 \\ \hline
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700 & 33255 & 571970 & 22 & 1307800 & 1444346 & 699564 \\ \hline
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900 & 25686 & 525058 & 24 & 1469500 & 1575438 & 544124 \\ \hline
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1100 & 20848 & 496844 & 21 & 1654500 & 1740838 & 445204 \\ \hline
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\end{tabular}
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\vspace*{5mm}
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In each case, the CPU time taken was less than the
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time it takes to run ``diff'' on the two files.\footnote{The wall
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clock time was approximately 2 minutes per run on a 50 MHz SPARC 10
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running SunOS, using rsh over loopback for communication. GNU diff
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took about 4 minutes.}
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The columns in the table are as follows:
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\begin{description}
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\item [block size] The size in bytes of the checksummed blocks.
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\item [matches] The number of times a block of $B$ was found in $A$.
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\item [tag hits] The number of times the 16-bit hash of the rolling
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checksum matched a hash of one of the checksums from $B$.
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\item [false alarms] The number of times the 32-bit rolling checksum
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matched but the strong checksum didn't.
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\item [data] The amount of file data transferred verbatim, in bytes.
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\item [written] The total number of bytes written by $\alpha$,
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including protocol overheads. This is almost all file data.
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\item [read] The total number of bytes read by $\alpha$, including
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protocol overheads. This is almost all checksum information.
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\end{description}
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The results demonstrate that for block sizes above 300 bytes, only a
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small fraction (around 5\%) of the file was transferred. The amount
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transferred was also considerably less than the size of the diff file
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that would have been transferred if the diff/patch method of updating
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a remote file was used.
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The checksums themselves took up a considerable amount of space,
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although much less than the size of the data transferred in each
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case. Each pair of checksums consumes 20 bytes: 4 bytes for the
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rolling checksum plus 16 bytes for the 128-bit MD4 checksum.
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The number of false alarms was less than $1/1000$ of the number of
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true matches, indicating that the 32-bit rolling checksum is quite
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good at screening out false matches.
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The number of tag hits indicates that the second level of the
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checksum search algorithm was invoked about once every 50
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characters. This is quite high because the total number of blocks in
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the file is a large fraction of the size of the tag hash table. For
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smaller files we would expect the tag hit rate to be much closer to
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the number of matches. For extremely large files, we should probably
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increase the size of the hash table.
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The next table shows similar results for a much smaller set of files.
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In this case the files were not packed into a tar file first. Rather,
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rsync was invoked with an option to recursively descend the directory
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tree. The files used were from two source releases of another software
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package called Samba. The total source code size is 1.7 MB and the
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diff between the two releases is 4155 lines long totalling 120 kB.
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\vspace*{5mm}
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\begin{tabular}{|l|l|l|l|l|l|l|} \hline
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{\bf block} & {\bf matches} & {\bf tag} & {\bf false} & {\bf data} & {\bf written} & {\bf read} \\
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{\bf size} & & {\bf hits} & {\bf alarms} & & & \\ \hline \hline
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300 & 3727 & 3899 & 0 & 129775 & 153999 & 83948 \\ \hline
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500 & 2158 & 2325 & 0 & 171574 & 189330 & 50908 \\ \hline
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700 & 1517 & 1649 & 0 & 195024 & 210144 & 36828 \\ \hline
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900 & 1156 & 1281 & 0 & 222847 & 236471 & 29048 \\ \hline
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1100 & 921 & 1049 & 0 & 250073 & 262725 & 23988 \\ \hline
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\end{tabular}
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\vspace*{5mm}
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\section{Availability}
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An implementation of rsync which provides a convenient interface
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similar to the common UNIX command rcp has been written and is
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available for download from http://rsync.samba.org/
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\end{document}
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