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An atomic system transition $t_i$ chains one -system state into another, making one ACID change on the system. The -usual atomic transitions are the single package install and remove -operations found in low-level package management code of PISI. Note -that in PISI, an upgrade operation is identical to a remove operation -followed by an install operation (which sets it apart from some other -packaging systems). +A system state $S_i$ is defined as the set of installed packages on +the system together with their versions, i.e. $S_i = \{ (x,v) : x +\text{ is installed}, v=version(x)\} $. An atomic system transition +$t_i$ chains one system state into another, making one ACID change on +the system. The usual atomic transitions are the single package +install, remove and reinstall (upgrade or downgrade) operations found +in low-level package management code of PISI. Note that in PISI, an +upgrade operation is identical to a remove operation followed by an +install operation (which sets it apart from some other packaging +systems). A package operation plan is thus naturally conceived of as a sequence of atomic system transitions. Given an initial state and a final @@ -116,39 +118,39 @@ look at the dependency condition. \subsection{Solving the simplest case with topological sorting} -We will now concentrate on a simple form of the problem which can -be solved with topological sorting. This form is not concerned with -versions. From initial set of packages $S_0$, we would like to +We will now concentrate on a simple form of the problem which can be +solved with topological sorting. This form is not concerned with +versions. From an initial set of packages $S_0$, we would like to install in addition a new set $A$ of packages obtaining $S_f = S_0 \cup A$. The only relations considered are of the form: $a$ Depends on $b$, or more briefly $aDb$. -The graph of all such simple dependency relations is a directed graph -(digraph) $G$. For each dependency relation $aDB$, there is an edge $a -\to b$ in $G$. Accessing graph $G$ usually requires a database operation and -is therefore expensive. +The graph of all such simple dependency relations is a digraph $G$. +For each dependency relation $aDb$, there is an edge $a \to b$ in $G$. +Accessing graph $G$ usually requires a database operation and is +therefore expensive. We now consider the digraph $G_A$ of the minimal set of simple dependency relations which contains all information required to -construct a plan to install packages $A$. $G_A$ is a vertex induced graph -such that the fringe of $A$, e.g. vertices with out-degree $0$ are -already installed. Vertices of $G_A$ are taken from $S_f$. First, let us -explain the labelling scheme. Already installed vertices are labelled -with 'i'. Packages to be added are labelled with 'a', and packages to -be installed due to dependencies are labelled with 'd'. We -construct the graph as follows +construct a plan to install packages $A$. $G_A$ is a vertex induced +graph such that the fringe of $A$, e.g. vertices with out-degree $0$ +depend only on packages that are already installed (or none). Vertices +of $G_A$ are taken from $S_f$. First, let us explain the labelling +scheme. Already installed vertices are labelled with 'i'. Packages to +be added are labelled with 'a', and packages to be installed due to +dependencies are labelled with 'd'. We construct the graph as follows \begin{algorithm} - %\caption{$\textsc{Par-Freq}(T_i, \epsilon, \textsc{Mine-Freq} )$} + \caption{$\textsc{Make-}G_A(G, A)$} \label{alg:cons-graph} \begin{algorithmic}[1] -\STATE $G_A \gets$ isolated vertex set $A$ labelled with 'a' +\STATE $G_A \gets$ vertex induced subgraph of $G$ by $A$ labelled with 'a' \REPEAT \STATE done $\gets$ true - \FOR{each $u \in V_A$ with out-degree $0$} + \FOR{each $u \in V(G_A)$ with out-degree $0$} \FOR{ $v \in adj(u) $ of $G$} - \IF{$v \notin V_A$} + \IF{$v \notin V(G_A)$} \STATE done $\gets$ false \IF{$v$ is installed} \STATE label $v$ with 'i' @@ -281,24 +283,29 @@ multi-package upgrade, the exact details of the goal state depend on the graph $G_f$ of the repository. Thus, we construct a graph $G_{A,f}$ that is a vertex-induced subgraph of $G_f$ such that it contains all information relevant to upgrading packages $A$. We begin -by a vertex induced graph by $A$. These are the packages that will be -upgraded in any case. Then, we make a pass on the vertices, and look -at all the outgoing edges, we compare whether this edge has changed in -any substantial way from the previous version. In particular, we are -interested in whether the predicate of the edge is valid for the -version of the same package in our current system. Every compared -vertex in this manner is marked done, and the edges not valid for the -current system pull new unmarked vertices into $G_f$, this continues -until there are no unmarked vertices left. Hence, the vertices of -$G_f$ are the packages that must be upgraded. +by a vertex induced subgraph of $G_f$ by $A$. These are the packages +that will be upgraded in any case. Then, we make a pass on the +vertices, and look at all the outgoing edges, we compare whether this +edge has changed in any substantial way from the previous version. In +particular, we are interested in whether the predicate of the edge is +valid for the version of the same package in our current system. Every +compared vertex in this manner is marked done, and the edges not valid +for the current system pull new unmarked vertices into $G_f$, this +continues until there are no unmarked vertices left. Hence, the +vertices of $G_f$ are the packages that must be upgraded. -To actually carry out the upgrade a strategy is to upgrade one by one -all the packages in some order. A good order is again the reverse -topological order order, in fact, the operation is merely a special case of a -multi-package installation code that can install from a remote -repository. However, in case no package depends on the packages to be -upgraded, then we can carry out a completely consistency-preserving -plan as discussed above. +To actually carry out the upgrade a strategy is to upgrade all the +packages in $G_{A,f}$ in some order. A good order is again the reverse +topological order order, in fact, the upgrade operation is merely a +special case of a multi-package installation code that can install +from a remote repository, since a multi-package installation can +contain upgrades in addition to new packages. However, in case no +package depends on the packages to be upgraded, then we can carry out +a completely consistency-preserving plan as discussed above. The +conflicts are resolved in the usual fashion, by removing those +packages in conflict with new packages that are installed. This can be +accomplished by invoking a remove operation prior to the upgrade +operation. \section{Examples} @@ -317,7 +324,6 @@ initial state:\\ In this case, we can find a consistency-preserving plan in terms of install and remove operations. - \\ plan:\\ remove $(a,1)$\\ @@ -339,10 +345,10 @@ chain. In fact, here there is no consistency-preserving plan in terms of atomic single package transitions: install, remove, upgrade. In these cases, it seems best to resort to upgrade in place, and in the reverse topological order of dependencies. - +\\ plan:\\ - upgrade $(c,1) \to (c,2)$ - upgrade $(b,1) \to (b,2)$ + upgrade $(c,1) \to (c,2)$\\ + upgrade $(b,1) \to (b,2)$\\ \subsection{A multi package remove}