%From dechter@ramat-aviv.ics.uci.edu Tue Jul 6 15:19:37 1999 %Date: Fri, 18 Jun 1999 10:36:55 -0700 %From: Rina Dechter %To: simon@cs.uchicago.edu %Cc: dolev@cs.bgu.ac.il %Subject: paper.tex \documentclass[12pt]{article} \usepackage{graphicx} %proof \usepackage{latexsym} %for Box symbol \def\QED{$\Box$} \newenvironment{proof} {\begin{trivlist}\item[\hskip\labelsep{\textbf{Proof\,}}]} {\leavevmode\unskip\nobreak\quad\hspace*{\fill}\QED\end{trivlist}} %ital operators \def\operator#1{\mathop{\it#1}\nolimits} \newcommand{\parent}{{\operator{parent}}} \newcommand{\children}{{\operator{children}}} \newcommand{\ancestors}{{\operator{ancestors}}} \newcommand{\predecessors}{{\operator{predecessors}}} \newcommand{\descendants}{{\operator{descendants}}} \newcommand{\mode}{{\operator{mode}}} \newcommand{\Domain}{{\operator{Domain}}} \newcommand{\tag}{{\operator{tag}}} \newcommand{\pointer}{{\operator{pointer}}} \newcommand{\direction}{{\operator{direction}}} \newcommand{\domain}{{\operator{domain}}} \newcommand{\Value}{{\operator{value}}} \newcommand{\val}{{\operator{val}}} \newcommand{\setofnodes}{{\operator{set\_of\_nodes}}} \newcommand{\Root}{{\operator{root}}} \newcommand{\Dim}{{\operator{dim}}} \newcommand{\shrink}[1]{} \newcommand{\tree}{minimally labeled tree} \newcommand{\lm}[1]{\mbox{$\ell_{#1}$}} \newcommand{\pl}[1]{\wp({#1})} %\setlength{\textwidth}{5.2in} % for 12pt %\setlength{\textheight}{8.5in} % for 12pt %\setlength{\topmargin}{0in} % for 12pt %\setlength{\headheight}{0in} %\setlength{\headsep}{0in} % DELIMITER PAIRS AND MATHEMATICAL FUNCTIONS \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\abs}[1]{\left| #1\right|} \newcommand{\card}[1]{\left| #1\right|} \newcommand{\ord}[1]{O\left( #1\right)} \newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\ang}[1]{\left\langle #1 \right\rangle} \newcommand{\prob}[1]{\Pr\left\{ #1 \right\}} % BINARY OPERATION, RELATION, AND ARROW SYMBOLS \newcommand{\propsubset} {\stackrel{\subset}{\scriptscriptstyle \not {\scriptstyle =}}} \newcommand{\dhrightarrow}{\rightarrow \hspace*{-15pt} \rightarrow} \renewcommand{\gets}{\leftarrow} \newcommand{\manyone}{\leq_{\rm m}} \newcommand{\turing}{\leq_{\rm T}} \newcommand{\encode}[1]{\langle #1 \rangle} % OTHER MATHEMATICAL SYMBOLS OR CONSTRUCTS \newcommand{\cents}{\hbox{\rm\rlap/c}} \newcommand{\half}{\frac{1}{2}} \def\squarebox#1{\hbox to #1{\hfill\vbox to #1{\vfill}}} \newcommand{\qed}{\hspace*{\fill} \vbox{\hrule\hbox{\vrule\squarebox{.667em}\vrule}\hrule}\smallskip} % THEOREM-LIKE ENVIRONMENTS \newtheorem{lemma}{Lemma} \newtheorem{definition}[lemma]{Definition} \newtheorem{claim}[lemma]{Claim} \newtheorem{theorem}[lemma]{Theorem} \newtheorem{corollary}[lemma]{Corollary} \newtheorem{proposition}[lemma]{Proposition} % PROGRAM-LIKE ENVIRONMENT % #1 - label_name #2 - caption_name #3 - text \newcommand{\Tabs}{xxx\= xxx\= xxx\= xxxx\= xxxx\= xxxx\= xxxx\= xxxx\= xxxx\= xxxx\= xxxx\= \kill} \newenvironment{program}[3]{\begin{figure*}[hbtp] \begin{center} \fbox{ \begin{minipage}{\textwidth} \begin{tabbing} \Tabs #3 \end{tabbing} \end{minipage} } \end{center} \caption{#2} \label{#1} \end{figure*}}{} % MACROS FOR ALGORITHMS OR PROGRAMS \newcommand{\Backward}{\mbox{\sc backward}} \newcommand{\Forward}{\mbox{\sc forward}} \newcommand{\True}{\mbox{\sc true}} \newcommand{\False}{\mbox{\sc false}} \newcommand{\On}{\mbox{\sc on}} \newcommand{\Off}{\mbox{\sc off}} %\input{psfig} \title{Self-stabilizing Distributed Constraint Satisfaction} \author{Zeev Collin\\ Computer Science Department\\ The Technion\\Haifa, Israel\\ zeev@cs.technion.ac.il \and Rina Dechter\\ Department of Information and Computer Science\\ University of California\\ Irvine, California, USA\\ dechter@ics.uci.edu \and Shmuel Katz\\ Computer Science Department\\ The Technion\\ Haifa, Israel\\ katz@cs.technion.ac.il } \date{31 December 1999} \begin{document} %% %% CJTCS frontmatter, Insert after \begin{document} %% \def\registered{\({}^{\ooalign% {\hfil\raise.07ex\hbox{{\tiny\textup{R}}}% \hfil\crcr\scriptsize\mathhexbox20D\cr}}\)} \begin{titlepage} \pagenumbering{roman} \null\vfil\vskip 60pt \begin{center} {\Huge\bf Chicago\nolinebreak[1] Journal of Theoretical Computer\nolinebreak[1] Science\par} \vskip 3em {\huge\it The MIT Press\par} \end{center} \par\vskip 1.5em {\Large \begin{center} Volume 1999, Article 10\\ \textit{Self-stabilizing Distributed Constraint Satisfaction} \end{center} } \par\noindent ISSN 1073--0486. MIT Press Journals, Five Cambridge Center, Cambridge, MA 02142-1493 USA; (617)253-2889; \emph{journals-orders@mit.edu, journals-info{\allowbreak}@mit.edu}. Published one article at a time in \LaTeX\ source form on the Internet. Pagination varies from copy to copy. For more information and other articles see \begin{itemize} \item \emph{http://mitpress.mit.edu/CJTCS/} \item \emph{http://www.cs.uchicago.edu/publications/cjtcs/} \item \emph{ftp://mitpress.mit.edu/pub/CJTCS} \item \emph{ftp://cs.uchicago.edu/pub/publications/cjtcs} \end{itemize} \par\noindent The \emph{Chicago Journal of Theoretical Computer Science} is abstracted or indexed in \emph{ Research Alert\registered, SciSearch\registered, Current Contents\registered/Engineering Computing \& Technology}, and \emph{CompuMath Citation Index\registered.} \par\noindent\copyright 1999 The Massachusetts Institute of Technology. Subscribers are licensed to use journal articles in a variety of ways, limited only as required to ensure fair attribution to authors and the journal, and to prohibit use in a competing commercial product. See the journal's World Wide Web site for further details. Address inquiries to the Subsidiary Rights Manager, MIT Press Journals; (617)253-2864; \emph{journals-rights@mit.edu}.\pagebreak[2] \par\noindent The \emph{Chicago Journal of Theoretical Computer Science} is a peer-reviewed scholarly journal in theoretical computer science. The journal is committed to providing a forum for significant results on theoretical aspects of all topics in computer science. \par \begin{trivlist} \item[\emph{Editor-in-Chief:}] Janos Simon \item[\emph{Consulting Editors:}] Joseph Halpern, Eric Allender, Raimund Seidel \item[\emph{Editors:}] \begin{tabular}[t]{lll} Martin Abadi & Greg Frederickson & John Mitchell \\ Pankaj Agarwal & Andrew Goldberg & Ketan Mulmuley \\ Eric Allender & Georg Gottlob & Gil Neiger \\ Tetsuo Asano & Vassos Hadzilacos & David Peleg \\ Laszl\'o Babai & Juris Hartmanis & Andrew Pitts \\ Eric Bach & Maurice Herlihy & James Royer \\ Stephen Brookes & Ted Herman & Alan Selman \\ Jin-Yi Cai & Stephen Homer & Nir Shavit \\ Anne Condon & Neil Immerman & Eva Tardos \\ Cynthia Dwork & Howard Karloff & Sam Toueg \\ David Eppstein & Philip Klein & Moshe Vardi \\ Ronald Fagin & Phokion Kolaitis & Jennifer Welch \\ Lance Fortnow & Stephen Mahaney & Pierre Wolper \\ Steven Fortune & Michael Merritt \end{tabular} \vspace{1ex} \item[\emph{Managing Editor:}] Michael J.\ O'Donnell \vspace{1ex} \item[\emph{Electronic Mail:}] \emph{chicago-journal@cs.uchicago.edu} \end{trivlist} \vfil\null \end{titlepage} \maketitle \pagenumbering{arabic} %% %% End frontmatter code %% %\input{title} \maketitle \begin{abstract} Distributed architectures and solutions are described for classes of constraint satisfaction problems, called {\em network consistency problems}. An inherent assumption of these architectures is that the communication network mimics the structure of the constraint problem. The solutions are required to be self-stabilizing and to treat arbitrary networks, which makes them suitable for dynamic or error-prone environments. We first show that even for relatively simple constraint networks, such as rings, there is no self-stabilizing solution that guarantees convergence from every initial state of the system using a completely uniform, asynchronous model (where all processors are identical). An {\em almost uniform}, asynchronous, network consistency protocol with one specially designated node is shown and proven correct. We also show that some restricted topologies such as trees can accommodate the uniform, asynchronous model when neighboring nodes cannot take simultaneous steps. \end{abstract} \section{Introduction} \label{introduction_sec} Consider the distributed version of the graph coloring problem, where each node must select a color (from a given set) that is different from any color selected by its neighbors. This NP-complete problem belongs to the class of {\em constraint satisfaction problems} (csp's), which present interesting challenges to distributed computation.\footnote{Obviously, there is no connection to Hoare's CSP (communicating sequential processes) notation \cite{H85}.} We call the distributed versions of this class of problems {\em network consistency problems}. Since constraint satisfaction is inherently intractable for the general case, the interesting questions for distributed models are those of feasibility rather than efficiency. We wish to answer the following main question: What types of distributed models admit a self-stabilizing algorithm, namely, one that converges to a solution, if such exists, from any initial state of the network? For models that do admit solutions, we present and prove the correctness of appropriate algorithms that converge to a solution for any specific problem. The motivation for addressing this question stems from attempting to solve constraint satisfaction problems in environments that are inherently distributed. For instance, an important family of constraint satisfaction problems in telecommunications involves scheduling transmissions in radio networks \cite{ramamathan}. A radio network is a set of $n$ stations that communicate by broadcasting and receiving signals. Typical examples of radio networks are packet radio networks and satellite networks. Every station has a transmission radius. The {\em broadcast scheduling problem} involves scheduling broadcast transmissions from different stations in an interference-free way. In the time-division multiplexing version, there is a fixed set of time slots, and each station must assign itself one time slot such that any two stations that are within the transmission radius of each other are assigned different time slots. The frequency-division multiplexing version is similar, with the fixed set of time slots replaced by a fixed set of frequencies. One can easily see that the problem translates naturally to a graph coloring problem where broadcasting radios are the nodes in the graph and any two nodes are connected if and only if their transmission radii overlap. Solving the broadcasting scheduling problem autonomously and distributedly by the radio stations themselves is highly desirable because the environment is inherently distributed: In some applications (e.g., in a military setting, when the radios are moving) no central control is feasible or practical. Moreover, a self-stabilizing distributed solution has the important virtue of being fault tolerant. Another motivating area, distributed artificial intelligence (DAI), has become very popular in recent years \cite{lesser}. The issue addressed is problem solving in a multiagent environment where agents have to cooperate to solve a problem and to carry out its solution in a distributed manner. As in the broadcast scheduling problem, the distributed environment is a physical necessity rather than a programmer's design choice. One possible architecture considered for solving the network consistency problem is neural networks. Such networks perform distributed computation with uniform units and are normally self-stabilizing. However, current connectionist approaches to constraint problems \cite{Ballard,Dahl} lack theoretical guarantees of convergence to a solution, and the conditions under which such convergence can be guaranteed (if at all) have not been systematically explored. This paper aims to establish such guarantees by studying the feasibility of solving a csp in uniform, self-stabilizing distributed architectures. Intuitively, a distributed algorithm is {\em self-stabilizing} \cite{Dijkstra} if it converges to a solution from any initial state of both the network and the algorithm. Our aim is to determine what types of distributed models admit self-stabilizing algorithms and, whenever possible, to present such an algorithm. Self-stabilization is a desirable property since it yields robustness in the face of dynamically changing environments. This is especially true if the solution treats any network configuration and, thus, within some time after a change to the network, converges to a solution. Some changes can be imposed externally (e.g., adding new components to the problem, changing the values of variables or buffers); others may occur to the system internally, by errors in the implementing hardware. The accepted model we use for self-stabilization \cite{Dijkstra} treats the part of the computation from after a perturbation to the next perturbation as a normally executing algorithm with abnormal initial values, including control locations. The implicit assumption is that perturbations occur infrequently, so that the system stabilizes and does most of its work in consistent states. In this paper, we characterize architectures that allow a self-stabilizing distributed solution for classes of constraint satisfaction problems, and we present algorithms when possible. As noted above, the self-stabilization can model dynamically changing constraint networks as well as transient network perturbations, thus increasing the robustness of the solutions. Following some background on constraint networks and self-stabilization (Section 2) we show that {\em uniform} networks, in which all nodes run identical procedures and any scheduling policy is allowed, cannot admit algorithms that guarantee convergence to a consistent solution from any initial state using an asynchronous model (Section 3). In Section 4, we depart slightly from uniformity by allowing one unit to execute a different algorithm. We call this model {\em almost uniform} and use it to present an asynchronous, network consistency protocol. It combines a subprotocol to generate a depth-first search (DFS) spanning tree, with a value-assignment subprotocol. The value assignment exploits the potential parallelism of the spanning tree, using a special form of backtracking appropriate for constraint problems. In Section 5, we show that some restricted topologies, such as trees, can accommodate the uniform, asynchronous model. Preliminary versions of some of these results first appeared in \cite{CDK:feasibility}. \section{Background} \label{model_sec} \subsection{Constraint networks} A csp is defined over a {\em constraint network} that consists of a finite set of {\it variables}, each associated with a {\it domain} of values, %$D_{1} ,\ldots ,D_{n},$ and a set of {\it constraints}. %, $\{C_{1}, \ldots ,C_{t}\}$. A {\em solution} is an assignment of a value to each variable from its domain such that all the constraints are satisfied. Typical constraint satisfaction problems are to determine whether a solution exists, to find one or all solutions, and to find an optimal solution relative to a given cost function. An example of a constraint satisfaction problem is the $k$-colorability problem mentioned in the introduction. The problem is to color, if possible, a given graph with $k$ colors only, such that any two adjacent nodes have different colors. A constraint satisfaction formulation of this problem associates the nodes of the graph with variables, the possible colors are their domains, and the inequality constraints between adjacent nodes are the constraints of the problem. Each constraint of a csp may be expressed as a relation, defined on some subset of variables, denoting their legal combinations of values. In addition, constraints can be described by mathematical expressions or by computable procedures. The structure of a constraint network is depicted by a constraint graph whose nodes represent the variables and in which any two nodes are connected if the corresponding variables participate in the same constraint. In the $k$-colorability formulation, the graph to be colored is the constraint graph. Constraint networks have proven successful in modeling mundane cognitive tasks such as vision, language comprehension, default reasoning, and abduction, as well as in applications such as scheduling, design, diagnosis, and temporal and spatial reasoning. In general, constraint satisfaction tasks are computationally intractable (NP-hard). Techniques for processing constraints can be classified into two categories \cite{dechter-ency}: (1) search and (2) consistency inference, and these techniques interact. Search algorithms traverse the space of partial instantiations, while consistency-inference algorithms reason through equivalent problems. Search is either systematic and complete, or stochastic and incomplete. Likewise, consistency-inference has complete solution algorithms (e.g., variable-elimination) or incomplete versions in the form of {\em local consistency} algorithms. Let us state this formally. \begin{definition} A {\em network of binary constraints} is a set of $n$ variables $X= \{X_1, \dots , X_n \}$, a domain $D_i$ of possible values for each variable $X_i$, $ 1 \leq i \leq n$, and a set of constraints $R_{S_1},\ldots,R_{S_t}$, where $S_i \subseteq X$. A {\em binary constraint}, denoted $R_{ij}$ over two variables $X_i$ and $X_j$, is a subset of the product of their domains, $R_{ij} \subseteq D_i \times D_j$. A {\em solution} is an assignment of a value to each variable, $x \bar = (x_1,\ldots,x_n)$, $x_i \in D_i$, such that $\forall i,j$, $1 \leq i,j \leq n, ~(x_i,x_j) \in R_{ij}$. A {\em constraint graph} has a node for each variable and links connecting pairs of variables that appear in the same constraint. \end{definition} General constraint satisfaction problems may involve constraints of any arity, but since network communication is only pairwise we focus on this subclass of problems. Figure \ref{CSP_fig}a presents a csp constraint graph, where each node represents a variable having values $\set{a,b,c}$, and each link is associated with a strict lexicographic order ($X_i\prec X_j$ in the lexicographic order if and only if $i 1)$. Since convergence to a solution is guaranteed from any initial configuration, the protocol also converges when initially all nodes are in identical states. We construct a fair execution of such a protocol that satisfies the restriction on the scheduler but for which the network never converges to a consistent solution, contradicting the self-stabilization property of the protocol. Assume the following execution: \[ \begin{array}{lllll} P_0, & P_q, & P_{2q}, & \ldots , & P_{(r-1)q}, \\ P_1, & P_{q+1}, & P_{2q+1},& \ldots , & P_{(r-1)q+1}, \\ \vdots \\ P_{q-1}, & P_{2q-1}, & P_{3q-1},& \ldots , & P_{rq-1}, \\ P_0, & \ldots \\ \vdots \end{array} \] Note that nodes $P_0, P_q, P_{2q}, \ldots ,P_{(r-1)q}$ move to identical states after their first activation, because their inputs, initial states, and transition functions are identical, and when each one of them is activated its neighbors are in identical states too. The same holds for any sequential activation of processors $\{ P_{iq+j} |~0 \! \leq \! i \! < \! r,~0 \! \leq \! j \! < \! q \}$. Thus, cycling through the above schedule assures that $P_0$ and $P_q$, for instance, move to identical states over and over again, an infinite number of times. Since a consistent solution requires their states to be different, the network never reaches a consistent solution, thus yielding a contradiction. Figure \ref{ring_fig} demonstrates such a counterexample execution for a ring with six nodes. The indicated nodes are scheduled in each configuration. Different colors refer to different states. \end{proof} \begin{figure}[tbh] \begin{center} \includegraphics[scale=.8]{aring_fig.eps} %\makebox[442pt][l]{ % \vbox to 106pt{ % \vfill % \special{psfile=ring_fig.ps} % } % \vspace{-\baselineskip} %} \end{center} \caption{The ring ordering problem ($n=6$)} \label{ring_fig} \end{figure} Theorem \ref{impossibility} is proven above for a centralized scheduler, but as noted earlier it holds also for a distributed scheduler. This theorem means that it is generally impossible to guarantee convergence to a consistent solution using a uniform protocol if no additional restrictions are made on the possible executions. In particular, convergence cannot be guaranteed for a class of sequential algorithms using ``repair'' methods, as in \cite{Minton}, if a completely random order of repair is allowed. It does not, however, exclude the possibility of uniform protocols for restricted scheduling policies or for special network topologies (not including a ring). In practice it is fair to assume that adversarial scheduling policies are not likely to occur or can be deliberately avoided. When using a distributed scheduler, convergence (to a solution) cannot be guaranteed even for {\em tree networks}. Consider, for instance, the coloring problem in a tree network constructed from two connected nodes, each having the domain \{{\sc black, white}\}. Since the two nodes are topologically identical, if they start from identical initial states and both of them are activated simultaneously, they can never be assigned different values and will never converge to a legal solution, although one exists. This trivial counterexample can be extended to a large class of trees, where there is no possible way to distinguish between two internal nodes. However, we show (Section \ref{tree_sec}) that for a central scheduler, a uniform self-stabilizing tree network consistency protocol does exist. Having proven that the network consistency problem cannot always be solved by using a uniform protocol, even with a central scheduler, we switch to a slightly more relaxed model of an {\em almost uniform} protocol, in which all nodes but one are identical. We denote the special node as node 0 (or $P_0$). Note that such a special processor can be determined by finding a leader in a distributed network. Thus, if a leader could be elected uniformly, it could be used to convert our almost uniform protocol to a uniform one. Since we cannot solve the consistency problem for a central scheduler with an almost uniform protocol, it follows from our impossibility result (as is well known \cite{angluin}) that a leader cannot be elected in an arbitrary anonymous (where all processors are identical) network. However, randomization can be used to break the symmetry and to elect a leader in uniform networks \cite{afek}. \section{Consistency-generation protocol} \label{protocol_sec} This section presents an almost uniform, self-stabilizing network consistency protocol. The completeness of this protocol (i.e., the guarantee of converging to a solution if one exists) stems from the completeness of the sequential constraint satisfaction algorithm it simulates. It can accommodate changes to constraints, as long as the resulting graph is connected and includes the special node (or one can be elected using randomization). We briefly review some basic sequential techniques for constraint satisfaction. \subsection{Sequential aspects of constraint satisfaction} The most common algorithm for solving a csp is {\em backtracking}. In its standard version, the algorithm traverses the variables in a predetermined order, provisionally assigning consistent values to a subsequence $(X_1 ,\ldots , X_i)$ of variables and attempting to append to it a new instantiation of $X_{i+1}$ such that the whole set is consistent ({\em forward} phase). If no consistent assignment can be found for the next variable $X_{i+1}$, a dead-end situation occurs; the algorithm backtracks to the most recent variable ({\em backward} phase), changes its assignment, and continues from there. A useful improvement of backtracking, {\em graph-based backjumping} \cite{D:enhancement}, consults the topology of the constraint graph to guide its backward phase. Specifically, instead of going back to the most recent variable instantiated, it {\em jumps back} several levels to the first variable connected to the dead-end variable. If the variable to which the algorithm retreats has no more values, it backs up further, to the most recent variable among those connected either to the original variable or to the new dead-end variable, and so on. It turns out that when using a depth-first search (DFS) on the constraint graph (to generate a DFS spanning tree) and then conducting backjumping in a preorder traversal of the DFS tree \cite{Even}, the jump-back destination of variable $X$ is the parent of $X$ in the DFS spanning tree \cite{dechter-ency}. The nice property of a DFS spanning tree that allows a parallel implementation is that any arc of the graph that is not in the tree connects a node to one of its tree ancestors (i.e., to a node residing along the path leading to it from the root). Consequently, the DFS spanning tree represents a useful decomposition of the graph: If a variable $X$ and all its ancestors in the tree are removed from the graph, the remaining subtrees rooted at the children of $X$ are disconnected. Figure \ref{CSP_fig}b presents a DFS spanning tree of the constraint graph presented in Figure \ref{CSP_fig}a. Note that if $X_2$ and its ancestor $X_1$ are removed from the graph, the network becomes two disconnected trees rooted at $X_3$ and $X_5$. This translates to a problem decomposition strategy: If all ancestors of variable $X$ are instantiated, then the solutions of all its subtrees are completely independent and can be performed in parallel \cite{FQ:lineal}. \subsection{General protocol description} \label{general_desc} The distributed version of the binary csp is called the network consistency (NC) problem. Our network consistency protocol is based on a distributed version of graph-based backjumping implemented on a variable ordering generated by a depth-first traversal of the constraint graph. The NC protocol is logically composed of two self-stabilizing subprotocols that can be executed interleaved, as long as one self-stabilizes for any configuration, and then establishes a condition guaranteeing the self-stabilization of the second \cite{DIM93}. The subprotocols are as follows: \begin{enumerate} \item DFS spanning tree generation \item value assignment (using the graph traversal mechanism) \end{enumerate} The second subprotocol assumes the existence of a DFS spanning tree in the network. However, the implementations of these subprotocols are unrelated to each other and, thus, can be independently replaced by any other implementation. Until the first subprotocol establishes a DFS spanning tree, the second subprotocol executes but, in all likelihood, does not lead to a proper assignment of values. %SSS We use a self-stabilizing DFS spanning tree protocol that is described in \cite{CD:dfs}. The spanning tree protocol is not affected by the interleaved second subprotocol, and thus the existence of a DFS spanning tree is eventually guaranteed. The convergence of the second subprotocol is also guaranteed starting from any configuration and assuming the existence of a DFS spanning tree (which is not changed by continued operation of the first subprotocol). Therefore, the combination is guaranteed to converge properly after the DFS spanning tree is completed. The basic idea of the second subprotocol is to decompose the network (problem) logically into several independent subnetworks (subproblems), according to the DFS spanning tree structure, and to instantiate these subnetworks (solve the subproblems) in parallel. Proper control over value instantiation is guaranteed by the graph traversal mechanism presented in Section \ref{graph_traversal}. We would like to emphasize that using graph-based backjumping rather than naive backtracking as the sequential basis for our distributed implementation is crucial for the success of our protocol. First, naive backtracking leads naturally to algorithms in which only one processor executes at a time. Second, it requires a total ordering of the processors generally encoded in their identifiers. Moreover, unless the nodes that are consecutive in the backtracking order are connected in the communication graph, passing activation from one node to another when going forward or upon a dead end seems feasible only when using node identifiers. Algorithm graph--based backjumping uses a partial order that is derived from the DFS spanning tree, thus providing more opportunities for parallelism and eliminating the need for node identifiers. \subsection{Self-stabilizing DFS spanning tree generation protocol} \label{dfs_section} First we describe an almost uniform, self-stabilizing protocol for generating a DFS spanning tree. The full algorithm was described in \cite{CD:dfs} and, thus, is not described in detail or proven here. Alternative self-stabilizing algorithms for generating a DFS spanning tree could be used instead and may yield a better space complexity, as discussed in Section \ref{space}. This subprotocol is the source of nonuniformity for the whole NC protocol. The root of the generated tree is the distinguished node 0 ($P_0$). Each processor, $P_i$, has (at most) one adjacent processor, $\parent(i)$, designated as its {\em parent} in the tree, and a set of {\em child} processors, $\children(i)$. The set of the processors that reside along the path from the root to $P_i$ in the DFS spanning tree is denoted by $\ancestors(i)$, while $\descendants(i)$ is the set of processors $P_j$ so that $P_i$ is in $\ancestors(j)$. The set of $P_i$'s neighboring processors that are in $\ancestors(i)$, except the parent of $P_i$, are called $P_i$'s predecessors and are denoted by $\predecessors(i)$. Figure \ref{neighborhood_fig} indicates the environment of an internal processor. The $\ancestors(i)$ set is empty if $i$ is the root, while the $\descendants(i)$ set is empty if $i$ is a leaf. \begin{figure}[tbh] \begin{center} \includegraphics{afig2a.eps} %\makebox[351pt][l]{ % \vbox to 117pt{ % \vfill % \special{psfile=fig2a.ps} % } % \vspace{-\baselineskip} %} \end{center} \caption{The neighborhood set of $i$ in a DFS-marked graph} \label{neighborhood_fig} \end{figure} The links of every processor $P$ are divided into the following categories (see Figure \ref{neighborhood_fig}): \begin{enumerate} \item {\em Tree-edges} are the edges that belong to the DFS spanning tree. \begin{enumerate} \item {\em Inlink} is the edge that connects $P$ to its parent. Every nonroot processor has one such link and the root has none. \item {\em Outlinks} are the edges that connect $P$ to its children. \end{enumerate} \item {\em Backward edges} are the edges that connect $P$ to its predecessors. \item {\em Descendant edges} are the edges that connect $P$ to its descendants, excluding its children. \end{enumerate} A distributed graph is called {\em DFS-marked} if there is a DFS spanning tree over the graph such that every processor in the system can identify the category of each of its adjacent edges with relation to this DFS spanning tree. Each node has a local numbering (ranking) of its edges. During the execution of the protocol the special node 0, which plays the role of the root, repeatedly assigns itself the label $\perp$ (the minimal element) and suggests labels to its neighbors. The label suggested to $j$ by its neighbor $i$ is constructed by concatenating $i$'s ranking of the edge from $i$ to $j$ to the right of $i$'s own label. Thus a label is a sequence of elements, each of which is the minimal element or an edge rank. Labels are compared using lexicographic order. The sequence in a label has a fixed maximal length, and the concatenation can lead to ``overflow'' where the leftmost element is removed (and this is needed for convergence). Every nonroot node chooses the smallest label suggested to it to be its label and the suggesting neighbor to be its parent, and suggests labels to its neighbors in a similar way. The communication register between $i$ and $j$ contains the following fields used by the DFS spanning tree generation protocol: \begin{description} \item[{$r_{ij}.\mbox{\it mark}$}] contains the label that is ``suggested'' to $j$ by $i$. \item[{$r_{ij}.\mbox{\it par}$}] is a boolean field that is set to \True\ if and only if $i$ chooses $j$ as its parent. \end{description} The output of the DFS spanning tree algorithm, after the network has converged, is a DFS-marked graph maintained in a distributed fashion. \subsection{Value assignment subprotocol} \label{graph_traversal} The second subprotocol assumes the existence of a DFS spanning tree in the network; namely, each nonroot node has a designated parent, children, and predecessors among its neighboring nodes (see Figure \ref{neighborhood_fig}). When the DFS spanning tree generation subprotocol stabilizes, each node has a minimal label and a designated parent. Using this information, each node can compute its children set, $\children(i)$, by selecting the neighbors whose $r_{ji}.\mbox{\it par}$ field is true, and its predecessors set, $\predecessors(i)$, by selecting the neighbors whose minimal label ($r_{ji}.\mbox{\it mark}$ without the last character) is a prefix of its own. This means that we can traverse the directed tree either toward the leaves or toward the root. The value assignment subprotocol presents a graph traversal mechanism that passes control to the nodes in the order of the value assignment of the variables (in DFS order), without losing the parallelism gained by the DFS structured network. Section \ref{activation} presents the basic idea of {\em privilege passing} that implements the graph traversal mechanism, while Section \ref{assignment} presents the value assignment strategy that guarantees convergence to a solution. Each node $i$ (representing the variable $X_i$) has a list of possible values, denoted as $\Domain_{i}$, and a pairwise relation $R_{ij}$ with each neighbor $j$. The domain and the constraints may be viewed as a part of the system or as inputs that are always valid (though they can be changed during the execution, forcing the network to readjust itself to the changes). The state register of each node contains the following fields: \begin{description} \item[{$\Value_i$}] is a field to which it assigns one of its domain values or the symbol ``$\star$'' (to denote a dead end). \item[{$\mode_i$}] is a field indicating the node's ``belief'' regarding the status of the network. A node's mode is \On\ if the value assignment of itself or one of its ancestors has changed since the last time it was in a forward phase (to be explained in Section \ref{assignment}); otherwise it is \Off. The modes of all nodes also give an indication of whether they have reached a consistent state (all in an \Off\ mode). \item[{\rm$\parent\_\tag$ and $\children\_\tag$}] are two boolean fields used for the graph traversal (Section \ref{activation}). \end{description} Additionally, each node has a sequence set of domain values that is implemented as an ordered list and is controlled by a local domain {$\pointer$} (to be explained later), and a local {$\direction$} field indicating whether the algorithm is in its forward or backward phase. \subsubsection{Graph traversal using privileges} \label{activation} The graph traversal is handled by a self-stabilizing {\em privilege passing mechanism}, according to which a node obtains the privilege to act, granted to it either by its parent or by its children. A node is allowed to change its state only if it is privileged. Our privilege passing mechanism is an extension of a mutual exclusion protocol for two nodes called {\em balance-unbalance} \cite{Dijkstra,DIM93}. Once a DFS spanning tree is established, this scheme is implemented by having every state register contain two fields: $\parent\_\tag$, referring to its inlink and $\children\_\tag$, referring to all its outlinks. A link is {\em balanced} if the $\children\_\tag$ and the $\parent\_\tag$ on its endpoints have the same value, and the link is {\em unbalanced} otherwise. A node becomes privileged if its inlink is unbalanced and {\em all} its outlinks are balanced. In other words, the following two conditions must be satisfied for a node $i$ to be privileged: \begin{enumerate} \item $\parent\_\tag_{i} \neq \children\_\tag_{\parent(i)}$ (the inlink is unbalanced) \item $\forall k \in \children(i) : \, \children\_\tag_{i} = \parent\_\tag_{k}$ (the outlinks are balanced) \end{enumerate} By definition, we consider the inlink of the root to be unbalanced and the outlinks of the leaves to be balanced. A node applies the value assignment subprotocol (described in Section \ref{assignment}) only when it is privileged; otherwise it leaves its state unchanged. As part of the execution of the subprotocol, the node passes the privilege. The privilege can be passed backward to the parent by balancing the inlink or forward to the children by unbalancing the outlinks (i.e., by changing the value of $\parent\_\tag$ or $\children\_\tag$, respectively). We use the following notation to define the set of configurations that are legally controlled relative to the graph traversal: \begin{itemize} \item Denote a {\em chain} to be a maximal sequence of unbalanced links, $e_{1}, e_{2} \ldots,\allowbreak e_{n}$, such that the following are true: \begin{enumerate} \item The inlink of the node whose outlink is $e_{1}$ is balanced, unless the node is the root. \item Every adjacent pair of links $e_{i}, e_{i+1}~(1\leq i < n)$ is an inlink and an outlink, respectively, of a common node. \item All the outlinks of the node whose inlink is $e_{n}$ are balanced. \end{enumerate} \item The chain begins at the node with $e_{1}$ as one of its outlinks, denoted as the {\em chain head}, and ends at the node with the inlink $e_{n}$, denoted as the {\em chain tail}. \item Denote a {\em branch} to be a path from the root to a leaf. \item A branch {\em contains} a chain (or a chain is {\em on} the branch) if all the links of the chain are in the branch. \item A configuration is {\em legally controlled} if it does not contain any nonroot chain heads, namely, every branch of the tree contains no more than one chain and its chain head is the root. \end{itemize} Figure \ref{legal_conf_fig} shows a legally controlled configuration. The DFS spanning tree edges are directed, and the values ($+$ or $-$) of the $\parent\_tag$ and the $\children\_tag$ of every node are specified above and below the node, respectively. The privileged nodes are black. In a legally controlled configuration, a node and its ancestors are not privileged at the same time and therefore cannot reassign their values simultaneously. The privileges travel backward and forward along the branches. We prove (Section \ref{correctness_sec}) that, using the graph traversal mechanism, the network eventually converges to a set of legally controlled configurations that are also legal with respect to the network consistency task. \begin{figure}[tbh] \begin{center} \includegraphics[scale=.95]{afig3.eps} %\makebox[397pt][l]{ % \vbox to 101pt{ % \vfill % \special{psfile=fig3.ps} % } % \vspace{-\baselineskip} %} \end{center} \caption{Legally controlled configuration} \label{legal_conf_fig} \end{figure} Once it has become privileged, a node cannot tell where the privilege came from (i.e., from its parent or from its children). Thus, a node uses its $\direction$ field to indicate the source of its privilege. Since during the legally controlled period no more than one node is privileged on every branch, the privileges travel along the branches backward and forward. The $\direction$ field of each node indicates the direction of the next expected wave. When passing the privilege to its children, the node assigns its $\direction$ field the \Backward\ value, expecting to get the privilege back during the next backward wave, while when passing the privilege to its parent it assigns the \Forward\ value, preparing itself for the next forward wave. Thus, upon receiving the privilege again, it is able to recognize the direction it came from: if $\direction = \Backward$, the privilege was recently passed toward the leaves, and therefore it can come only from its children; if $\direction = \Forward$, the privilege was recently passed toward the root, and therefore it can come only from its parent. The value of the $\direction$ field can be improper upon the initialization of the system. However, after the first time a node passes the privilege, its $\direction$ field remains properly updated. Figure \ref{priv-proc} presents the privilege passing procedures for node $i$. \begin{program}{priv-proc}{Privilege passing procedures}{\small procedure {\bf pass-privilege-to-parent} \\ \small Begin \\ \small 1.\> \small $\direction_{i} \gets$ \Forward \>\>\>\>\>\>\>\> \small \{ prepare for the next wave \}\\ \small 2.\> \small $\parent\_tag_{i} \gets \children\_tag_{\parent(i)}$\>\>\>\>\>\>\>\> \small \{ balance inlink \} \\ \small End. \\ \\ \small procedure {\bf pass-privilege-to-children} \\ \small Begin \\ \small 1.\> \small $\direction_{i} \gets$ \Backward \>\>\>\>\>\>\>\> \small \{ prepare for the next wave \}\\ \small 2.\> \small $\children\_tag_{i} \gets \small \neg \parent\_tag_{k\in \children(i)}$\>\>\>\>\>\>\>\> \small \{ unbalance outlinks \} \\ \small End.} \end{program} \subsubsection{Value assignment} \label{assignment} The value assignment has forward and backward phases, corresponding to the two phases of the sequential backtracking algorithm. During the forward phase, nodes in different subtrees assign themselves (in parallel) values consistent with their predecessors or verify the consistency of their assigned values. When a node senses a dead end (i.e., it has no consistent value to assign), it assigns its $\Value$ field a $\star$ and initiates a backward phase. Since the root has no ancestors, it does not check consistency and is never dead-ended. It only assigns a new value at the end of a backward phase, when needed, and then initiates a new forward phase. When the network is consistent (all the nodes are in an \Off\ mode), the forward and backward phases continue, where the forward phase is used to verify the consistency of the network, while the backward phase just returns the privilege to the root to start a new forward wave. Once consistency is violated, the node sensing the violation relative to its predecessors moves to an \On\ mode and initiates a new value assignment. A more elaborate description follows. An internal node can be in one of three situations: \begin{itemize} \item {Node {$i$} is activated by its parent, which is in an \On\ mode} (this is the forward phase of value assignments). In that case, some change of value in one of its predecessors might have occurred. It therefore finds the first value in its domain that is consistent with all its predecessors, puts itself in \On\ mode, and passes the privilege to its children. If no consistent value exists, it assigns itself the $\star$ value (a dead end) and passes the privilege to its parent (initiating a backward phase). \item {Node {$i$} is activated by its parent, which is in an \Off\ mode} (this is the forward phase of consistency verification). In that case, it verifies the consistency of its current value with its predecessors. If it is consistent, it stays in (or moves to) an \Off\ mode and passes the privilege to its children. If not, it tries to find the next value in its domain that is consistent with all its predecessors, and continues as in the previous case. A leaf, having no children, is always activated by its parent and always passes the privilege back to its parent (initiating a backward phase). \item {Node {$i$} is activated by its children} (backward phase). If one of the children has a $\star$ value, $i$ selects the next value in its domain that is consistent with all its predecessors, and passes the privilege back to its children. If no consistent value is available, it assigns itself a $\star$ and passes the privilege to its parent. If all children have a consistent value, $i$ passes the privilege to its parent. \end{itemize} Due to the privilege passing mechanism, when a parent sees one of its children in a dead end, it still has to wait until {\em all} of them have given it the privilege. This is done to guarantee that all subtrees have a consistent view regarding their predecessors' values. %ccc adding: This requirement limits the amount of parallelism considerably. It can be relaxed in various ways to allow more parallelism. %ccc The algorithms performed by a nonroot node ($i \neq 0$) and the root once they become privileged and after reading the neighbors' states are presented in Figures \ref{update} and \ref{procedures}. %\input{protocol} \begin{program}{update}{Value assignment subprotocols for root and nonroot nodes}{ \footnotesize {\bf root 0:} \\ \footnotesize Begin \\ %1. \> \footnotesize if $\exists k \! \in \! \children(0): \; \Value_{k} = \star$ then \\ \footnotesize 1. \> \footnotesize if $\neg${\bf consistent($\Value_0,\children(0)$)} then \\ \footnotesize 2. \> \> \footnotesize $\mode \gets$ \On \\ \footnotesize 3. \> \> \footnotesize $\Value \gets$ {\bf next-value} \\ \footnotesize 4. \> \footnotesize else\> \> \> \> \> \footnotesize \{ all children are consistent \} \\ \footnotesize 5. \> \> \footnotesize $\mode \gets$ \Off\ \\ \footnotesize 6. \> \footnotesize {\bf pass-privilege-to-children} \\ \footnotesize End. \\ \\ \footnotesize {\bf\boldmath nonroot $i$:} \\ \footnotesize Begin \\ \footnotesize 1. \> \footnotesize if $\direction$ = \Forward\ then \>\>\>\>\>\>\> \footnotesize \{ forward phase \}\\ \footnotesize 2. \> \> \footnotesize if $\mode_{\parent(i)} =$ \On\ then \>\>\>\>\>\> \footnotesize \{ a change in a value assignment occurred \} \\ \footnotesize 3. \> \> \> \footnotesize $\pointer \gets 0$ \>\>\>\>\> \footnotesize \{ reset domain pointer \}\\ \footnotesize 4. \> \> \footnotesize else \> \> \> \> \> \> \footnotesize \{ parent's mode is \Off \} \\ \footnotesize 5. \> \> \> \footnotesize if {\bf consistent($\Value_i,\predecessors(i)$)} then \\ \footnotesize 6.\> \> \> \> \footnotesize $\mode \gets$ \Off \\ \footnotesize 7. \> \footnotesize if $(\pointer=0)\vee\neg${\bf consistent($\Value_i,\predecessors(i)$)} $\vee$ \\ \pushtabs \footnotesize xxx\= \footnotesize if $(\pointer=0)$\= \kill \footnotesize \>\> \footnotesize $\vee (\direction=\Backward\wedge\neg${\bf consistent($\Value_i,\children(i)$)}) then \\ \poptabs \footnotesize 8.\> \> \footnotesize $\Value \gets$ {\bf compute-next-consistent-value} \\ \footnotesize \> \> \> \> \> \> \> \> \> \footnotesize \{ privilege passing \}\\ \footnotesize 9.\> \footnotesize if {\bf leaf($i$)} $\vee\; (\Value=\star )\vee (\direction=\Backward\ \wedge$ \footnotesize {\bf consistent($\Value_i,\children(i)$)}) \\ \footnotesize 10.\>\> \footnotesize then {\bf pass-privilege-to-parent} \\ %\pushtabs %xxxx\= if $(leaf(i)\vee value=\star\vee (direction=\Backward\wedge %children-consistent))$ \= then \kill \footnotesize 11.\>\> \footnotesize else {\bf pass-privilege-to-children} \\ %\poptabs \footnotesize End. } \end{program} The procedure {\em compute-next-consistent-value} (Figure \ref{procedures}) tests each value located after the domain pointer for consistency. More precisely, the domain value is checked against each of $\predecessors (i)$'s values, and the next domain value consistent with the values of the predecessors is returned. The pointer's location is readjusted accordingly (i.e., to the found value), and the mode of the node is set to \On. If no consistent value is found, the value returned is $\star$ and the pointer is reset to the beginning of the domain. The predicate {\em consistent($\val,\setofnodes$)} is \True\ if the value of $\val$ is consistent with the $\Value$ fields of $\setofnodes$ and none of them is dead-ended (has the value $\star$). %Following are the procedures for processor $P_i$ %(we assume that its domain is implemented as an array $Domain_{i}$): \begin{program}{procedures}{Consistency procedure}{ procedure {\bf\small compute-next-consistent-value} \\ \small Begin \\ \small 1.\> \small $\mode_i\gets\On$ \\ \small 2.\> \small while $\pointer \leq$ endof $\Domain_i$ do \\ \small 3.\> \> \small $\pointer \gets \pointer+1$ \\ \small 4.\> \> \small if {\bf consistent($\Domain_i [\pointer],\; \predecessors(i)$)} then \\ \small 5.\> \> \> \small return $\Domain_i [\pointer]$ \> \> \> \> \> \> \small \{ a consistent value was found \} \\ \small 6.\> \small $\pointer \gets 0$ \\ \small 7.\> \small return $\star$ \> \> \> \> \small \{ no consistent value exists \} \\ \small End. \\ } \end{program} The algorithm performed by the root $P_0$, when it is privileged, is simpler. The root does not check consistency. All it does is assign a new value at the end of each backward phase, when needed, and then initiate a new forward phase. The procedure {\em next-value} increments the domain pointer's location and returns the value indicated by the domain pointer. If the end of the domain list is reached, the pointer is reset to the first (smallest) value. The value assignment subprotocol can be regarded as uniform, since each node may have both the root's protocol and the nonroot's protocol and decide between them based on the role assigned to it by the DFS spanning tree protocol. \subsubsection{Proof of self-stabilization} \label{correctness_sec} To prove the correctness of our NC protocol, we first prove that the graph traversal is self-stabilizing, namely, that the system eventually reaches a legally controlled configuration (even if the values in the nodes are not yet consistent), and from that point it remains legally controlled. Assuming the system is legally controlled, we show that if a legal assignment exists, it is eventually reached and thereafter remains unchanged. Thus the system reaches a legal set of configurations and stays there---and therefore is self-stabilizing. Note that a nonroot node is privileged when its inlink is unbalanced (thus it is on a chain) and all its outlinks are balanced. In other words, a nonroot node is privileged if and only if it is a chain tail. The root is privileged if and only if it is not a chain head. Also note that passing the privilege by a node affects only the chains on the branches containing that node, because it has no interaction with other branches. To prove the self-stabilization of the privilege passing mechanism, we first prove some of its properties. \begin{lemma} \label{pass-priv} In every infinite fair execution, every nonroot node that is on a chain eventually passes the privilege to its parent. \end{lemma} \begin{proof} We prove the lemma by induction on the node's height, $h$ (i.e., its distance from the nearest leaf), and on the possible value assignments from the domain of the node. {\em Base:} $h =0$. The node is a leaf and, therefore, when activated, can pass the privilege only to its parent. {\em Step:} Assume node $i$, whose height is $h>0$, is on a chain. Node $i$ eventually becomes privileged because, if any of $i$'s outlinks are unbalanced, then the corresponding children are on chains and the induction hypothesis holds for them, namely, they eventually pass the privileges to $i$. Note that a node that passes its privilege to the parent (to $i$ in our case) does not become privileged again unless its parent had become privileged first and passed the privilege to its children, since outlinks are unbalanced by a privileged node only. If, when becoming privileged, $i$ passes the privilege to its parent, the claim is proven. Otherwise, whenever $i$ passes the privilege to its children the same argument holds, so $i$ eventually becomes privileged again. Moreover, $i$'s domain pointer is advanced every time it passes the privilege to its children. Therefore, after a finite number of such passings, bounded by the size of $\domain_i$, the domain pointer reaches a $\star$ and then, following the code in Figures \ref{update} and \ref{procedures}, $i$ passes the privilege to its parent. \end{proof} \begin{theorem} \label{acm} The graph traversal mechanism is self-stabilizing with respect to the set of legally controlled configurations. Namely, it satisfies the following assertions: \begin{enumerate} \item {Reachability:} Starting from any initial configuration, the system eventually reaches a legally controlled configuration. \item {Closure:} If $c$ is a legally controlled configuration and $c \rightarrow c'$, then $c'$ is also a legally controlled configuration. \end{enumerate} \end{theorem} \begin{proof} We prove the theorem by showing that all the nonroot chain heads in the network eventually disappear. Note that passing a privilege by the root makes the root a chain head but does not increase the number of nonroot chain heads. Passing a privilege by any nonroot node does not create any chain head. When a nonroot node passes the privilege, it is a chain tail (its outlinks are balanced). Thus, if the privilege is passed to the node's children, none of them become a chain head since their parent is still on the same chains. On the other hand, if the privilege is passed to its parent, the node balances its inlink, which cannot possibly create a new chain head. Thus the number of nonroot chain heads in the network never increases. Moreover, Lemma \ref{pass-priv} implies that every nonroot node that is on a chain, and particularly any nonroot chain head, eventually passes the privilege to its parent and stops being a chain head. Therefore, the number of nonroot chain heads steadily decreases until no nonroot chain heads are left, and hence the network is legally controlled. Since no nonroot chain heads are ever created, the network remains legally controlled forever. \end{proof} The self-stabilization property of the NC protocol is inherited from its subprotocols: DFS spanning tree generation and value assignment. Once the self-stabilization of privilege passing is established, it assures that the control is a correct, distributed implementation of DFS-based backjumping, which guarantees the convergence of the network to a legal solution, if one exists, and if not, it repeatedly checks all the possibilities. \subsection{Complexity analysis} \label{complexity} \subsubsection{Time complexity} A worst-case complexity estimate of the value assignment subprotocol, once a DFS tree already exists, can be given by counting the number of state changes until the network becomes {\em legally controlled} and by adding to it the number of state changes before a legal {\em consistent configuration} is reached. These two counts bound the sequential performance of the value assignment protocol and thus, also, its worst-case parallel time. The bound is tight since it can be realized when the depth of the DFS tree equals $n$. In that case only one node is activated at a time. We next bound the sequential time and the parallel time, as a function of the depth of the DFS tree, $m$. We show that in some cases an optimal speedup is realizable. Let $T^1_m$ stand for the maximal number of privilege passings in the subnetwork with a DFS spanning subtree of depth $m$, before its root completes a full round of assigning all of its domain values (if necessary) and passing the privilege forward to its children for every assigned value (for a nonroot node it is the number of privilege passings in its subtree before the node passes the privilege backward). Let $b$ be the maximal branching degree in the DFS spanning tree, and let $k$ bound the domain sizes. Since every time the root of a subtree becomes privileged, it either tries to assign a new value or passes the privilege backward, $T^1_m$ obeys the following recurrence: \begin{eqnarray*} T^1_m & = & k \cdot b \cdot T^1_{m-1}, \\ T^1_0 & = & k. \end{eqnarray*} Solving this recurrence yields $T^1_m = b^m k^{m+1}$, which is the worst-case number of privilege passings before reaching the legally controlled configuration where only the root is privileged. The worst-case number of additional state changes toward a consistent solution (if one exists) is bounded by the worst-case time of sequential graph-based backjumping on the DFS tree-ordering. Let $T^2_m$ stand for the maximal number of value reassignments in the subnetwork with a DFS spanning subtree of depth $m$, before it reaches a solution. This equals the search space explored by the sequential algorithm. Since any assignment of a value to the root node generates $b$ subtrees of depth $m-1$ or less that can be solved independently, $T^2_m$ obeys the same recurrence as $T^1_m$ and results in the same expression: $T^2_m = b^m k^{m+1}$. Thus, the overall worst-case time complexity of the value assignment subprotocol is $T_m = T^1_m+T^2_m = \ord{b^m k^{m+1}}$. Note that when the tree is balanced, we have $T_m = \ord{(n/b)\cdot k^{m+1}}$ since $n= O(b^{m+1})$. Next, we evaluate the parallel time of the value assignment subprotocol, assuming that the privilege passing mechanism has already stabilized into legally controlled configurations (namely, the root is activated). For this analysis we assume that the DFS tree is balanced. Clearly, when the tree is not balanced, the speedup reduces as a function of $b$. Consider now two cases. Assume that the network is backtrack-free. In that case, since there are no dead ends, the sequential time obeys the recurrence \begin{eqnarray*} T^2_m & = &k + b \cdot T^2_{m-1}, \\ T^2_0 & = & 1, \end{eqnarray*} yielding \begin{eqnarray*} T^2_m &=& b^m + k \cdot (b^{m+1} -1 )/(b-1)\\ &=& O(n/b \cdot k). \end{eqnarray*} The parallel time obeys the recurrence \begin{eqnarray*} T^2_m & = & k+ T^2_{m-1}, \\ T^2_0 & = & 1, \end{eqnarray*} yielding $T^2_m = m \cdot k +1 $. The overall speedup in this case is bounded by $n/(b \cdot m)$ where $m = \log_b n$. Consider now the general case while still assuming that the tree is balanced. The sequential complexity, as shown earlier, is $O(n/b \cdot k^{m+1} )$. In that case, since the subtrees are traversed in parallel, the parallel complexity obeys the recurrence \begin{eqnarray*} T^2_m & = & k \cdot T^2_{m-1}, \\ T^2_0 & = & k, \end{eqnarray*} yielding $T^2_m = k^{m+1}$. In this case a speedup of $(n/b)$ seems realizable. In summary, we have shown that, as in the sequential case, our protocol's complexity is exponentially proportional to the depth of the DFS spanning tree; that is, the system has a better chance for a ``quick'' convergence when the DFS spanning tree is of a minimal depth. There is no gain in speedup when the depth of the tree is $n$. However, for balanced trees having a depth of $m = log_b n$, the speedup lies between $n/b$ and $n/(b \cdot m)$. \subsubsection{Space complexity} \label{space} Each node needs to have the information about the constraints with its neighbors. Assuming that the maximum degree in the graph is $d$ and since a constraint can be expressed in $O(k^2)$, each node needs space for $O(d \cdot k^2)$ values. Among the subprotocols, the DFS subprotocol requires the most space, $O(n \cdot \log d )$ bits. Thus there is an overall space requirement for each processor of $O(n \cdot \log d + d \cdot k^2)$ values, using our subprotocols. In \cite{DJPV98} a DFS spanning tree can be accomplished using only $O(\log d)$ space per node, but the time needed for stabilization may be longer. In order to store the whole network information, a processor needs space for $O(n^2 \cdot k^2)$ values, so our distributed treatment is clearly superior to a centralized solution. Note that the log encoding common in the analysis of space requirements may not be feasible in practice for this context, because the communication registers are fixed, values must be communicated, and constraints must be changed during execution. \subsubsection{Speedup of incremental change} Our model allows adaptation to change without external intervention. Moreover, in many cases a change may be locally, and thus quickly, stabilized. For example, suppose, after the network has converged to a solution, that the domain of a variable is made smaller by some external event or that a new constraint is introduced. If these newly enforced restrictions are consistent with the current assignment, there is no change to the solution. Alternatively, if a current assignment does not satisfy the new constraint, at least one processor detects the problem, since it repeatedly checks consistency, and tries to choose an alternative value that satisfies its local environment. If it succeeds and if the neighbors are happy with the new assignment, change stops. It is clear, however, that in the worst case, local changes may cause a complete execution of the algorithm that may be time exponential. \subsubsection{Adding arc consistency} The average performance of the NC protocol can be further improved by adding to it a uniform self-stabilizing {\em arc consistency} subprotocol \cite{MF:complexity}. A network is said to be {\em arc consistent} if, for every value in each node's domain, there is a consistent value in all its neighbors' domains. Arc consistency can be achieved by a repeated execution of a ``relaxation procedure," where each node reads its neighbors' domains and eliminates any of its own values for which there is no consistent value in one of its neighbors' domains. This protocol is clearly self-stabilizing, since the domain sizes are finite, and they can only shrink or be left intact by each activation. As a result, after a finite number of steps all the domains remain unchanged. Since arc consistency can be applied in a uniform and self-stabilizing manner, it suggests that various constraint propagation methods can be incorporated to improve backjumping \cite{dfrost94a}, while maintaining properties of self-stabilization. In particular, the well-known technique of forward checking can be used along with arc consistency during the value assignment subprotocol. If, as a result, a future variable's domain becomes empty, that information can be propagated back to the current privileged variable. The details and impact of such improvements remain to be worked out. \section{Network consistency for trees} \label{tree_sec} It is well known that the sequential network consistency problem on trees is tractable and can be achieved in linear time \cite{MF:complexity}. A special algorithm for this task is composed of an arc consistency phase (explained in the previous section) that can be efficiently implemented on trees, followed by a backtrack-free value assignment in an order created by some {\em rooted tree}. Since the DFS spanning tree subprotocol of our general algorithm was the source of its nonuniformity, we reexamine the possibility that for trees, a rooted directed tree can be imposed via a uniform protocol. We have already shown that when using a distributed scheduler, a uniform, network consistency protocol for trees is not feasible. Therefore, the only avenue not yet explored is whether under a central scheduler such a protocol does exist. We next show that this conjecture is indeed correct. In principle a uniform tree consistency (TC) protocol can be extracted from the general NC protocol by replacing the DFS spanning tree protocol with a uniform rooted tree protocol to direct an undirected tree, since in trees any rooted tree is a DFS tree. Since the arc consistency protocol and the value assignment protocol are already uniform, the resulting TC protocol is uniform. Nevertheless, we show that for trees, the value assignment protocol can be simplified as well, while there is no need for a special privilege-passing mechanism. The proposed TC protocol consists of the three subprotocols: tree directing, arc consistency, and tree value assignment. When the variables' domains are arc consistent and the tree has been directed, value assignment is eventually guaranteed by having each node follow the rule (of the {\em tree value assignment} protocol), ``Choose a value consistent with your parent's assignment.'' Such a value must exist, since otherwise the value assigned by the parent would have been removed by the arc consistency procedure. Since, as we show, the tree-directing protocol is self-stabilizing, and since the arc consistency protocol is self-stabilizing as well, the value assignment protocol eventually converges to a consistent solution. To direct the tree uniformly, we must exploit the topology of the tree to break the symmetry reflected by the identical codes and the lack of identifiers. For this task we use a distributed protocol for finding the {\em centers} of a tree \cite{Korach,KPBG94}. A center of a tree is a node whose maximal distance from the leaves is minimal. Consider a sequential algorithm that works in phases, so that in every phase the leaves of the previous phase are removed from the tree. In the last phase the tree has either one or two connected nodes left. These nodes are the centers of the tree. Note that if we regard a center as the root of the tree, the children of every node are removed from the tree in earlier phases than the node itself (except in the case of two centers that are removed in the same, last, phase of the algorithm), which means that the removal phase of a node is greater than the removal phase of any of its children, and the removal phase of its parent is greater than (or, in the case of two centers, equals) its own. We denote the removal phase of a node in this sequential algorithm as its {\em level}. The level of the leaves is 0. Another way to define the level of a node is its maximal distance to a leaf without passing through a center. Our protocol distributedly simulates the above algorithm. If only one center exists, it declares itself as a root, %(by setting its $root$ field to \True), and all the incident edges are directed toward it. When two centers exist, one of them becomes a root and the link that connects them is directed accordingly. The choice of which center becomes the root is not deterministic and depends on the scheduling order and the initial values. This approach yields a simple, uniform, tree-directing protocol that simulates the above description. Every node $i$ has the following fields: \begin{description} \item{$l_i$} is the level of $i$, a variable that eventually indicates the phase of the sequential algorithm in which $i$ is removed from the tree. %(so that $l = 0$ means that $i$ is a leaf in %the original tree). \item{$\Root_i$} is a boolean field that indicates whether $i$ is the root. %Eventually only one of the centers has the value %\True in this field. \item{$\parent_i$} is a variable assigned the number of the edge leading to the neighbor that becomes the parent of $i$.% (the enumeration is made by the %$\alpha_i$ function, as in the general protocol, and is local to $i$). \shrink{ When the network stabilizes, namely, when all the $l$ values converge, every node except the root has only one neighbor that is eligible to be its parent, and no two nodes are parents of each other. } \end{description} The protocol works by having each node repeatedly compute its own $l$-value by adding 1 to the second-largest value among the $l$-values of its neighbors. Each node except the centers ultimately chooses as its parent the neighbor to which it is still connected whenever it becomes a leaf, namely, that neighbor whose level is greater than its own. A node views itself as a center when one of the following two conditions is satisfied: \begin{enumerate} \item Its $l$-value is greater than all of its neighbors', which means that it is a single center. \item Its $l$-value is equal to that of its largest neighbor---the other center. In this case, the node checks whether the other center is already the root. If so, it chooses the other center to be its parent; otherwise it becomes the root. \end{enumerate} Because the $l$-values converge to the correct values of the level of the node, the above interpretation is ultimately accurate. Clearly, once the $l$-values converge, each node properly chooses its parent, and the direction of the tree is completed. Therefore, it is sufficient to prove the convergence of the $l$-values to the levels of the nodes. A proper convergence of the $l$-values can be proved by induction on the levels of the nodes. For more details of the tree-directing algorithm and its proof see \cite{collin:full}. The parallel time can be linearly bounded by the {\em diameter} ($\Dim$) of the tree, where the diameter is the longest path between any two leaves of the tree. Since in the worst case the diameter of a tree equals the number of nodes, $n$, the space that is required in this case to hold the level of each node is $\ord{\log{\Dim}}$. A different self-stabilizing tree-directing algorithm is presented in \cite{pinkas-dechter}. In that algorithm, any node of the tree may become the root, depending on the initial configuration and the schedule. Although that algorithm is usually better in its space requirements than the one presented above, forcing a center to become the root, as is done here, yields a more balanced tree. \section{Conclusions} \label{conclusions_sec} The results presented in this paper establish theoretical bounds on the capabilities of distributed self-stabilizing architectures to solve constraint satisfaction problems. The paper focuses on the feasibility of solving the network consistency problem using self-stabilizing distributed protocols, namely, guaranteeing convergence to a consistent solution, if such exists, from {\em any} initial configuration. We have proved that, when any scheduler is allowed, a uniform protocol (one in which all nodes are identical) cannot solve the network consistency problem, even if only one node is activated at a time (i.e., when using a central scheduler). Consequently, although such protocols have obvious advantages, they cannot guarantee convergence to a solution. On the other hand, distinguishing one single node from the rest is sufficient to guarantee such a convergence even when sets of nodes are activated simultaneously. A protocol for solving the problem under such conditions is presented. Note that the negative results are established under a model requiring convergence for every central or distributed schedule and is not applicable to many cases where the schedule is restricted. We demonstrated that when the network is restricted to trees, a uniform, self-stabilizing protocol exists for solving the problem with any central scheduler, where only one neighboring node is activated at a time. Note also that the restriction to a central scheduler is not as severe as it might at first appear. Any protocol that works with a central scheduler can also be implemented with a distributed scheduler that obeys the restriction that two neighboring nodes are never activated together. It is still an open question whether a uniform protocol is feasible for general graphs under restricted scheduling policies (e.g., round-robin). Regarding time complexity, we showed that in the worst case the distributed and the sequential protocols have the same complexity bound: exponential in the depth of the DFS tree. On the average, however, a speedup between $n/b$ and $n/(b \cdot m)$ is possible, where $n$ is the number of nodes and $m$ is the DFS's tree depth. We also argued that when the environment undergoes local change, the solution to the network consistency problem can often be repaired quickly (but not for all cases), due to the inherent local computation of the distributed architecture. \section*{Acknowledgments} We thank Shlomo Moran for help in the proof of the root-finding protocol and Arun Jagota for his advice on related work and for useful discussions. \subsection*{Acknowledgment of support} Dechter was partially supported by the National Science Foundation, grant number IIS9610015, and by the Air Force Office of Scientific Research, grant number AFOSR-90-0136. Katz was partially supported by the Argentinian Research Fund at the Technion-Israel Institute of Technology. \bibliography{collin} \bibliographystyle{alpha} \end{document}