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\section{The Graph Library} \subsection{Overview} Graphs are the most fundamental data structure in the \MLRISC{} system, and in fact in many optimizing compilers. \MLRISC{} now contains an extensive library for working with graphs. All graphs in \MLRISC{} are modeled as edge- and node-labeled directed multi-graphs. Briefly, this means that nodes and edges can carry user supplied data, and multiple directed edges can be attached between any two nodes. Self-loops are also allowed. A node is uniquely identified by its \sml{node_id}, which is simply an integer. Node ids can be assigned externally by the user, or else generated automatically by a graph. All graphs keep track of all node ids that are currently used, and the method \sml{new_id : unit -> node_id} generates a new unused id. A node is modeled as a node id and node label pair, $(i,l)$. An edge is modeled as a triple $i \edge{l} j$, which contains the \newdef{source} and \newdef{target} node ids $i$ and $j$, and the edge label $l$. These types are defined as follows: \begin{SML} type 'n node = node_id * 'n type 'e edge = node_id * node_id * 'e \end{SML} \subsubsection{The graph signature} All graphs are accessed through an abstract interface of the polymorphic type \sml{('n,'e,'g) graph}. Here, \sml{'n} is the type of the node labels, \sml{'e} is the type of the edge labels, and \sml{'g} is the type of any extra information embedded in a graph. We call the latter \sml{graph info}. Formally, a graph $G$ is a quadruple $(V,L,E,I)$ where $V$ is a set of node ids, $L : V -> 'a$ is a node labeling function from vertices to node labels, $E$ is a multi-set of labeled-edges of type $V * V * 'e$, and $I: 'g$ is the graph info. The interface of a graph is packaged into a record of methods that manipulate the base representation: \begin{SML} signature \mlrischref{graphs/graph.sig}{GRAPH} = sig type node_id = int type 'n node = node_id * 'n type 'e edge = node_id * node_id * 'e exception Graph of string exception Subgraph exception NotFound exception Unimplemented exception Readonly datatype ('n,'e,'g) graph = GRAPH of ('n,'e,'g) graph_methods withtype ('n,'e,'g) graph_methods = \{ name : string, graph_info : 'g, (* selectors *) (* mutators *) (* iterators *) \} end \end{SML} A few exceptions are predefined in this signature, which have the following informal interpretation. Exception \sml{Graph} is raised when a bug is encountered. The exception \sml{Subgraph} is raised if certain semantics constraints imposed on a graph are violated. The exception \sml{NotFound} is raised if lookup of a node id fails. The exception \sml{Unimplemented} is raised if a certain feature is accessed but is undefined on the graph. The exception \sml{Readonly} is raised if the graph is readonly and an update operation is attempted. \subsubsection{Selectors} Methods that access the structure of a graph are listed below: \begin{methods} nodes : unit -> $'n$ node list & Return a list of all nodes in a graph em \\ edges : unit -> $'e$ edge list & Return a list of all edges in a graph \\ order : unit -> int & Return the number of nodes in a graph. The graph is empty if its order is zero \\ size : unit -> int & Return the number of edges in a graph \\ capacity : unit -> int & Return the maximum node id in the graph, plus 1. This can be used as a new id \\ succ : node\_id -> node\_id list & Given a node id $i$, return the node ids of all its successors, i.e. $\{ j | i \edge{l} j \in E\}$. \\ pred : node\_id -> node\_id list & Given a node id $j$, return the node ids of all its predecessors, i.e. $\{ i | i \edge{l} j \in E\}$. \\ out\_edges : node\_id -> $'e$ edge list & Given a node id $i$, return all the out-going edges from node $i$, i.e. all edges whose source is $i$. \\ in\_edges : node\_id -> $'e$ edge list & Given a node id $j$, return all the in-coming edges from node $j$, i.e. all edges whose target is $j$. \\ has\_edge : node\_id * node\_id -> bool & Given two node ids $i$ and $j$, find out if an edge with source $i$ and target $j$ exists. \\ has\_node : node\_id -> bool & Given a node id $i$, find out if a node of id $i$ exists. \\ node\_info : node\_id -> $'n$ & Given a node id, return its node label. If the node does not exist, raise exception \sml{NotFound}. \\ \end{methods} \subsubsection{Graph hierarchy} A graph $G$ may in fact be a subgraph of a \newdef{base graph} $G'$, or obtained from $G'$ via some transformation $T$. In such cases the following methods may be used to determine of the relationship between $G$ and $G'$. An \newdef{entry edge} is an edge in $G'$ that terminates at a node in $G$, but is not an edge in $G$. Similarly, an \newdef{exit edge} is an edge in $G'$ that originates from a node in $G$, but is not an edge in $G$. An \newdef{entry node} is a node in $G$ that has an incoming entry edge. An \newdef{exit node} is a node in $G$ that has an out-going exit edge. If $G$ is not a subgraph, all these methods will return nil. \begin{methods} entries : unit -> node\_id list & Return the node ids of all the entry nodes. \\ exits : unit -> node\_id list & Return the node ids of all the exit nodes. \\ entry\_edges : node\_id -> $'e$ edge list & Given a node id $i$, return all the entry edges whose sources are $i$. \\ exit\_edges : node\_id -> $'e$ edge list & Given a node id $i$, return all the exit edges whose targets are $i$. \end{methods} \subsubsection{Mutators} Methods to update a graph are listed below: \begin{methods} new\_id : unit -> node\_id & Return a unique node id guaranteed to be absent in the current graph. \\ add\_node : 'n node -> unit & Insert node into the graph. If a node of the same node id already exists, replace the old node with the new. \\ add\_edge : 'e edge -> unit & Insert an edge into the graph. \\ remove\_node : node\_id -> unit & Given a node id $n$, remove the node with the node id from the graph. This also automatically removes all edges with source or target $n$. \\ set\_out\_edges : node\_id * 'e edge list -> unit & Given a node id $n$, and a list of edges $e_1,\ldots,e_n$ with sources $n$, replace all out-edges of $n$ by $e_1,\ldots,e_n$. \\ set\_in\_edges : node\_id * 'e edge list -> unit & Given a node id $n$, and a list of edges $e_1,\ldots,e_n$ with targets $n$, replace all in-edges of $n$ by $e_1,\ldots,e_n$. \\ set\_entries : node\_id list -> unit & Set the entry nodes of a graph. \\ set\_exits : node\_id list -> unit & Set the exit nodes of a graph. \\ garbage\_collect : unit -> unit & Reclaim all node ids of nodes that have been removed by \sml{remove_node}. Subsequent \sml{new_id} will reuse these node ids. \\ \end{methods} \subsubsection{Iterators} Two primitive iterators are supported in the graph interface. Method \sml{forall_nodes} iterates over all the nodes in a graph, while method \sml{forall_edges} iterates over all the edges. Other more complex iterators can be found in other modules. \begin{methods} forall\_nodes : ($'n$ node -> unit) -> unit & Given a function $f$ on nodes, apply $f$ on all the nodes in the graph. \\ forall\_edges : ($'e$ edge -> unit) -> unit & Given a function $f$ on edges, apply $f$ on all the edges in the graph. \end{methods} \subsubsection{Manipulating a graph} Since operations on the graph type are packaged into a record, an ``object oriented'' style of graph manipulation should be used. For example, if \sml{G} is a graph object, then we can obtain all the nodes and edges of \sml{G} as follows. \begin{SML} val GRAPH g = G val edges = #edges g () val nodes = #nodes g () \end{SML} We can view \sml{#edges g} as sending the message to \sml{G}. While all this seems like mere syntactic deviation from the usual signature/structure approach, there are two crucial differences which we will exploit: \emph{(i)} records are first class objects while structures are not (consequently late binding of ``methods'' and cannot be easily simulated on the structure level); \emph{(ii)} recursion is possible on the type level, while recursive structures are not available. The extra flexibility of this choice becomes apparent with the introduction of views later. \subsubsection{Creating a Graph} A graph implementation has the following signature \begin{SML} signature \mlrischref{graphs/graphimpl.sig}{GRAPH_IMPLEMENTATION} = sig val graph : string * 'g * int -> ('n,'e,'g) graph end \end{SML} The function \sml{graph} takes a string (the name of the graph), graph info, and a default size as arguments and create an empty graph. The functor \sml{DirectedGraph}: \begin{SML} functor DirectedGraph(A : ARRAY_SIG) : GRAPH_IMPLEMENTATION \end{SML} implements a graph using adjacency lists as internal representation. It takes an array type as a parameter. For graphs with node ids that are dense enumerations, the \sml{DynamicArray} structure should be used as the parameter to this functor. The structure \sml{DirectedGraph} is predefined as follows: \begin{SML} structure \mlrischref{graphs/digraph.sml}{DirectedGraph} = DirectedGraph(DynamicArray) \end{SML} For node ids that are sparse enumerations, the structure \sml{HashArray}, which implements integer-keyed hash tables with the signature of arrays, can be used as argument to \sml{DirectedGraph}. For graphs with fixed sizes determined at creation time, the structure \sml{Array} can be used (see also functor \mlrischref{library/undoable-array.sml}{\sml{UndoableArray}}, which creates arrays with undoable updates, and transaction-like semantics.) \subsubsection{Basic Graph Algorithms} \subsubsection{Depth-/Breath-First Search} \begin{SML} val dfs : ('n,'e,'g) graph -> (node_id -> unit) -> ('e edge -> unit) -> node_id list -> unit \end{SML} The function \sml{dfs} takes as arguments a graph, a function \sml{f : node_id -> unit}, a function \sml{g : 'e edge -> unit}, and a set of source vertices. It performs depth first search on the graph. The function \sml{f} is invoked whenever a new node is being visited, while the function \sml{g} is invoked whenever a new edge is being traversed. This algorithm has running time $O(|V|+|E|)$. \begin{SML} val dfsfold : ('n,'e,'g) graph -> (node_id * 'a -> 'a) -> ('e edge * 'b -> 'a) -> node_id list -> 'a * 'b -> 'a * 'b val dfsnum : ('n,'e,'g) graph -> (node_id * 'a -> 'a) -> { dfsnum : int array, compnum : int array } \end{SML} The function \sml{bfs} is similar to \sml{dfs} except that breath first search is performed. \begin{SML} val bfs : ('n,'e,'g) graph -> (node_id -> unit) -> ('e edge -> unit) -> node_id list -> unit val bfsdist : ('n,'e,'g) graph -> node_id list -> int array \end{SML} \subsubsection{Preorder/Postorder numbering} \begin{SML} val preorder_numbering : ('n,'e,'g) graph -> int -> int array val postorder_numbering : ('n,'e,'g) graph -> int -> int array \end{SML} Both these functions take a tree $T$ and a root $v$, and return the preorder numbering and the postorder numbering of the tree respectively. \subsubsection{Topological Sort} \begin{SML} val topsort : ('n,'e,'g) graph -> node_id list -> node_id list \end{SML} The function \sml{topsort} takes a graph $G$ and a set of source vertices $S$ as arguments. It returns a topological sort of all the nodes reachable from the set $S$. This algorithm has running time $O(|S|+|V|+|E|)$. \subsubsection{Strongly Connected Components} \begin{SML} val strong_components : ('n,'e,'g) graph -> (node_id list * 'a -> 'a) -> 'a -> 'a \end{SML} The function \sml{strong_components} takes a graph $G$ and an aggregate function $f$ with type \begin{SML} node_id list * 'a -> 'a \end{SML} \noindent and an identity element \sml{x : 'a} as arguments. Function $f$ is invoked with a strongly connected component (represented as a list of node ids) as each is discovered. That is, the function \sml{strong_components} computes \[ f(SCC_n,f(SCC_{n-1},\ldots, f(SCC_1,x))) \] where $SCC_1,\ldots,SCC_n$ are the strongly connected components in topological order. This algorithm has running time $O(|V|+|E|)$. \subsubsection{Biconnected Components} \begin{SML} val biconnected_components : ('n,'e,'g) graph -> ('e edge list * 'a -> 'a) -> 'a -> 'a \end{SML} The function \sml{biconnected_components} takes a graph $G$ and an aggregate function $f$ with type \begin{SML} 'e edge list * 'a -> 'a \end{SML} \noindent and an identity element \sml{x : 'a} as arguments. Function $f$ is invoked with a biconnected component (represented as a list of edges) as each is discovered. That is, the function \sml{biconnected_components} computes \[ f(BCC_n,f(BCC_{n-1},\ldots, f(BCC_1,x))) \] where $BCC_1,\ldots,BCC_n$ are the biconnected components. This algorithm has running time $O(|V|+|E|)$. \subsubsection{Cyclic Test} \begin{SML} val is_cyclic : ('n,'e,'g) graph -> bool \end{SML} Function \sml{is_cyclic} tests if a graph is cyclic. This algorithm has running time $O(|V|+|E|)$. \subsubsection{Enumerate Simple Cycles} \begin{SML} val cycles : ('n,'e,'g) graph -> ('e edge list * 'a -> 'a) -> 'a ->'a \end{SML} A simple cycle is a circuit that visits each vertex only once. The function \sml{cycles} enumerates all simple cycles in a graph $G$. It takes as argument an aggregate function $f$ of type \begin{SML} 'e edge list * 'a -> 'a \end{SML} and an identity element $e$, and computes as result the expression \[ f(c_n,f(c_{n-1},f(c_{n-2},\ldots, f(c_1,e)))) \] where $c_1,\ldots,c_n$ are all the simple cycles in the graph. All cycles $c_1,\ldots,c_n$ are guaranteed to be distinct. A cycle is represented as a sequence of adjacent edges, i.e. in the order of \[ v_1 -> v_2, v_2 -> v_3, v_3 -> v_4, \ldots, v_{n-1} -> v_n, v_n -> v_1 \] Our implementation works by first decomposing the graph into its strongly connected components, then uses backtracking to enumerate simple cycles in each component. \subsubsection{Minimal Cost Spanning Tree} \begin{SML} signature \mlrischref{graphs/spanning-tree.sig}{MIN_COST_SPANNING_TREE} = sig exception Unconnected val spanning_tree : \{ weight : 'e edge -> 'a, < : 'a * 'a -> bool \} -> ('n, 'e, 'g) graph -> ('e edge * 'a -> 'a) -> 'a -> 'a end structure \mlrischref{graphs/kruskal.sml}{Kruskal} : MIN_COST_SPANNING_TREE \end{SML} Structure \sml{Kruskal} implements Kruskal's algorithm for computing a minimal cost spanning tree of a graph. The function \sml{spanning_tree} takes as arguments: \begin{itemize} \item a \sml{weight} function which when given an edge returns its weight \item an ordering function \sml{<}, which is used to compare the weights \item a graph $G$ \item an accumulator function $f$, and \item an identity element $x$ \end{itemize} The function \sml{spanning_tree} computes \[ f(e_{n},f(e_{n-1},\ldots, f(e_1,x))) \] where $e_1,\ldots,e_n$ are the edges in a minimal cost spanning tree of the graph. The exception \sml{Unconnected} is raised if the graph is unconnected. \subsubsection{Abelian Groups} Graph algorithms that deal with numeric weights or distances are parameterized with respect to the signatures \sml{ABELIAN_GROUP} or \sml{ABELIAN_GROUP_WITH_INF}. These are defined as follows: \begin{SML} signature \mlrischref{graphs/groups.sig}{ABELIAN_GROUP} = sig type elem val + : elem * elem -> elem val - : elem * elem -> elem val : elem -> elem val zero : elem val < : elem * elem -> bool val == : elem * elem -> bool end signature \mlrischref{graphs/groups.sig}{ABELIAN_GROUP_WITH_INF} = sig include ABELIAN_GROUP val inf : elem end \end{SML} Signature \sml{ABELIAN_GROUP} specifies an ordered commutative group, while signature \sml{ABELIAN_GROUP_WITH_INF} specifies an ordered commutative group with an infinity element \sml{inf}. \subsubsection{Single Source Shortest Paths} \begin{SML} signature \mlrischref{graphs/shortest-paths.sig}{SINGLE_SOURCE_SHORTEST_PATHS} = sig structure Num : ABELIAN_GROUP_WITH_INF val single_source_shortest_paths : \{ graph : ('n,'e,'g) graph, weight : 'e edge -> Num.elem, s : node_id \} -> \{ dist : Num.elem array, pred : node_id array \} end functor \mlrischref{graphs/dijkstra.sml}{Dijkstra}(Num : ABELIAN_GROUP_WITH_INF) : SINGLE_SOURCE_SHORTEST_PATHS \end{SML} The functor \sml{Dijkstra} implements Dijkstra's algorithm for single source shortest paths. The function \linebreak \sml{single_source_shortest_paths} takes as arguments: \begin{itemize} \item a graph $G$, \item a \sml{weight} function on edges, and \item the source vertex $s$. \end{itemize} It returns two arrays \sml{dist} and \sml{pred} indexed by vertices. These arrays have the following interpretation. Given a vertex $v$, \begin{itemize} \item \sml{dist}[$v$] contains the distance of $v$ from the source $s$ \item \sml{pred}[$v$] contains the predecessor of $v$ in the shortest path from $s$ to $v$, or -1 if $v=s$. \end{itemize} Dijkstra's algorithm fails to work on graphs that have negative edge weights. To handle negative weights, Bellman-Ford's algorithm can be used. The exception \sml{NegativeCycle} is raised if a cycle of negative total weight is detected. \begin{SML} functor \mlrischref{graphs/bellman-ford.sml}{BellmanFord}(Num : ABELIAN_GROUP_WITH_INF) : sig include SINGLE_SOURCE_SHORTEST_PATHS exception NegativeCycle end \end{SML} \subsubsection{All Pairs Shortest Paths} \begin{SML} signature \mlrischref{graphs/shortest-paths.sig}{ALL_PAIRS_SHORTEST_PATHS} = sig structure Num : ABELIAN_GROUP_WITH_INF val all_pairs_shortest_paths : \{ graph : ('n,'e,'g) graph, weight : 'e edge -> Num.elem \} -> \{ dist : Num.elem Array2.array, pred : node_id Array2.array \} end functor \mlrischref{graphs/floyd-warshall.sml}{FloydWarshall}(Num : ABELIAN_GROUP_WITH_INF) : ALL_PAIRS_SHORTEST_PATHS \end{SML} The functor \sml{FloydWarshall} implements Floyd-Warshall's algorithm for all pairs shortest paths. The function \sml{all_pairs_shortest_paths} takes as arguments: \begin{itemize} \item a graph $G$, and \item a \sml{weight} function on edges \end{itemize} It returns two 2-dimensional arrays \sml{dist} and \sml{pred} indexed by vertices $(u,v)$. These arrays have the following interpretation. Given a pair $(u,v)$, \begin{itemize} \item \sml{dist}[$u,v$] contains the distance from $u$ to $v$. \item \sml{pred}[$u,v$] contains the predecessor of $v$ in the shortest path from $u$ to $v$, or $-1$ if $u=v$. \end{itemize} This algorithm runs in time $O(|V|^3+|E|)$. An alternative implementation is available that uses Johnson's algorithm, which works better for sparse graphs: \begin{SML} functor \mlrischref{graphs/johnson.sml}{Johnson}(Num : ABELIAN_GROUP_WITH_INF) : sig include ALL_PAIRS_SHORTEST_PATHS exception Negative Cycle end \end{SML} \subsubsection{Transitive Closure} \begin{SML} signature \mlrischref{graphs/trans-closure.sml}{TRANSITIVE_CLOSURE} = sig val acyclic_transitive_closure : { + : ('e * 'e -> 'e), simple : bool } -> ('n,'e,'g) graph -> unit val acyclic_transitive_closure2 : \{ + : 'e * 'e -> 'e, max : 'e * 'e -> 'e \} -> ('n,'e,'g) graph -> unit val transitive_closure : ('e * 'e -> 'e) -> ('n,'e,'g) graph -> unit structure \mlrischref{graphs/trans-closure.sml}{TransitiveClosure} : TRANSITIVE_CLOSURE \end{SML} Structure \sml{TransitiveClosure} implements in-place transitive closures on graphs. Three functions are implemented. Functions \sml{acyclic_transitive_closure} and \sml{acyclic_transitive_closure2} can be used to compute the transitive closure of an acyclic graph, whereas the function \sml{transitive_closure} computes the transitive closure of a cyclic graph. All take a binary function \begin{SML} + : 'e * 'e -> 'e \end{SML} defined on edge labels. Transitive edges are inserted in the following manner: \begin{itemize} \item \sml{acyclic_transitive_closure}: given $u \edge{l} v$ and $v \edge{l'} w$, if the flag \sml{simple} is false or if the transitive edge $u \rightarrow w$ does not exists, then $u \edge{l + l'} w$ is added to the graph. \item \sml{acyclic_transitive_closure2}: given $u \edge{l} v$ and $v \edge{l'} w$, the transitive $u \edge{l + l'} w$ is added to the graph. Furthermore, all parallel edges \[ u \edge{l_1} w, \ldots, u \edge{l_n} w \] are coalesced into a single edge $u \edge{l} w$, where $l = {\tt max}_{i = 1 \ldots n} l_i$ \end{itemize} \subsubsection{Max Flow} The function \sml{max_flow} computes the maximum flow between the source vertex \sml{s} and the sink vertex \sml{t} in the \sml{graph} when given a \sml{capacity} function. \begin{SML} signature \mlrischref{graphs/max-flow.sig}{MAX_FLOW} = sig structure Num : ABELIAN_GROUP val max_flow : \{ graph : ('n,'e,'g) graph, s : node_id, t : node_id, capacity : 'e edge -> Num.elem, flows : 'e edge * Num.elem -> unit \} -> Num.elem end functor \mlrischref{graphs/max-flow.sml}{MaxFlow}(Num : ABELIAN_GROUP) : MAX_FLOW \end{SML} The function \sml{max_flow} returns its result in the follow manner: The function returns the total flow as its result value. Furthermore, the function \sml{flows} is called once for each edge $e$ in the graph with its associated flow $f_e$. This algorithm uses Goldberg's preflow-push approach, and runs in $O(|V|^2|E|)$ time. \subsubsection{Min Cut} The function \sml{min_cut} computes the minimum (undirected) cut in a \sml{graph} when given a \sml{weight} function on its edges. \begin{SML} signature \mlrischref{graphs/min-cut.sig}{MIN_CUT} = sig structure Num : ABELIAN_GROUP val min_cut : \{ graph : ('n,'e,'g) graph, weight : 'e edge -> Num.elem \} -> node_id list * Num.elem end functor \mlrischref{graphs/min-cut.sml}{MinCut}(Num : ABELIAN_GROUP) : MIN_CUT \end{SML} The function \sml{min_cut} returns a list of node ids denoting one side of the cut $C$ (the other side of the cut is $(V - C)$ and the weight cut. \subsubsection{Max Cardinality Matching} \begin{SML} val matching : ('n,'e,'g) graph -> ('e edge * 'a -> 'a) -> 'a -> 'a * int \end{SML} The function \sml{BipartiteMatching.matching} computes the maximal cardinality matching of a bipartite graph. As result, the function iterates over all the matched edges and returns the number of matched edges. The algorithm runs in time $O(|V||E|)$. \subsubsection{Node Partition} \begin{SML} signature NODE_PARTITION = sig type 'n node_partition val node_partition : ('n,'e,'g) graph -> 'n node_partition val !! : 'n node_partition -> node_id -> 'n node val == : 'n node_partition -> node_id * node_id -> bool val union : 'n node_partition -> ('n node * 'n node -> 'n node) -> node_id * node_id -> bool val union': 'n node_partition -> node_id * node_id -> bool end \end{SML} \subsubsection{Node Priority Queue} \begin{SML} signature NODE_PRIORITY_QUEUE = sig type node_priority_queue exception EmptyPriorityQueue val create : (node_id * node_id -> bool) -> node_priority_queue val fromGraph : (node_id * node_id -> bool) -> ('n,'e,'g) graph -> node_priority_queue val isEmpty : node_priority_queue -> bool val clear : node_priority_queue -> unit val min : node_priority_queue -> node_id val deleteMin : node_priority_queue -> node_id val decreaseWeight : node_priority_queue * node_id -> unit val insert : node_priority_queue * node_id -> unit val toList : node_priority_queue -> node_id list end \end{SML} \subsection{Views}\label{sec:views} Simply put, a view is an alternative presentation of a data structure to a client. A graph, such as the control flow graph, frequently has to be presented in different ways in a compiler. For example, when global scheduling is applied on a region (a subgraph of the CFG), we want to be able to concentrate on just the region and ignore all nodes and edges that are not part of the current focus. All transformations that are applied on the current region view should be automatically reflected back to the entire CFG as a whole. Furthermore, we want to be able to freely intermix graphs and subgraphs of the same type in our program, without having to introducing sums in our type representations. The \sml{subgraph_view} view combinator accomplishes this. \sml{Subgraph} takes a list of nodes and produces a graph object which is a view of the node induced subgraph of the original graph. All modification to the subgraph are automatically reflected back to the original graph. From the client point of view, a graph and a subgraph are entirely indistinguishable, and furthermore, graphs and subgraphs can be freely mixed together (they are the same type from ML's point of view.) This transparency is obtained by selective method overriding, composition, and delegation. For example, a generic graph object provides the following methods for setting and looking up the entries and exits from a graph. \begin{SML} set_entries : node_id list -> unit set_exits : node_id list -> unit entries : unit -> node_id list exits : unit -> node_id list \end{SML} For example, a CFG usually has a single entry and a single exit. These methods allow the client to destinate one node as the entry and another as the exit. In the case of subgraph view, these methods are overridden so that the proper conventions are preserved: a node in a subgraph is an entry (exit) iff there is an in-edge (out-edge) from (to) outside the (sub-)graph. Similarly, the methods \sml{entry_edges} and \sml{exit_edges} can be used return the entry and exit edges associated with a node in a subgraph. \begin{SML} entry_edges : node_id -> 'e edge list exit_edges : node_id -> 'e edge list \end{SML} These methods are initially defined to return \sml{[]} in a graph and subsequently overridden in a subgraph. \subsubsection{ Update Transparency } Suppose a view $G'$ is created from some base graphs or views. \newdef{Update transparency} refers to the fact that $G'$ behaves consistently according to its conventions and semantics when updates are performed. There are 4 different type of update transparencies: \begin{itemize} \item\newdef{update opaque} A update opaque view disallows updates to both itself and its base graphs. \item\newdef{globally update transparent} A globally update transparent view allows updates to its base graphs but not to itself. Changes will then be automatically reflected in the view. \item\newdef{locally update transparent} A locally update transparent view allows updates to itself but not to its base graphs. Changes will be automatically reflected to the base graphs. \item\newdef{fully update transparent} A fully update transparent view allows updates through its methods or through its base graphs'. \end{itemize} \subsubsection{Structural Views}\label{sec:structural-views} \subsubsection{Reversal} \begin{SML} val \mlrischref{graphs/revgraph.sml}{ReversedGraphView.rev_view} : ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} This combinator takes a graph $G$ and produces a view $G^R$ which reverses the direction of all its edges, including entry and exit edges. Thus the edge $i \edge{l} j$ in $G$ becomes the edge $j \edge{l} i$ in $G^R$. This view is fully update transparent. \subsubsection{Readonly} \begin{SML} val \mlrischref{graphs/readonly.sml}{ReadOnlyGraphView.readonly_view} : ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} This function takes a graph $G$ and produces a view $G'$ in which no mutator methods can be used. Invoking a mutator method raises the exception \sml{Readonly}. This view is globally update transparent. \subsubsection{Snapshot} \begin{SML} functor \mlrischref{graphs/snap-shot.sml}{GraphSnapShot}(GI : GRAPH_IMPLEMENTATION) : GRAPH_SNAPSHOT signature GRAPH_SNAPSHOT = sig val snapshot : ('n,'e,'g) graph -> \{ picture : ('n,'e,'g) graph, button : unit -> unit \} end \end{SML} The function \sml{snapshot} can be used to keep a cached copy of a view a.k.a the \sml{picture}. This cached copy can be updated locally but the modification will not be reflected back to the base graph. The function \sml{button} can be used to keep the view and the base graph up-to-date. \subsubsection{Map} \begin{SML} val \mlrischref{graphs/isograph.sml}{IsomorphicGraphView.map} : ('n node -> 'n') -> ('e edge -> 'e') -> ('g -> 'g') -> ('n,'e,'g) graph -> ('n','e','g') graph \end{SML} The function \sml{map} is a generalization of the \sml{map} function on lists. It takes three functions \begin{SML} f : 'n node -> 'n g : 'e edge -> 'e h : 'g -> g' \end{SML} and a graph $G=(V,L,E,I)$ as arguments. It computes the view $G'=(V,L',E',I')$ where \begin{eqnarray*} L'(v) & = & f(v,L(v)) \mbox{\ for all $v \in V$} \\ E' & = & { i \edge{g(i,j,l)} j | i \edge{l} j \in E } \\ I' & = & h(I) \end{eqnarray*} \subsubsection{Singleton} \begin{SML} val \mlrischref{graphs/singleton.sml}{SingletonGraphView.singleton_view} : ('n,'e,'g) graph -> node_id -> ('n,'e,'g) graph \end{SML} Function \sml{singleton_view} takes a graph $G$ and a node id $v$ (which must exists in $G$) and return an edge-free graph with only one node ($v$). This view is opaque. \subsubsection{Node id renaming} \begin{SML} val \mlrischref{graphs/renamegraph.sml}{RenamedGraphView.rename_view} : int -> ('n,'e,'g) graph -> ('n','e','g') graph \end{SML} The function \sml{rename_view} takes an integer $n$ and a graph $G$ and create a fully update transparent view where all node ids are incremented by $n$. Formally, given graph $G=(V,E,L,I)$ it computes the view $G'=(V',E',L',I)$ where \begin{eqnarray*} V' & = & { v + n | v \in V } \\ E' & = & { i+n \edge{l} j+n | i \edge{l} j \in E } \\ L' & = & \lambda v. L(v-n) \end{eqnarray*} \subsubsection{Union and Sum} \begin{SML} val \mlrischref{graphs/uniongraph.sml}{UnionGraphView.union_view} : ('g * 'g') -> 'g'') -> ('n,'e,'g) graph * ('n,'e,'g') graph -> ('n','e','g'') graph GraphCombinations.unions : ('n,'e,'g) graph list -> ('n,'e,'g) graph GraphCombinations.sum : ('n,'e,'g) graph * ('n,'e,'g) graph -> ('n,'e,'g) graph GraphCombinations.sums : ('n,'e,'g) graph list -> ('n,'e,'g) graph \end{SML} Function \sml{union_view} takes as arguments a function $f$, and two graphs $G=(V,L,E,I)$ and $G'=(V',L',E',I')$, it computes the union $G+G'$ of these graphs. Formally, $G \union G'=(V'',L'',E'',I'')$ where \begin{eqnarray*} V'' & = & V \union V' \\ L'' & = & L \overrides L' \\ E'' & = & E \union E' \\ I'' & = & f(I,I') \end{eqnarray*} The function \sml{sum} constructs a \newdef{disjoint sum} of two graphs. \subsubsection{Simple Graph View} \begin{SML} val \mlrischref{graphs/simple-graph.sml}{SimpleGraph.simple_graph} : (node_id * node_id * 'e list -> 'e) -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} Function \sml{simple_graph} takes a merge function $f$ and a multi-graph $G$ as arguments and return a view in which all parallel multi-edges (edges with the same source and target) are combined into a single edge: i.e. any collection of multi-edges between the same source $s$ and target $t$ and with labels $l_1,\ldots,l_n$, are replaced by the edge $s \edge{l_{st}} t$ in the view, where $l_{st} = f(s,t,[l_1,\ldots,l_n])$. The function $f$ is assumed to satisfy the equality $l = f(s,t,[l])$ for all $l$, $s$ and $t$. \subsubsection{No Entry or No Exit} \begin{SML} val \mlrischref{graphs/no-exit.sml}{NoEntryView.no_entry_view} : ('n,'e,'g) graph -> ('n,'e,'g) graph NoEntryView.no_exit_view : ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} The function \sml{no_entry_view} creates a view in which all entry edges (and thus entry nodes) are removed. The function \sml{no_exit_view} is the dual of this and creates a view in which all exit edges are removed. This view is fully update transparent. It is possible to remove all entry and exit edges by composing these two functions. \subsubsection{Subgraphs} \begin{SML} val \mlrischref{graphs/subgraph.sml}{SubgraphView.subgraph_view} : node_id list -> ('e edge -> bool) -> ('n,'e,'g) graph -> (n','e','g') graph \end{SML} The function \sml{subgraph_view} takes as arguments a set of node ids $S$, an edge predicate $p$ and a graph $G=(V,L,E,I)$. It returns a view in which only the visible nodes are $S$ and the only visible edges $e$ are those that satisfy $p(e)$ and with sources and targets in $S$. $S$ must be a subset of $V$. \begin{SML} val \mlrischref{graphs/subgraph-p.sml}{Subgraph_P_View.subgraph_p_view} : node_id list -> (node_id -> bool) -> (node_id * node_id -> bool) -> ('n,'e,'g) graph -> ('n','e','g') graph \end{SML} The function \sml{subgraph_view} takes as arguments a set of node ids $S$, a node predicate $p$, an edge predicate $q$ and a graph $G=(V,L,E,I)$. It returns a view in which only the visible nodes $v$ are those in $S$ satisfying $p(v)$, and the only visible edges $e$ are those that satisfy $q(e)$ and with sources and targets in $S$. $S$ must be a subset of $V$. \subsubsection{Trace} \begin{SML} val \mlrischref{graphs/trace-graph.sml}{TraceView.trace_view} : node_id list -> ('n,'e,'g) graph -> ('n','e','g') graph \end{SML} \begin{wrapfigure}{r}{3in} \begin{Boxit} % \psfig{figure=../pictures/eps/trace.eps,width=2.8in} \includegraphics[width=2.8in]{../pictures/pdf/trace} \end{Boxit} \label{fig:trace-view} \caption{A trace view} \end{wrapfigure} A \newdef{trace} is an acyclic path in a graph. The function \sml{trace_view} takes a trace of node ids $v_1,\ldots,v_n$ and a graph $G$ and returns a view in which only the nodes are visible. Only the edges that connected two adjacent nodes on the trace, i.e. $v_i -> v_{i+1}$ for some $i = 1 \ldots n-1$ are considered be within the view. Thus if there is an edge $v_i -> v_j$ in $G$ where $j \ne i+1$ this edge is not considered to be within the view --- it is considered to be an exit edge from $v_i$ and an entry edge from $v_j$ however. Trace views can be used to construct a CFG region suitable for trace scheduling \cite{trace-scheduling,bulldog}. Figure \ref{fig:trace-view} illustrates this concept graphically. Here, the trace view is formed from the nodes \sml{A, C, D, F} and \sml{G}. The solid edges linking the trace is visible within the view. All other dotted edges are considered to be either entry of exit edges into the trace. The edge from node \sml{G} to \sml{A} is considered to be both since it exits from \sml{G} and enters into \sml{A}. \subsubsection{Acyclic Subgraph} \begin{SML} val \mlrischref{graphs/acyclic-graph.sml}{AcyclicSubgraphView.acyclic_view} : node_id list -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} \begin{wrapfigure}{r}{3in} \begin{Boxit} \includegraphics[width=2.8in]{../pictures/pdf/subgraph} % \psfig{figure=../pictures/eps/subgraph.eps,width=2.8in} \end{Boxit} \label{fig:acyclic-subgraph-view} \caption{An acyclic subgraph} \end{wrapfigure} The function \sml{acyclic_view} takes an ordered list of node ids $v_1,\ldots,v_n$ and a graph $G$ as arguments and return a view $G'$ such that only the nodes $v_1,\ldots,v_n$ are visible. In addition, only the edges with directions consistent with the order list are considered to be within the view. Thus an edge $v_i -> v_j$ from $G$ is in $G'$ iff $1 \le i < j \le n$. Acyclic views can be used to construct a CFG region suitable for DAG scheduling. Figure \ref{fig:acyclic-subgraph-view} illustrates this concept graphically. \subsubsection{Start and Stop} \begin{SML} val \mlrischref{graphs/start-stop.sml}{StartStopView.start_stop_view} : \{ start : 'n node, stop : 'n node, edges : 'e edge list \} -> ('n,'e,'g) graph -> ('n','e','g') graph \end{SML} The function \sml{start_stop_view} \subsubsection{Single-Entry/Multiple-Exits} \begin{SML} \mlrischref{graphs/SEME.sml}{SingleEntryMultipleExit.SEME} exit : 'n node -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} The function \sml{SEME} converts a single-entry/multiple-exits graph $G$ into a single entry/single exit graph. It takes an exit node $e$ and a graph $G$ and returns a view $G'$. Suppose $i \edge{l} j$ is an exit edge in $G$. In view $G$ this edge is replaced by a new normal edge $i \edge{l} e$ and a new exit edge $e \edge{l} j$. Thus $e$ becomes the sole exit node in the new view. \subsubsection{Behavioral Views} \subsubsection{Behavioral Primitives} Figure \ref{fig:behavioral-view-primitives} lists the set of behavioral primitives defined in structure \mlrischref{graphs/wrappers.sml}{\sml{GraphWrappers}}. These functions allow the user to attach an action $a$ to a mutator method $m$ such that whenever $m$ is invoked so does $a$. Given a graph $G$, the combinator \begin{SML} do_before_\(xxx\) : f -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} \noindent returns a view $G'$ such that whenever method $xxx$ is invoked in $G'$, the function $f$ is called. Similarly, the combinator \begin{SML} do_after_\(xxx\) : f -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} \noindent creates a new view $G''$ such that the function $f$ is called after the method is invoked. \begin{Figure} \begin{boxit} \begin{SML} do_before_new_id : (unit -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_new_id : (node_id -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_add_node : ('n node -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_add_node : ('n node -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_add_edge : ('e edge -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_add_edge : ('e edge -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_remove_node : (node_id -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_remove_node : (node_id -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_set_in_edges : (node_id * 'e edge list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_set_in_edges : (node_id * 'e edge list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_set_out_edges : (node_id * 'e edge list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_set_out_edges : (node_id * 'e edge list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_set_entries : (node_id list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_set_entries : (node_id list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_before_set_exits : (node_id list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_set_exits : (node_id list -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} \end{boxit} \label{fig:behavioral-view-primitives} \caption{Behavioral view primitives} \end{Figure} Frequently it is not necessary to know precisely by which method a graph's structure has been modified, only that it is. The following two methods take a notification function $f$ and returns a new view. $f$ is invoked before a modification is attempted in a view created by \sml{do_before_changed}. It is invoked after the modification in a view created by \sml{do_after_changed}. \begin{SML} do_before_changed : (('n,'e,'g) graph -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph do_after_changed : (('n,'e,'g) graph -> unit) -> ('n,'e,'g) graph -> ('n,'e,'g) graph \end{SML} Behavioral views created by the above functions are all fully update transparent.

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