• Motivation
• API
  • Types
  • Contracts
  • constructGraph :: EdgeADT -> Array -> Array -> Graph
  • findPathsBetweenTwoVertices :: FindPathSettings -> Vertex -> Vertex -> Graph -> Array
  • searchGraphEdges :: TraversalSpecs -> Vertex -> Graph -> SearchResults
    • Search algorithm
    • Description
    • Semantics
    • Contracts
  • searchGraphEdgesGenerator
    • Contracts
  • Provided searches
  • Tests
  • Examples
• Tips and gotchas
• Related

Status

• 0.7.0 is fine
• 0.8.0 fails with babel (because of generators? to be investigated)

Motivation

We had trouble finding a library for graph data structure which addresses our requirements. The principal need and motivator for now is model-based testing (MBT): a reactive system is modelled as a state-machine, which itself can be modelled as a multidigraph, or quiver. We could not find libraries which can handle such constructs. To generate input sequences for model testing, we need to examine the associated graph in a certain number of ways. This library aims at gathering such ways.

As such it is not and does not intend to be a all-purpose graph manipulation library. It just has what we need for model-based testing. Typically, the model coverage criteria used in MBT for transition-based models are :

• all paths: Every path traversed at least once (exhaustive testing of control structure)
  • all transition pairs: Every pair of adjacent transitions traversed at least once
    • all transitions: Every transition of the model traversed at least once
  • all states: Every state is visited at least once
  • all one-loop paths: Visit all the loop-free paths plus all the paths that loop once
    • all round trips: Test each loop (iterate only once), but no need to check all paths preceding or following a loop (transition coverage required)
    • all loop-free paths: Every loop-free (no repetition of config/ states) path traversed at least once

We currently offer :

• enumeration of all edge paths between two vertices, with a configurable maximum for edge repetition
• generic graph edge search, which can be customized to implement some of the criteria presented thereabove

This was chosen for pragmatic reasons. From this included algorithm, it is possible to implement most of the above-mentioned coverage criteria.

API

Types

/**
 * @typedef {Object} Edge Edge must be an object (we use referential equality so this is to avoid surprises with
 * equality of native types)
 */
/**
 * @typedef {Array<Edge>} EdgePath
 */
/**
 * @typedef {Object} Vertex Vertices must be an object (we use referential equality so this is to avoid surprises with
 * equality of native types)
 */
/**
 * @typedef {Object} EdgeADT
 * @property {function(Edge) : Vertex} getEdgeTarget
 * @property {function(Edge) : Vertex} getEdgeOrigin
 * @property {function(Vertex, Vertex) : Edge} constructEdge
 */
/**
 * @typedef {Object} Graph
 * @property {function (Vertex) : Array<Edge>} outgoingEdges
 * @property {function (Vertex) : Array<Edge>} incomingEdges
 * @property {function (Vertex) : Array<Vertex>} getNeighbours
 * @property {function (Vertex) : Array<Vertex>} vertices
 * @property {function (Vertex) : Array<Vertex>} edges
 * @property {function (Vertex) : Array<Vertex>} settings
 * @property {function (Edge) : Vertex} getEdgeOrigin
 * @property {function (Edge) : Vertex} getEdgeTarget
 * @property {function(Vertex, Vertex) : Edge} constructEdge
 * @property {function (Vertex) : ()} showVertex
 * @property {function (Vertex) : ()} showEdge
 * @property {function () : ()} clear free any resources that was used
 */
/**
 * @typedef {Object} FindPathSettings
 * @property {Number} [maxNumberOfTraversals=1] a number greater or equal to 0. Set to 1 by default
 * @property {String} [strategy='BFS'] search strategy : depth-first or breadth-first (default)
 */
/**
 * @typedef {() => Store<P>} StoreFactory Store constructor
 * @constructor
 */
/**
 * @typedef {Object} StoreInterface
 * @property {Store<P> | StoreFactory} empty empty store or constructor for an empty store
 * @property {function (Array<P>, Store<P>) : Store<P>} add adds a list of values into a store
 * @property {function (Store<P>) : {popped, newStore: Store<P>}} takeAndRemoveOne empty store. removes one value
 * from the store and returns that value and the new store without that value
 * @property {function (Store<P>): Boolean} isEmpty predicate which returns true iff the store is empty
 * @template P
 */

/**
 * @typedef {Map} GraphTraversalState `graphTraversalState` is shared state between search, result accumulation and
 * visiting functions. As such it must be used carefully. Ideally it is not necessary. If it is necessary then only
 * one function should modify it while the others read from it. That eliminates the need to think about order of
 * function application.
 */
/**
 * @typedef {{value:*, isProduced:Boolean}} SearchOutput
 */
/**
 * @typedef {Object} SearchSpecs
 * @property {function (Edge, Graph, PathTraversalState, GraphTraversalState) : {graphTraversalState : GraphTraversalState, isGoalReached : Boolean, output: SearchOutput} } evaluateGoal predicate which assesses whether a given goal is reached, or if instead the search should continue. To assess the goal, the provided information is the edge being visited, and the current edge traversal state (roughly the sequence of edges visited so far). If the goal is reached, this means we have some results for the search, and those results are aggregated somewhere in the graph traversal state for posterior extraction through `showResults`.
 * @property {*} initialGoalEvalState seed for the reducer associated to goal evaluation
 * @property {function (GraphTraversalState) : Search<Result>} showResults returns the accumulated results which
 * have been stored in the graph traversal state
 */
/**
 * @typedef {*} PathTraversalState
 */
/**
 * @typedef {{initialPathTraversalState:*, visitEdge : ReducerEdge}} VisitSpecs
 */
/**
 * @typedef {function (Edge, Graph, PathTraversalState, GraphTraversalState) : {isTraversableEdge :Boolean,  pathTraversalState:PathTraversalState}} ReducerEdge
 * function which visit the edge, updates the state of the edge traversal, and evaluates if the visited edge
 * should be included in the search (`isTraversableEdge` set to true) or not.
 */
/**
 * @typedef {*} Result
 */

Contracts

Both edges and vertices must be objects! The good behaviour of the library is not guaranteed for edges or vertices having native types.

constructGraph :: EdgeADT -> Array -> Array -> Graph

A graph is constructed from its array of edges and its vertices. An edge itself must contain two vertices. A vertex can be anything.

A complete description should be :

• V a set of vertices or nodes,
• A a set of edges or lines,
• source :: e -> V, assigning to each edge its source node,
• target :: e -> V, assigning to each edge its target node.

This happens to coincide exactly with the signature of our graph constructor -- though instead of sets, we use arrays.

The array of vertices is mandatory, and must contain any and every vertex part of the graph. This means in particular that no edge can be based on vertices which do not figure in the passed vertex array. This is checked by contract, by means of referential equality.

findPathsBetweenTwoVertices :: FindPathSettings -> Vertex -> Vertex -> Graph -> Array

Finds all paths between two vertices in a graph. Note that a path here is not an array of vertices but an array of edges. This is a requirement as we commonly deal in MBT with graphs with several guards between two states, with same-state transitions, and other circles.

If such paths exist, they are computed and returned in the form of array in which all elements are unique, i.e. there are no two same paths, with sameness defined by referential equality of the contained edges.

It is possible to configure the maximum number of occurrences of a given edge in a path. Sameness is defined by referential equality. The default value is set to 1 (no repetition of a given edge - this ensures loop-free paths). Settings that parameter to a value greater than 1 allows to have some control over the traversal of the graph cycles.

searchGraphEdges :: TraversalSpecs -> Vertex -> Graph -> SearchResults

Search algorithm

We did not bother much with a sophisticated algorithm. A collection of search algorithms can be found in the assets directory of this repository. We used an iterative version of a brute-force enumeration algorithm, adapted for the need of enumerating edges, accounting for loops, cycles, and multi-edges. The algorithm can be found in Enumeration algorithm, p14

Description

The function findPathsBetweenTwoVertices uses under the hood the search algorithm implemented by searchGraphEdges. searchGraphEdges returns results from the graph search starting from a given vertex , based on the configuration passed in TraversalSpecs.

TraversalSpecs contains three properties (search, store, visit) respectively dictating what are and how to build search results (for instance a result is a path between given origin and target and is to be aggregated in an array), the traversal order (for instance breadth-first), and the traversal criteria (for instance do not pass through the same edge twice).

The search property contains the following properties : initialGoalEvalState, evaluateGoal, showResults, respectively specifying :

• the initial value for the search state (generally an empty container of results)
• when a search is successful (isGoalReached), what/if to output as a result (output), and the update of the search state (graphTraversalState).
• the final output of the search. This is generally a data structure containing all results.

The visit property contains the following properties : initialPathTraversalState, visitEdge, respectively specifying :

• the initial value for the traversal state
• whether a traversal candidate edge should indeed be traversed (isTraversableEdge) or alternatively the search should backtrack, and the update of the traversal state (pathTraversalState).

The store property contains the following properties :

• empty : empty store
• add : adds elements to the store
• takeAndRemoveOne: remove an element from the store and returns a store with the same elements except the removed element
• isEmpty: predicate which returns true iff the store is empty

For instance, for the findPathsBetweenTwoVertices function:

• the store is a queue or a stack according to whether we want a BFS or DFS search
• visitEdge accumulates all traversable edges in an array (with initialPathTraversalState initially an empty array), forming an edge path. A traversable edge being an edge such that when added to the current edge path, does not feature more than a given number (maxNumberOfTraversals) of edge repetition.
• isGoalReached returns true iff the final edge of the current edge path is the target vertex. graphTraversalState, initially an empty array (initialGoalEvalState, accumulates results, i.e . the paths we have found which fulfiil our search criteria). Finally showResults simply outputs the accumulated array.

As such, the searchGraphEdges is fairly generic, and can implement a large variety of graph (edge) searches. Among the thing which can be done :

• probabilistic searches, or prefix searches can be done by configuring the store property, all else being equal

The invariant part of the function, is that for any traversed edge, the outgoing edges from that edge will be enumerated as potential traversable edges. What is a traversable edge and how those edges are eventually accumulated into search results is the parameterizable part of the function.

Semantics

• we start with initializing the search state, the traversal state and an empty store
• we start the enumeration with adding to the store the outgoing edges of the initial vertex
• for any edge in the store, we removed the edge from the store and visit it
  • if the edge is not traversable, we continue reviewing edges in the store
  • if the edge is traversable, it is traversed
    • we evaluate if the search goal is reached
      • if the search goal is reached, we update the search state and possibly compute and output the result of the search
      • if the search goal is not reached, we add the outgoing edges for the traversed edge to the store, and update the traversal state
    • we continue reviewing edges in the store

Contracts

The following contracts apply : type contracts, graph contracts, edge contracts, vertex contracts.

We could consider it an implicit contract that the search must terminate, i.e. that the search parameters (in particular isGoalReached and isTraversableEdge) have to be configured in a way that the store eventually returns to being empty. This is not enforced but important as a graph may have loops in which case the enumeration of edge paths would be infinite.

searchGraphEdgesGenerator

The searchGraphEdgesGenerator follows the same algorithm as searchGraphEdges, except for the fact that when a search result is found, that result is yielded to the function caller. searchGraphEdgesGenerator is indeed a generator, which when called returns an iterator. That iterator, for each next call will return a new search result, till the enumeration is finished (emptied store). Note that the generator returns the list of all found results. As usual, the return value of a generator cannot be retrieved in a for..of loop. We provide the getIteratorReturnValue function to that purpose.

We included this version of the search because :

• for large graphs, or graph featuring loops, the enumeration may be expensive, and we might want to instead process results one by one, instead of computing them synchronously and process the whole results
  • this is typically the case for testing, where we want to have tests failing as soon as possible
• in particular the enumeration may be infinite, and we might be interested in only some of the results, and want to stop the iteration at discretion
• we can interface it for further processing with :
  • transducers (for instance transducers-js)
  • iterator combinators for processing (for instance ixjs, the pull version of rxjs)
  • property-based testing libraries (jsverify, or jscheck)

Contracts

Same contracts as searchGraphEdges, except that the implicit contract that the search must terminate can be relaxed.

Provided searches

For the sake of our needs, we provided two parameter presets (a preset is a configuration of isTraversableEdge, and isGoalReached):

• ALL_n_TRANSITIONS : will reject edge paths which features more than n repetitions of any given edge. A successful search is the one generating an edge path whose final edge has a given target vertex
• ALL_TRANSITIONS : ALL_n_TRANSITIONS with n = 1.

This is specific to our state transducer library and our need to automatically generate test sequences.

In that context, note that ALL_TRANSITIONS helps enumerating a set of paths (input tests) which includes all transitions from a given vertex to a target vertex. That set of paths however is not the minimal set of such paths. It is in fact the maximal set. As there is no unicity of the minimum set, we chose to enumerate the maximal set and let the user pick from that set the sequences he favors.

Tests

Tests can be run with npm run test

Examples

Examples can be found in the test directory

Tips and gotchas

• edges which are not reachable from the starting vertex won't be reached. That sounds obvious but it is easy to forget.
• the GraphTraversalState can be used to implement early termination of the search. Typically visit would update some flag in the graph traversal state, and isTraversableEdge would not produce any further edges after consulting that flag.

Related

• search-algorithms
• enumeration algorithms