SwiftNodes

👩🏻‍🚀 This project is still a tad experimental. Contributors and pioneers welcome!

SwiftNodes offers a concurrency safe graph data structure together with graph algorithms. A graph stores values in identifiable nodes which can be connected via edges.

• Why?
• How?
• Included Algorithms
• Architecture
• Development Status
• Roadmap

Graphs may be the most fundamental mathematical concept besides numbers. They have wide applications in problem solving, data analysis and visualization. And although such data structures fit well with the language, graph implementations in Swift are lacking – in particular, comprehensive graph algorithm libraries.

SwiftNodes and its included algorithms were extracted from Codeface. But SwiftNodes is general enough to serve other applications as well – and extensible enough for more algorithms to be added.

• Usability, safety, extensibility and maintainability – which also imply simplicity.
• In particular, the API is supposed to feel familiar and fit well with official Swift data structures. So one question that guides its design is: What would Apple do?

We put the above qualities over performance. But that doesn't mean we neccessarily end up with suboptimal performance. The main compromise SwiftNodes involves is that nodes are value types and can not be referenced, so they must be hashed. But that doesn't change the average case complexity and, in the future, we might even be able to avoid that hashing in essential use cases by exploiting array indices.

🚧 Disclaimer: This section is now particularly outdated and the rewrite of all documentation is next on the roadmap.

The following explanations touch only parts of the SwiftNodes API. We recommend exploring the DocC reference, unit tests and production code. The code in particular is actually small and easy to grasp.

A Graph<NodeID: Hashable, NodeValue> holds values of type NodeValue in nodes of type GraphNode<NodeID: Hashable, NodeValue>. Nodes are unique and have IDs of type NodeID:

var graph = Graph<String, Int> { "id\($0)" }  // NodeID == String, NodeValue == Int
let node = graph.insert(1)                    // node.id == "id1", node.value == 1

let nodeForID1 = graph.node(for: "id1")       // nodeForID1.id == "id1"
let valueForID1 = graph.value(for: "id1")     // valueForID1 == 1

When inserting a value, a Graph must determine the ID of the node that would store the value. So the Graph initializer takes a closure returning a NodeID given a NodeValue.

Side Note: The reason, there's an explicit node type at all is that a) values don't need to be unique, but nodes in a graph are, and b) a node holds caches for quick access to its neighbours. The reason there is an explicit edge type at all is that edges have a count (they are "weighted") and may hold their own values in the future.

You may generate NodeIDs independent of NodeValues:

var graph = Graph<UUID, Int> { _ in UUID() }  // NodeID == UUID, NodeValue == Int
let node1 = graph.insert(42)
let node2 = graph.insert(42)  // node1.id != node2.id, same value in different nodes

If NodeID and NodeValue are the same type, you can omit the closure and the Graph will assume the value is itself used as the node ID:

var graph = Graph<Int, Int>()  // NodeID == NodeValue == Int
let node1 = graph.insert(42)   // node1.value == node1.id == 42
let node2 = graph.insert(42)   // node1.id == node2.id because 42 implies the same ID

And if your NodeValue is itself Identifiable by IDs of type NodeID, then you can also omit the closure and Graph will use the ID of a NodeValue as the NodeID of the node holding that value:

struct IdentifiableValue: Identifiable { let id = UUID() }
var graph = Graph<UUID, IdentifiableValue>()  // NodeID == NodeValue.ID == UUID
let node = graph.insert(IdentifiableValue())  // node.id == node.value.id

var graph = Graph<String, Int> { "id\($0)" }
let node1 = graph.insert(1)
let node2 = graph.insert(2)
let edge = graph.addEdge(from: node1.id,  to: node2.id)

An edge is directed and points from the node with ID edge.originID to the node with ID edge.destinationID.

Every edge has an integer count accessible via edge.count. It is more specifically a "count" rather than a "weight", as it increases when the same edge is added again. By default, a new edge has count 1 and adding it again increases its count by 1. But you can specify a custom count when adding an edge:

graph.addEdge(from: node1.id, to: node2.id, count: 40)  // edge count is 40
graph.addEdge(from: node1.id, to: node2.id, count: 2)   // edge count is 42

To work with a Graph constant (for example as a property on a Sendable reference type), you need to initialize the whole graph, complete with its values and edges. There are more ways to do so than we can exemplify here, so have a look at the graph initializers in code.

When you don't need to specify edge counts, the initializers allow to specify edges as tuples of node IDs. And when you're passing in values and edges anyway, you can omit the type parameters. Furthermore, you can often omit the closure that determines node IDs, as described earlier. All this together can make initialization as simple as it gets:

let graph = Graph(values: [-7, 0, 5, 42], 
                  edges: [(-7, 0), (0, 42)])

A GraphEdge<NodeID: Hashable, NodeValue> has its own ID type which combines the edge's originID- and destinationID node IDs. In the context of a Graph or GraphEdge, you can create edge IDs like so:

let edgeID = Edge.ID(node1.id, node2.id)

This leads to 3 ways of removing an edge:

let edge = graph.addEdge(from: node1.id, to: node2.id)

graph.removeEdge(with: edge.id)
graph.removeEdge(with: .init(node1.id, node2.id))
graph.removeEdge(from: node1.id, to: node2.id)

Graph offers many ways to query its nodes, node IDs, values and edges. Have a look into Graph.swift to see them all. In addition, a GraphNode has caches that enable quick access to its neighbours:

node.descendantIDs  // IDs of all nodes to which there is an edge from node
node.ancestorIDs    // IDs of all nodes from which there is an edge to node
node.neighbourIDs   // all descendant- and ancestor IDs
node.isSink         // whether node has no descendants
node.isSource       // whether node has no ancestors

Like the official Swift data structures, Graph is a pure struct and inherits the benefits of value types:

• You decide on mutability by using var or let.
• You can use a Graph as a @State or @Published variable with SwiftUI.
• You can use property observers like didSet to observe changes in a Graph.
• You can easily copy a whole Graph.

Many algorithms produce a variant of a given graph. Rather than modifying the original graph, SwiftNodes suggests to copy it. You copy a Graph like any other value. But right now, SwiftNodes lets you add and remove only edges – not nodes. So, to create a subgraph with a subset of the nodes of a graph, you can use graph.subGraph(nodeIDs:...):

var graph = Graph<Int, Int>()
	/* then add a bunch of nodes and edges ... */
let subsetOfNodeIDs: Set<Int> = [0, 3, 6, 9, 12]
let subGraph = graph.subGraph(nodeIDs: subsetOfNodeIDs)

A Graph is also Sendable if its value- and id type are. SwiftNodes is thereby ready for the strict concurrency safety of Swift 6. You can safely share Sendable Graph values between actors. Remember that, to declare a Graph property on a Sendable reference type, you need to make that property constant (use let).

Many graph algorithms do associate little intermediate results with individual nodes. The literature often refers to this as "marking" a node. The most prominent example is marking a node as visited while traversing a potentially cyclic graph. Some algorithms write multiple different markings to nodes.

When we made SwiftNodes concurrency safe (to play well with the new Swift concurrency features), we removed the possibility to mark nodes directly, as that had lost its potential for performance optimization. See how the included algorithms now use hashing to associate markings with nodes.

SwiftNodes has begun to accumulate some graph algorithms. The following overview also links to Wikipedia articles that explain what the algorithms do. We recommend also exploring them in code.

graph.findComponents() returns multiple sets of node IDs which represent the components of the graph.

graph.findStronglyConnectedComponents() returns multiple sets of node IDs which represent the strongly connected components of the graph.

graph.makeCondensationGraph() creates the condensation graph of the graph, which is the graph in which all strongly connected components of the original graph have been collapsed into single nodes, so the resulting condensation graph is acyclic.

graph.findTransitiveReductionEdges() finds all edges of the transitive reduction (the minimum equivalent graph) of the graph. You can also use filterTransitiveReduction() and filteredTransitiveReduction() to create a graph's minimum equivalent graph.

Right now, all this only works on acyclic graphs and might even hang or crash on cyclic ones.

graph.findEssentialEdges() returns the IDs of all "essential" edges. You can also use graph.filterEssentialEdges() and graph.filteredEssentialEdges() to remove all "non-essential" edges from a graph.

Edges are essential when they correspond to edges of the MEG (the transitive reduction) of the condensation graph. In simpler terms: Essential edges are either in cycles or they are essential to the reachability described by the graph – i.e. they cannot be removed without destroying the only path between some nodes.

Note that only edges of the condensation graph can be non-essential and so edges in cycles (i.e. in strongly connected components) are all considered essential. This is because it's algorithmically as well as conceptually hard to decide which edges in cycles are "non-essential". We recommend dealing with cycles independently of using this function.

graph.findNumberOfNodeAncestors() returns a [NodeID: Int] containing the ancestor count for each node ID of the graph. The ancestor count is the number of all (recursive) ancestors of the node. Basically, it's the number of other nodes from which the node can be reached.

This only works on acyclic graphs right now and might return incorrect results for nodes in cycles.

Ancestor counts can serve as a proxy for topological sorting.

Here is the architecture (composition and essential dependencies) of the SwiftNodes code folder:

The above image was created with Codeface.

From version/tag 0.1.0 on, SwiftNodes adheres to semantic versioning. So until it has reached 1.0.0, its API may still break frequently, and we express those breaks with minor version bumps.

SwiftNodes is already being used in production, but Codeface is still its primary client. SwiftNodes will move to version 1.0.0 as soon as either one of these conditions is met:

• Basic practicality and conceptual soundness have been validated by serving multiple real-world clients.
• We feel it's mature enough (well rounded and stable API, comprehensive tests, complete documentation and solid achievement of design goals).

1. Review, update and complete all documentation, including API comments.
2. Round out and add algorithms (starting with the needs of Codeface):
   1. Make existing algorithms compatible with cycles (two algorithms are still not). meaning: don't hang or crash, maybe throw an error!
   2. Move to version 1.0.0 if possible
   3. Add general purpose graph traversal algorithms (BFT, DFT, compatible with potentially cyclic graphs)
   4. Add better ways of topological sorting
   5. Approximate the minimum feedback arc set, so Codeface can guess "faulty" or unintended dependencies, i.e. the fewest dependencies that need to be cut in order to break all cycles.
3. Possibly optimize performance – but only based on measurements and only if measurements show that the optimization yields significant acceleration. Optimizing the algorithms might be more effective than optimizing the data structure itself.
   • What role can @inlinable play here?
   • What role can lazy play here?