NoGraphs: Graph analysis on the fly

NoGraphs simplifies the analysis of graphs that can not or should not be fully computed, stored or adapted, e.g. infinite graphs, large graphs and graphs with expensive computations. (Here, the word graph denotes the thing with vertices and edges, not with diagrams.)

The approach: Graphs are computed and/or adapted in application code on the fly (when needed and as far as needed). Also, the analysis and the reporting of results by the library happen on the fly (when, and as far as, results can already be derived).

Think of it as graph analysis - the lazy (evaluation) way.

Documentation

• Homepage of the documentation
• Installation guide
• Tutorial (contains many examples)
• API reference

Feature overview

• Unidirectional traversal algorithms: DFS, BFS, topological search, Dijkstra, A* and MST.
• Bidirectional search algorithms: BFS and Dijkstra.
• Results: Reachability, depth, distance, and paths. Paths can be calculated with vertices, edges, or attributed edges, and can be iterated in both directions. Additionally, for DFS: forest, all kinds of edge types, both entering and leaving events, and DFS tree edges or all paths or all walks.
• Flexible graph notion:
  • Infinite directed multigraphs with loops and attributes (this includes multiple adjacency, cycles, self-loops, directed edges, weighted edges and attributed edges).
  • Infinite graphs are supported, but need to be locally finite (i.e., a vertex has only finitely many outgoing edges).
• Generic API:
  • Vertices: Can be anything except for None. Hashable vertices can be used directly, unhashable vertices can be used together with hashable identifiers.
  • Edge weights and distances: Wide range of data types supported (float, int, Decimal, mpmath.mpf and others), e.g., for high precision computations.
  • Edge attributes: Any object, e.g, a container with further data.
  • Identity and equivalence of vertices can be chosen / defined.
  • Bookkeeping: Several sets of bookkeeping data structures are predefined, optimized for different situations (data types used by the application, hashing vs. indexing, collections for Python objects or C native data types,...); Adaptable and extendable, e.g., specialized collections of 3rd party libraries can be integrated easily and runtime efficiently
• Flexible API: The concept of implicit graphs that NoGraphs is based on allows for an API that eases operations like graph pruning, graph abstraction, the typical binary graph operations (union, intersection, several types of products), the computation of search-aware graphs, the combination of problem reduction with lazy evaluation, and traversals of vertex equivalence classes on the fly. Bookkeeping data can be pre-initialized and accessed during computations.
• Typing: The API can be used fully typed (optionally).
• Implementation: Pure Python (>=3.9). It introduces no further dependencies.
• CI tests: For all supported versions of Python and both supported interpreters CPython and PyPy, both code and docs, 100% code coverage.
• Runtime and memory performance: Have been goals (CPython). In its domain, it often even outperforms Rust- and C-based libraries. Using PyPy improves its performance further.

Extras (outside of the core of NoGraphs)

• Computation of exact solutions for (small) traveling salesman problems (shortest / longest route, positive / zero / negative edge weights, graph does not need to be complete)
• Dijkstra shortest paths algorithm for infinitely branching graphs with locally sorted edges.
• Gadget functions for test purposes. They make the easy task of adapting existing explicit test graphs a no brainer, may they be stored in edge indices or edge iterables or in arrays.

Examples with further algorithms

• Depth-limited search
• Iterative deepening depth-first search
• Critical path in a weighted, acyclic graph
• Longest path between two vertices in a weighted, acyclic graph
• Longest path between two vertices in a weighted graph or in an unweighted graph
• Strongly connected components of a graph
• Biconnected components of a connected undirected graph

Example

Our graph is directed, weighted and has infinitely many edges. These edges are defined in application code by the following function. For a vertex i (here: an integer) as the first of two parameters, it yields the edges that start at i as tuples (end_vertex, edge_weight). What a strange graph - we do not know how it looks like...

>>> def next_edges(i, _):
...     j = (i + i // 6) % 6
...     yield i + 1, j * 2 + 1
...     if i % 2 == 0:
...         yield i + 6, 7 - j
...     elif i % 1200000 > 5:
...         yield i - 6, 1

We would like to find out the distance of vertex 5 from vertex 0, i.e., the minimal necessary sum of edge weights of any path from 0 to 5, and (one of) the shortest paths from 0 to 5.

We do not know which part of the graph is necessary to look at in order to find the shortest path and to make sure, it is really the shortest. So, we use the traversal strategy TraversalShortestPaths of NoGraphs. It implements the well-known Dijkstra graph algorithm in the lazy evaluation style of NoGraphs.

>>> import nographs as nog
>>> traversal = nog.TraversalShortestPaths(next_edges)

We ask NoGraphs to traverse the graph starting at vertex 0, to calculate paths while doing so, and to stop when visiting vertex 5.

>>> traversal.start_from(0, build_paths=True).go_to(5)
5

The state data of this vertex visit contains our results:

>>> traversal.distance
24
>>> traversal.paths[5]
(0, 1, 2, 3, 4, 10, 16, 17, 11, 5)

We learn that we need to examine the graph at least till vertex 17 to find the shortest path from 0 to 5. It is not easy to see that from the definition of the graph...

A part of the graph, the vertices up to 41, is shown in the following picture. Arrows denote directed edges. The edges in red show shortest paths from 0 to other vertices.

And now?

Can you imagine...

• An infinite generator of primes, defined by just a graph and a call to a standard graph algorithm?
• Or a graph that defines an infinite set of Towers of Hanoi problems in a generic way, without fixing the number of towers, disk sizes, and the start and goal configuration - and a specific problem instance is solved by just one library call?
• Or a way for computing an exact solution for traveling salesman problems, that is based on just a graph and a call of the Dijkstra single source shortest path algorithm?
• Or graphs that are dynamically computed based on other graphs, or on analysis results about other graphs, or even on partial analysis results for already processed parts of the same graph?

Let's build it.

Welcome to NoGraphs!