HAL Id: hal-01422227
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Submitted on 24 Dec 2016
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Strongly Connected Components in graphs, formal proof of Tarjan1972 algorithm
Jean-Jacques Levy, Ran Chen
To cite this version:
Jean-Jacques Levy, Ran Chen. Strongly Connected Components in graphs, formal proof of Tarjan1972 algorithm. Groupe de travail LTP du GDR GPL , Nov 2016, Orsay, France. 2016. �hal-01422227�
Strongly Connected Components
in graphs, formal proof
of Tarjan1972 algorithm
jean-jacques.levy@inria.fr
Plan
2
.. joint work (in progress) with Ran Chen
•
pre-/post-conditions•
imperative programs•
conclusion•
motivationMotivation
3
•
to be fully published in articles or journals•
Coq proofs seem to me unreadable (and by normal human ?)•
algorithms on graphs = a good testbed•
nice algorithms should have simple formal proofs•
how to publish formal proofs ?•
Why3 allows mix of automatic and interactive proofsA one-pass
linear-time algorithm
The algorithm
(1/2)
5
2 3 8
4 9
1 7
5 6 • depth-first search algorithm
• with pushing non visited vertices into a working stack
• and computing oldest vertex reachable by at most a single « back-edge »
• when that oldest vertex is equal to currently visited vertex, a new strongly connected component is in the working stack on top of current vertex.
The algorithm
(2/2)
6 2 3 8 4 9 1 7 5 6 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5 1 2 3 4 5 6 1 1 2 2 3 3 4 4 8 8 1 2 3 4 5 6 9 7les valeurs successives de la pile
0 1 2 3 4 5 6 se ns cro issa nt du ra ng
The algorithm
(2/2)
7 2 3 8 4 9 1 7 5 6 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5 1 2 3 4 5 6 1 1 2 2 3 3 4 4 8 8 1 2 3 4 5 6 9 7les valeurs successives de la pile
0 1 2 3 4 5 6 se ns cro issa nt du ra ng 1 0 0 0 4 4 4 3 3
Proof in algorithms book
8
•
Christmas tree structure of strongly connected components•
give the program•
consider the spanning trees (forrest)•
2-3 lemmas about ancestors in spanning treesThe program
(1/3)
9
• a functional version with lists and finite sets
The program
(2/3)
10
• blacks, grays are sets of vertices; sccs is a set of sets of vertices
• naming conventions:
x, y, z for vertices; b for black sets; s for stacks; cc for connected components;
The program
(3/3)
Pre/Post-conditions
(2/3)
x stack x s blacks ✓ b sccs ✓ sccs n y m m rank y stack m rank x stackInvariants
(1/4)
blacks \ grays = ;
(set of sccs) ✓ blacks
Invariants
(2/4)
cc1 sccs s grays blacks in cre asi ng ra nk cc2 ccnInvariants
(4/4)
cc1 sccs s grays blacks in cre asi ng ra nk cc2 ccnAssertions
x0 2 s2 ^ y0 62 s2 ^ edge x0 y0
• Coq proof: there exists x0, y0 with
y0 is white
x0 = x
x0 6= x impossible because no black to white
impossible because successors are black
y0 2 s3 = stack
x0 = x
x0 6= x impossible because crossedge
impossible because m1 rank y’ s1 < rank x s1
Towards imperative
program
Missing
•
implementation of graphs•
vertices as integers in an array•
successors as lists for every vertexConclusion
31
•
readable proofs ?•
simple algorithms should have simple proofsto be shown with a good formal precision
•
compare with other proof systems•
further algorithms (in next talks ?) • graphs represented with arrays + lists• topological sort, articulation points, sccK, sscT
