topology in distributed computing: a primer 1 / 16 sergey velder spbsu itmo
TRANSCRIPT
Topology in Distributed Computing: A Primer1 / 16
Sergey Velder
SPbSU ITMO
Overview2 / 16
Applications of topology to the theoryof distributed computing were discovered in
• Maurice Herlihy and Nir Shavit papers (1994, 1999)
• Michael Saks and Fotios Zaharoglou papers (1993, 2000)
They were awarded Gödel prize in 2004.
Hypergraphs3 / 16
In a hypergraph an edge can connect any number of vertices.
n-hypergraph (hypergraph with dimension n) is a hypergraph
where any edge connects at most n + 1 vertices.
Hypergraphs may have or not have an orientation.
1-hypergraphs are called graphs.
It is convenient to consider edges as simplices.
2-hypergraph example:
Simplicial complexes4 / 16
If a hypergraph (as a set of edges on vertices) is closedw. r. t. taking subsets of edges then it is calleda simplicial complex.A vertex-to-vertex map of complexes that is a hypergraphhomomorphism is called simplicial.Simplicial map is piecewise linear on geometric complexes.A subcomplex-to-subcomplex map that preserves intersections(M(P Q) = M(P) M(Q)) is called a carrier map.
3-complex Carrier map
Complex meaning5 / 16
Vertex color is a process ID.Vertex value is a process state.
Simplex is a global state.
Complex is a set of global states.
Binary consensus problem (n = 3)6 / 16
Input complex Output complex
Binary consensus problem (n = 3)6 / 16
Input complex Output complex
Carrier map
All 0 inputsAll 0 inputs
All 0 outputsAll 0 outputs
Binary consensus problem (n = 3)6 / 16
Input complex Output complex
Carrier map
All 1 inputsAll 1 inputs All 1 outputsAll 1 outputs
Binary consensus problem (n = 3)6 / 16
Input complex Output complex
Carrier map
Mixed 0-1 inputsMixed 0-1 inputs All 1 outputsAll 1 outputs
All 0 outputsAll 0 outputs
An example of protocol type7 / 16
Protocol complex
Vertex defines process ID and view(complete log of messages sent and received).Simplex defines compatible set of views.Each execution defines a simplex.
view = my_input_value;for (i = 0; i < r; i++){ broadcast(view); view += messages_received;}return δ(view)
Round 08 / 16
Single input Protocol complex
Round 19 / 16
Single input Protocol complex
Protocol complex evolution10 / 16
Round 2
Protocol complex evolution10 / 16
Round 0 Round 1 Round 2
Transformations11 / 16
δ
Input complex Protocol complex Output complex
Lower bound strategy is to find topological obstruction to δ
Must be simplicial
Consensus12 / 16
Subcomplex of all-0 inputs must map here
δ
Consensus12 / 16
δ
Subcomplex of all-1 inputs must map here
Path-connectedness13 / 16
δ
A protocol cannot solve consensus if its complex is path-connected.
Path-connectedness14 / 16
If protocol complex path remains path-connected…
• forever —then consensus is impossible;
• for r rounds —then we have a round-complexity lower bound;
• for time t —then we have a time-complexity lower bound.
Summary15 / 16
Combinatorial and algorithmic argumentscomplement one another.
Algorithmic is about what we can do.
Combinatorial is about what we can’t do.
Bibliography16 / 16
• M. Herlihy, N. Shavit. Applications of Algebraic Topology to Concurrent Computation.In Applications on Advanced Architecture Computers (ed. G. Astfalk), pp. 255–263 (1996)
• M. Saks, F. Zaharoglou. Wait-Free k-set Agreement is impossible: The Topology of Public Knowledge.In Proceedings 25th Annual ACM STOC,pp. 101–110 (1993)
• E. Borowsliy, E. Gafni. Generalized FLP Impossibility Result for t-resilient Asynchronous Computations.In Proceedings 25th Annual ACM STOC,pp. 91–100 (1993)
• M. J. Atallah, M. Blanton (eds.). Algorithms and Theory of Computation Handbook. General Concepts and Techniques, CRC Press (1998)