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)