Topology in Distributed Computing: A Primer 1 / 16 Sergey Velder SPbSU ITMO

Overview 2 / 16 Applications of topology to the theory of 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.

Hypergraphs 3 / 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 complexes 4 / 16 If a hypergraph (as a set of edges on vertices) is closed w. r. t. taking subsets of edges then it is called a simplicial complex. A vertex-to-vertex map of complexes that is a hypergraph homomorphism 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 -complexCarrier map

Complex meaning 5 / 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 inputs All 0 outputs

Binary consensus problem ( n = 3 ) 6 / 16 Input complex Output complex Carrier map All 1 inputs All 1 outputs

Binary consensus problem ( n = 3 ) 6 / 16 Input complex Output complex Carrier map Mixed 0-1 inputs All 1 outputs All 0 outputs

An example of protocol type 7 / 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 0 8 / 16 Single input Protocol complex

Round 1 9 / 16 Single input Protocol complex

Protocol complex evolution 10 / 16 Round 2

Protocol complex evolution 10 / 16 Round 0 Round 1 Round 2

Transformations 11 / 16 δ Input complex Protocol complex Output complex Lower bound strategy is to find topological obstruction to δ Must be simplicial

Consensus 12 / 16 Subcomplex of all-0 inputsmust map here δ

Consensus 12 / 16 δ Subcomplex of all-1 inputsmust map here

Path-connectedness 13 / 16 δ A protocol cannot solve consensus if its complex is path-connected.

Path-connectedness 14 / 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.

Summary 15 / 16 Combinatorial and algorithmic arguments complement one another. Algorithmic is about what we can do. Combinatorial is about what we cant do.

Bibliography 16 / 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)Applications of Algebraic Topology to Concurrent Computation 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)Wait-Free k-set Agreement is impossible: The Topology of Public Knowledge E. Borowsliy, E. Gafni. Generalized FLP Impossibility Result for t-resilient Asynchronous Computations. In Proceedings 25th Annual ACM STOC, pp. 91–100 (1993)Generalized FLP Impossibility Result for t-resilient Asynchronous Computations M. J. Atallah, M. Blanton (eds.). Algorithms and Theory of Computation Handbook. General Concepts and Techniques, CRC Press (1998)