The Role of Communication Complexity in Distributed Computing Rotem Oshman

Background: Distributed Computing

Distributed Computing Typical model: – Local computation for free – Charge for “communication”

Distributed Lower Bounds

Shared Memory Processes communicate by accessing objects in shared memory – Read/write registers – Read-modify-write: CAS, T&S, … Typically asynchronous: – Schedule = sequence of process IDs – Adversarially chosen – Sometimes processes may crash

Shared Memory Lower Bounds

Message Passing Processes communicate by sending messages – Over some network graph, often complete Fully synchronous, fully asynchronous, or anywhere in between Processes can crash, recover, cheat, lie,… Many successful applications of CC

Some differences… Complexity measure Problems Input Distributed Computing Comm. Complexity #rounds (limited bandwidth)total #bits SearchDecision Number-In-HandNumber-on- Forehead (usually)

Message-Passing Models MESSAGE-PASSING LOCAL CONGEST SHARED BLACKBOARD #rounds total CC

Talk Overview I.Lower bound techniques a.CONGEST (#rounds): reductions from 2-party communication complexity b.Total CC with private channels II.Shared blackboard a.Number-in-hand b.“Not-quite-number-in-hand”

The CONGEST Model

CONGEST Lower Bounds for Arbitrary Graphs … by reduction from 2-party disjointness

2-Party Reductions

Example: Approximate Diameter

Lower Bound

1 2 …

1 2 …

Approximate Diameter

Multi-Player NIH Communication with Private Channels

The Message-Passing Model

The Coordinator Model

Message-Passing vs. Coordinator Secure multi-party computation!

Message-Passing Lower Bounds

Symmetrization [Phillips, Verbin, Zhang ’12]

Symmetrization Example: Bitwise-XOR

Set Disjointness ?

Symmetrization vs. Disjointness

[BEOPV’13] Lower Bound Outline

Info Cost for Multi-Player

Reduction from D ISJ to graph connectivity [Based on WZ’13] (Players)(Elements)

Number-In-Hand Shared Blackboard

Why Should We Care? Some fundamental question still open Natural model for distributed computing – Single-hop wireless network – More generally, abstracts away network topology – Related to MapReduce, etc. [Hegeman and Pemmaraju’14]

Example: NIH Multi-Party Disjointness

“Not-Quite Number in Hand”

Simulating the Algorithm

More complicated….

Upper Bound on Subgraph Detection

Detecting Triangles

Triangles to 3-Party NOF Disjointness

3-Party NOF Disjointness

Conclusion MESSAGE-PASSING LOCAL CONGEST SHARED BLACKBOARD #rounds total CC

Directions for Future Research Exploiting asynchrony and faults to get stronger communication lower bounds

Example 1: Dynamic Networks

Example 2: Byzantine Consensus