1. Chapter 15.5 from “Distributed Algorithms” by Nancy A. Lynch
Minimum Spanning Tree
Kevin Schaffer
Chapter 15.5 from “Distributed Algorithms” by Nancy A. Lynch

2. GHS Named for Gallager, Humblet, and Spira
Synchronous version is SynchGHS
Components are combined into larger components through minimum-weight outgoing edges (MWOEs)
Assuming all weights to be unique ensures a unique MST
Combining is done in levels; at level k all components have at least 2k nodes

3. Difficulties in GHS
Processes i and j might be in the same component but not realize it since process j might not have received notification
Components might be combined in an unbalanced way leading to a O(n2) message complexity
Possible interference from components at different levels

4. How GHS Works Processes in a component work together to find the MWOE
Once found, the component and the one attached to it across the MWOE combine into a new component
Every component has a level, but levels are not synchronized
Initially components are individual nodes with level 0
Process continues until there is one component

5. Combining Components
Components C and C’ can combine in one of two ways
Merge: If level(C) = level(C’) and C and C’ share a common MWOE then they form a new component whose level is level(C) + 1
Absorb: If level(C) < level(C’) and the MWOE of C connects to a node in C’ then C is added to C’
Ensures that components do not become unbalanced if some lag behind

6. Messages in GHS
initiate: Starts the process to find an MWOE and carries the component identifier
report: Convergecast MWOE information
test, accept, reject: Test whether neighbors belong to the same component
changeroot, connect: Combine components

7. Complexity of GHS Message complexity: O(n log n + |E|)
test messages that lead to rejection and reject messages: O(|E|)
All other messages: O(n log n)
Time complexity: O(n log n(l + d))

8. SimpleMST Works more like SynchGHS
Processes wait until all their neighbors are at the same level before continuing
Much simpler than GHS
Higher message complexity: O(|E| log n)
Same time complexity