Minimum Spanning Trees GallagherHumbletSpira GHS Algorithm 1 Weighted

Minimum Spanning Trees Gallagher-Humblet-Spira (GHS) Algorithm 1

Weighted Graph G=(V, E), |V|=n, |E|=m 2

Spanning tree Any tree T=(V, E’) (connected acyclic graph) spanning all the nodes of G 3

Minimum-weight spanning tree (MST) A spanning tree s. t. the sum of its weights is minimized: For MST : is minimized 4

Spanning tree fragment: Any (connected) sub-tree of a MST 5

Minimum weight outgoing edge (MWOE) An edge adjacent to the fragment with smallest weight and that does not create a cycle 6

Two important properties for building MST Property 1: The union of a fragment and any of its MWOE is a fragment of some MST (so called blue rule). Property 2: If the weights are distinct then the MST is unique 7

Property 1: The union of a fragment F T and any of its MWOE is a fragment of some MST. Proof: Distinguish two cases: 1. the MWOE belongs to T 2. the MWOE does not belong to T In both cases, we can prove the claim. 8

Case 1: Fragment MWOE MST T 9

Trivially, if then is a fragment Fragment MWOE MST T 10

Case 2: Fragment MST T MWOE 11

If then add to and remove Fragment MST T 12

Fragment Obtain T’ and since But w(T’) w(T), since T is an MST w(T’)=w(T), i. e. , T’ is an MST 13

Fragment MST T’ thus is a fragment of T’ END OF PROOF 14

Property 2: If the weights are distinct then the MST is unique Proof: Basic Idea: Suppose there are two MSTs Then there is another MST of smaller weight contradiction! 15

Suppose there are two MSTs 16

Take the smallest weight edge not in intersection 17

Cycle in RED MST 18

Cycle in RED MST e’: any red edge not in BLUE MST ( since blue tree is acyclic) 19

Cycle in RED MST Since is not in the intersection, (the weight of is the smallest) 20

Cycle in RED MST Delete and add in RED MST we obtain a new tree with smaller weight contradiction! END OF PROOF 21

Prim’s Algorithm (sequential version) Start with a node as an initial fragment Repeat Augment fragment with a MWOE Until no other edge can be added to 22

Fragment 23

Fragment MWOE 24

Fragment MWOE 25

Fragment MWOE 26

Fragment 27

Theorem: Proof: Prim’s algorithm gives an MST Use Property 1 repeatedly END OF PROOF 28

• Prim’s algorithm (distributed version) Works by repeatedly applying the blue rule to a single fragment, to yield the MST for G • Works with both asynchronous and synchronous nonanonymous, uniform models (and also with non-distinct weights) Algorithm (asynchronous high-level version): Let vertex r be the root as well as the first fragment REPEAT • r broadcasts a message on the current fragment to search for the MWOE of the fragment (each vertex in the fragment searches for its local (i. e. , adjacent) MWOE) • convergecast (i. e. , reverse broadcast towards r) the MWOE of the appended subfragment (i. e. , the minimum of the local MWOEs of itself and its descendents) • the MWOE of the fragment is then selected by r and added to the fragment, by sending an add-edge message on the appropriate path • then, the root and nodes adjacent to that just entered in the fragment are notified the edge has been added UNTIL there is only one fragment left 29

Local description of asynchronous Prim Each processor stores: 1. The state of any of its incident edges, which can be either of {basic, branch, reject} 2. Its own state, which can be either {in, out} 3. Local MWOE 4. MWOE for each branch-down edge 5. Parent channel (route towards the root) 6. MWOE channel (route towards the MWOE of its appended subfragment) 30

Type of messages in asynchronous Prim 1. Search MWOE: coordination message initiated by the root 2. Test: check the status of a basic edge 3. Reject, Accept : response to Test 4. Report(weight): report to the parent node the MWOE of the appended subfragment 5. Add edge: say to the fragment node adjacent to the fragment’s MWOE to add it 6. Connect: perform the union of the found MWOE to the fragment (this changes the status of the corresponding end-node from out to in) 7. Connected: notify the root and adjacent nodes that connection has taken place Message complexity = O(n 2) 31

Synchronous Prim It will work in O(n 2) rounds…think to it by yourself… 32

Kruskal’s Algorithm (sequential version) Initially, each node is a fragment Repeat • Find the smallest MWOE e of all fragments • Merge the two fragments adjacent to e Until there is only one fragment left 33

Initially, every node is a fragment 34

Find the smallest MWOE 35

Merge the two fragments 36

Find the smallest MWOE 37

Merge the two fragments 38

Merge the two fragments 39

Resulting MST 40

Theorem: Kruskal’s algorithm gives an MST Proof: Use Property 1, and observe that no cycle is created. END OF PROOF 41

Synchronous GHS Algorithm Distributed version of Kruskal’s Algorithm • Works by repeatedly applying the blue rule to multiple fragments • Works with non-uniform models, distinct weights, synchronous start Initially, each node is a fragment Repeat in parallel: (Synchronous Phase) • Each fragment – coordinated by a fragment root node - finds its MWOE • Merge fragments adjacent to MWOE’s Until there is only one fragment left 42

Local description of syncr. GHS Each processor stores: 1. The state of any of its incident edges, which can be either of {basic, branch, reject} 2. Identity of its fragment (the weigth of a core edge – for single-node fragments, the proc. id ) 3. Local MWOE 4. MWOE for each branching-out edge 5. Parent channel (route towards the root) 6. MWOE channel (route towards the MWOE of its appended subfragment) 43

Type of messages 1. New fragment(identity): coordination message sent by the root at the end of a phase 2. Test(identity): for checking the status of a basic edge 3. Reject, Accept : response to Test 4. Report(weight): for reporting to the parent node the MWOE of the appended subfragment 5. Merge: sent by the root to the node incident to the MWOE to activate union of fragments 6. Connect(My Id): sent by the node incident to the MWOE to perform the union 44

Phase 0: Initially, every node is a fragment… … and every node is a root of a fragment 45

Phase 1: Find the MWOE for each node 46

Phase 1: Merge the nodes and select a new root Root Symmetric MWOE Asymmetric MWOE The new root is adjacent to a symmetric MWOE 47

Rule for selecting a new root in a fragment Fragment 2 Fragment 1 root MWOE root Merging 2 fragments 48

Rule for selecting a new root in a fragment Merged Fragment root Higher ID Node on MWOE (non-anonymity) 49

Rule for selecting a new root in a fragment root root Merging more than 2 fragments 50

Rule for selecting a new root in a fragment Merged Fragment Root Higher ID Node on symmetric MWOE asymmetric 51

Remark: In merged fragments there is exactly one symmetric MWOE (n-1 edges vs n arrows) zero Impossible Creates a fragment with no MWOE two Impossible Creates a fragment with two MWOEs 52

After merging has taken place, the new root broadcasts New fragment(w(e)) to the new fragment, and afterwards a new phase starts e is the symmetric MWOE of the merged fragments is the identity of the new fragment 53

In our example, at the end of phase 1 each fragment has its new identity. Root End of phase 1 54

At the beginning of each new phase each node in fragment finds its MWOE MWOE 55

To discover its own MWOE, each node sends a Test message containing its identity over its basic edge of min weight, until it receives an Accept test( ) accept test( ) reject 56

Then it knows its local MWOE 57

Then each node sends a Report with the MWOE of the appended subfragment to the root with convergecast (the global minimum survives in propagation) MWOE 58

The root selects the minimum MWOE and sends along the appropriate path a Merge message, which will become a Connect message at the proper node MWOE 59

Phase 2 of our example: After receiving the new identity, find again the MWOE for each fragment 60

Phase 2: Merge the fragments Root 61

At the end of phase 2 each fragment has its own unique identity. Root End of phase 2 62

Phase 3: Find the MWOE for each fragment 63

Phase 3: Merge the fragments Root 64

Phase 3: New fragment FINAL MST 65

Correctness • To guarantee correctness, phases must be syncronized • But at the beginning of a phase, each fragment can have a different number of nodes, and thus the MWOE selection is potentially asynchronous… • But each fragment can have at most n nodes, has height at most n-1, and each node has at most n-1 incident edges… • So, the MWOE selection requires at most 3 n rounds, and the Merge message requires at most n rounds. • Then, the Connect message must be sent at round 4 n+1 of a phase, and so at round 4 n+2 a node knows whether it is a new root • Finally, the New fragment message will require at most n rounds. A fixed number of 5 n+2 total rounds can be used to complete each phase (in some rounds nodes do nothing…)! 66

Complexity Phase Smallest Fragment size (#nodes) 67

Algorithm Time Complexity Maximum possible fragment size Number of nodes Maximum # phases: Total time = Phase time • #phases = 68

Algorithm Message Complexity Thr: Synchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages: 1. Test-Reject msgs: at most 2 for each edge; 2. Each node sends/receives at most a single: New Fragment, Test-Accept, Report, Merge, Connect message for each phase. Since from previous lemma we have at most log n phases, the claim follows. END OF PROOF 69

Asynchronous Version of GHS Algorithm • Simulates the synchronous version • Works with uniform models, distinct weights, asynchronous start • Every fragment F has a level L(F)≥ 0: at the beginning, each node is a fragment of level 0 • Two types of merges: absorption and join 70

Local description of asyncr. GHS Like the synchronous, but: 1. Identity of a fragment is now given by an edge weight plus the level of the fragment; 2. A node has its own status, which can be either of {sleeping, finding, found} 71

Type of messages Like the synchronous, but now: 1. New fragment(weight, level, status): coordination message sent just after a merge 2. Test(weight, level): to test an edge 3. Connect(weight, level): to perform the union 72

Absorption Fragment MWOE If then is absorbed by (cost of merging proportional to ) 73

The combined level is New fragment MWOE and a New fragment(weight, level, status) message is broadcasted to nodes of F 1 by the node of F 2 on which the merge took place 74

Join Fragment MWOE If and F 1 and F 2 have a symmetric MWOE, then F 1 joins with F 2 (cost of merging proportional to ) 75

The combined level is New fragment MWOE and a New fragment(weight , L(F 2)+1, finding ) message is broadcasted to all nodes of F 1 and F 2, where weigth is that of the edge on which the merge took place 76

Remark: a joining requires that fragment levels are equal…how to control this? If L(F 1)>L(F 2), then F 1 waits until L(F 1)= L(F 2) (this is obtained by letting F 2 not replying to Test messages from F 1 ) Fragment Test 77

Algorithm Message Complexity Lemma: A fragment of level L contains at least 2 L nodes. Proof: By induction. For L=0 is trivial. Assume it is true up to L=k-1, and let F be of level k. But then, either: 1. F was obtained by joining two fragments of level k-1, each containing at least 2 k-1 nodes by inductive hypothesis F contains at least 2 k-1 + 2 k-1 = 2 k nodes; 2. F was obtained after absorbing another fragment F’ of level<k apply recursively to FF’, until case (1) applies. END OF PROOF 78

Algorithm Message Complexity (2) Thr: Asynchronous GHS requires O(m+n logn) msgs. Proof: We have the following messages: 1. Connect msgs: at most 2 for each edge; 2. Test-Reject msgs: at most 2 for each edge; 3. Each node sends/receives at most a single: New Fragment, Test-Accept, Merge, Report message each time the level of its fragment increases ; and since from previous lemma each node can change at most log n levels, the claim follows. END OF PROOF 79

Homework Execute asynchronous GHS on the following graph: assuming that system is pseudosynchronous: Start from 1 and 5, and messages sent from odd (resp. , even) nodes are received after 1 (resp. , 2) round(s) 80