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3 ........
APPENDIX: THE ALGORITHM
Local Variables at nodes
All variables are integer values except where noted otherwise. The initial values of all of the variables are arbitrary, except for the variable level, whose initial value is oo.
cycle =
current level of the exploration in the local algorithm.
count =
counter of unacknowledged LOCAL_BROADCAST messages.
desired_level
=
the last level to be processed in the iteration
initial_level level =
the first level to be processed in the sub-iteration.
current estimate of the node's level in shortest path tree.
status =
estimate of whether node has frozen descendents, active descendents, or only
inactive descendents; only meaningful for nodes in local trees. Possible values
are (Active,Frozen,Inactive}.
state (e)
estimate of whether there are frozen descendents, active descendents, or only
inactive descendents on the further side of tree edge e. If e is not a tree edge,
state (e )=unused. It is kept for each incident edge e. Possible values are
{Frozen,Active,Inward,Inactive,Unused).
the level of synch nodes.
synch_level =
synchs_number =
estimate of the number of active synchs which are descendents of that node; is
meaningful only for nodes in the global tree.
Messages sent during the algorithm
GL OBALBR OAD CAST (s, i, d ) =
message by which the root triggers execution of the next sub-iteration. The
message carries the following parameters: synch_Ievel, initiallevel,
desired_level.
GLOBAL_ACK(c ,m) =
message by which a node reports on its inedge about the termination of the
The message carries the following parameters: cycle,
sub-iteration.
synchs_number.
LOCAL_BROADCAST(c ) =
message by which a synch node triggers the local exploration process. The
parameter c is the cycle number.
LOCAL_ACK(c ,s ) =
message by which a node reports about the termination of the local exploration process. The parameters of the message are c (the cycle number) and s.
The latter determines the state of the edge.
Procedures used in the algorithm:
XMIT LOCAL_BROADCAST =
generates LOCAL_BROADCAST messages.
XMIT GLOBAL_BROADCAST =
generates GLOBAL_BROADCAST messages.
SYNCHCONTROL =
procedure executed by the synch nodes after termination of each cycle.
ROOT_CONTROL =
procedure executed by the root node after termination of each sub-iteration.
Initialization of the algorithm (by the root node)
begin
if
the set of incident edges is empty then
STOP /* the algorithm is complete */
else
begin
cycle := 1;
level := 0;
synchlevel := 0;
desired_level := V04;
initial_level := 1;
f or all edges e , state(e) := active;
Call Procedure XMIT_LOCAL_BROADCAST;
end
end /* of the initialization */
The node algorithm
Procedure XMIT_LOCAL_BROADCAST
begin
count :- 0;
status := inactive;
f or each edge e such that state(e) = active or frozen
begin
count :=-count +1;
send LOCALBROADCAST(cycle) on edge e;
end
if count =0 then
begin
status := inactive;
send LOCAL_ACK(cycle, inactive) on inedge;
end
end /* of the procedure */
Response to receipt of LOCAL_BROADCAST(p) on edge e:
begin
p < level then /* node joins new tree at lower level */
begin
if count > 0 and level boo then send ACK(cycle,unused) on inedge;
/ * terminating branch of old tree if ack awaited */
level :-p;
cycle := p;
inedge :=e;
state(e):=inward;
synchjevel := 0; /* initial value, to ensure that level > synchlevel */
4 e, state(e' ) := active;
f or all edges e'
if node_degree > V 0 4 then status := Frozen
else status := active;
send LOCAL_ACK(cycle,status) on inedge;
end
else /*p > level */
if e = inedge then
if p > level then /* broadcast must be propagated outward*/
begin
if
cycle := p;
Call Procedure XMIT_LOCAL BROADCAST;
end
else send LOCAL_ACK(cycle,status) on inedge;
/* in this case p = level; this means that this is the check for frozen nodes and the
current node is in the last layer*/
else /* eqinedge*/
begin
send LOCAL_ACK(p,unused) on edge e;
state(e) := unused;
end
/* this occurs for p>level, eytinedge and negatively acknowledges a broadcast when old
tree is retained */
end /* of the response
Response to receipt of LOCAL_ACK(p,s) on edge e
begin
if p = cycle then
/* outdated acks satisfy p > cycle; they are ignored */
begin
if s=frozen then status :=frozen;
else if s=active and status:;frozen then status :=active;
state(e) := s;
count := count -1;
if count= 0 then
if level > synch_level then /* the node is not a synch */
send LOCAL_ACK(cycle, status) on inedge;
else Call Procedure SYNCH_CONTROL
end
end / * of the response */
Procedure SYNCH CONTROL
begin
if level=--0 and status = frozen then status :-- active;
/* the root ignores freezing on its local iteration */
if status = active and cycle < desired_level then
begin
cycle :- cycle + 1;
Call Procedure XMITLOCAL_BROADCAST
end
else /* local exploration process has been freezed or terminated */
begin
if status = inactive then synchs_number := 0;
else synchs_number := 1;
if level > 0 then
send GLOBAL_ACK (cycle,synchs_number) on inedge
else if level= 0 then
/* it is the end of the local cycle at the the root */
Call Procedure ROOT_CONTROL;
end
end /* of the procedure */
-.
Procedure XMIT GLOBAL BROADCAST
begin
count := 0;
synchs_number := 0; /* lower bound */
cycle := oo; / * upper bound */
f or each edge e such that state(e) =active or frozen
begin
count :- count +1;
send GLOBALBROADCAST(synchlevel,initial_level, desiredlevel) on e;
end
end /* of the procedure */
Response to receipt of GLOBAL.BROADCAST (s,i,d) on edge e :
begin
if
there exists no incident edge e with state(e)=active or frozen then
send GLOBAL_ACK(cycle,0) on inedge;
else
begin
synch_level := s;
initial_level := i;
desiredjevel := d;
if level < synch_level then Call Procedure XNMIT_GLOBAL_B
else /* level = synch_level */
ROADCAST
begin
if initial_level=cycle+1 then cycle := cycle +1;
/ *otherwise test for still frozen local tree */
Call Procedure XMIT_LOCAL_BROADCAST;
/ * this invokes execution of the local exploration process*/
end
end
end /* of the response */
Response to receipt of GLOBAL.ACK (c,m) on edge e:
begin
synchs_number := synchs_number + m;
if
m > 0 then
begin
cycle := min {cycle,c};
state(e) := active;
end
else state(e) :- inactive;
count := count- 1;
if count = 0 then
if level > 0 then send GLOBAL_ACK(cycle,synchs_number) on inedge
else Call Procedure ROOT_CONTROL;
end /* of the response */
Procedure ROOTCONTROL
begin
if synchs_number =0 then STOP ; /* algorithm is complete; BFS tree is given by
inedges pointing in and inactive edges pointing out */
else
begin /* next sub-iteration */
initial_level := cycle + 1 ; /* starting at the height of lowest tree */
if cycle = desired_level then /* otherwise freezing exists and synch_level and
desired level need not be changed*/
if synchs_number < V 0° 2 and desiredlevel - synch_level < V°0 4 then
desiredjevel := synchlevel + V0°4;
/ *few synchs, make group long */
else
if
desired_level = synch_level + V 0° 2 then
desired_level := desired_level +1;
/ * the estimate of active synchs may be too high; test actual number of
active synch nodes by processing just one layer */
else
begin /* new iteration */
synchlevel : cycle;
desired_level := synch_level + V0°2;
end
Call Procedure XMIT_GLOBALBROADCAST
end
end /* of the Procedure */
FIGURE 1A
BRORDCRSTTO FIND A NEW LEVEL
FIGURE 1B TESTING NODES RT NEW LEUEL
FIGURE 1C ACKNOWLEDGEMENTS
FIGURE 10 COORDINRTION BEFORE NEXT LAYER
I--
---
-
--
SYNCH
SYN(i
NODE
FIGURE 28: LOCAL BORDCAST
NOOE
NODE
Fl6URE 2C: LOCKL ACKNOWLE61EMENT
NODE
ELOW LOCA. TREE
FIGURE 3A EXAMPLE OF FAST TREE TAKING EXTRA NODES
NODE
NODE
FIGURE 3B: LOCAL BROADCAST INSLOW TREE RECAPTURING NODE
URE: LOL A NOEW
FIGURE 3C: LOCAL ACKNOWLEDGEMENT
TREE
INSLOW TREE