### D4R Algorithm examples

D4R is the distributed dynamic deadlock detection and resolution algorithm. It is used to detect deadlock for bounded scheduling of PN (and CPN). Here, we provide some examples of the D4R algorithm.

A paper describing the algorithm was presented at ICASSP 2007. The following two figures show the algorithm state variables at each node, and the algorithm state transitions.

The first example is simple, but important -- the algorithm detects a deadlock for the most basic of cases. Node A continually tries to write 2 samples to queue Q, and node B continually tries to read 2 samples from Q. Q starts with a capacity of only 1. We know that until Q has a capacity of 2, A cannot write the 2 tokens. This example shows that the D4R algorithm detects that Q is too small, and causing an artificial deadlock.

 Node A() { while (true) Q.put(2); } Node B() { while (true) Q.get(2); }

Note that this is not the only possible order of node executions, but that all orders will eventually lead to Q being lengthened.

The second example is more complicated, and illustrates an artificial deadlock containing 3 nodes. Queue Q must grow to size 2 before the program can function normally. Again, this example demonstrates that this problem is detected and resolved.

 Node A() {while (true) { Q.put(2); Q.put(1); }} Node B() {while (true) { P.get(1); R.put(1); }} Node C() {while (true) { R.get(1); Q.get(2); }}

Again, the nodes could have executed in many different orders -- this order is one possible execution order. Note that step 6, node C knows that a deadlock exists, however the culpable queue is not located until step 7.