The AC-3 Algorithm

AC-3 repeatedly revises arcs until no domain changes or a domain becomes empty.

r1c15	r1c23	r1c3	r1c4	r1c57	r1c6	r1c7	r1c8	r1c9
r2c16	r2c2	r2c3	r2c41	r2c59	r2c65	r2c7	r2c8	r2c9
r3c1	r3c29	r3c38	r3c4	r3c5	r3c6	r3c7	r3c86	r3c9
r4c18	r4c2	r4c3	r4c4	r4c56	r4c6	r4c7	r4c8	r4c93
r5c14	r5c2	r5c3	r5c48	r5c5	r5c63	r5c7	r5c8	r5c91
r6c17	r6c2	r6c3	r6c4	r6c52	r6c6	r6c7	r6c8	r6c96
r7c1	r7c26	r7c3	r7c4	r7c5	r7c6	r7c72	r7c88	r7c9
r8c1	r8c2	r8c3	r8c44	r8c51	r8c69	r8c7	r8c8	r8c95
r9c1	r9c2	r9c3	r9c4	r9c58	r9c6	r9c7	r9c87	r9c99

queue = every directed peer arc
while queue is not empty:
    (Xi, Xj) = remove an arc
    if REVISE(Xi, Xj):
        if D(Xi) is empty: fail
        add (Xk, Xi) for other neighbors Xk

Sudoku trace

Propagation establishes D(r5c5)={5}. Revising neighboring arcs removes 5 from peers: r5c2:{2,5}→{2} and r7c5:{3,5}→{3}. Because those domains changed, their incoming arcs return to the queue.

Why the queue matters

A deletion can create a new singleton, which can cause more deletions. AC-3 continues until this cascade reaches a fixed point.

Trace: After r5c2 becomes {2}, list its row, column, and box peers whose domains must be reconsidered.