D.3.3 ALGORITHMS FOR INCREMENTAL ... - SecureChange

20 F. Massacci and L.M.S. Tran
Algorithm 2. Updating incrementally DAT.
1 procedure updateDAT(G : EvolutionaryHyperGraph, t : node)
2 begin
3 makeQemtpy();
4 enqueue(Q, t);
5 while Q ≠ ∅ do
6 n ←dequeue(Q);
7 i ←0;
8 DAT[n] ←∅ ;
9 for each 〈 x, n, p 〉 ∈G do {incoming edges of n}
10 i ←i + 1; {i th incoming edges of n}
11 if n is an observable node then
12 DAT[n] ←concat(n, DAT[n], map(〈 x, i, p 〉, DAT[x]));
13 else if n is a goal node then
14 DAT[n] ←concat(n, DAT[n], DAT[x]);
15 else
16 DAT[n] ←join(n, DAT[n], DAT[x]);
17 for each 〈 n, y, p 〉 ∈G do {outgoing edges of n}
18 enqueue(Q, y);
19 end
The inputs of Algo. 2 are the evolutionary hypergraph whose nodes’ DAT have
been generated by Algo. 1, and a changed node t. The algorithm initializes a queue
Q holding nodes whose DATs need to be updated (line 3). At the beginning, Q holds
t. Then, the while loop at line 5 sequentially extracts nodes n from Q to update
DAT(n). Once node n extracted, its DAT is reset to empty, then the for loop at line
9 re-calculate DAT[n] as exactly as Algo. 1 does. After that, the for loop at line
17 puts all ancestors of n to Q to further update their DATs.
Theorem 1 The algorithm 1 always terminates in time polynomial in the number
of nodes of the hypegraph and the number of alternative solutions. When terminated,
all DATs of all nodes in the input graph are initialized as Formula 9.
Proof (Theorem 1) Let us consider the first point in the theorem about the polynomial
complexity of the algorithm. Algo. 1 has two for loops (line 9, 17) nesting
inside a while loop (line 4). The two for loops are proportional to the maximum
incoming edges of an arbitrary node. Suppose that the graph has n node, then the
maximum incoming edges is n − 1. Hence the complexity of these for loops are
O(n) if we ignore the complexities of join and concat.
The outside while loop terminates when the queue Q empty. At beginning,
the initialization at line 3 pushes all leaf nodes to the queue. Later on, only nodes
y which REACH[y] equals 0 are enqueued (line 19, 20). When a node y is being
processed, its corresponding REACH[y] is marked as -1 (line 8). Thus every node
is processed exactly once due to the check at line 7. Hence the complexity of the
algorithm is O(n 2 ).