Buffering in the Layout Environment - Computer Engineering ...

More documents

Recommendations

Info

ts(a)(b)Figure 28.3: Routing graph.to solution α 2 = (c 2 , d 2 , m 2 , v) if c 1 ≥ c 2 and d 1 ≥ d 2 .The partial solutions are maintained in a priority queue Q initialized with the solution (C t , 0, 0, t)at the sink t. Each time, the top solution in Q, which has the minimum delay, is extracted for expansion.A solution (c, d, m, u) is expanded to its neighbor node v if there is an edge (u, v) ∈ E.The expanded solution is (c+C(u, v), d+R(u, v)·(c+C(u, v)/2), m, v) where the delay increaseis based on the Elmore delay model. If a solution (c, d, m, v) is at node v where m(v) = 0 andp(v) = 1, buffers of each type are inserted there to generate new partial solutions. If the buffer typeis b ∈ B, its corresponding buffered solution is (C(b), d + R(b) · c + K(b), m, v) with m(v) = b.If a solution reaches the source node as (c, d, m, s), the driver is added by updating the solution as(0, d + R d · c, m, s). When a solution with the driver is at the top of the Q, it is the minimum delaysolution.The pseudo code of this algorithm is given in Figure 28.4. Please note that pruning is performedin many steps to remove inferior solutions so that the runtime can be improved. The complexity ofthis algorithm is O(|B||V |(|E| + |B||V |) log |B||V |) [1].4
Algorithm: Buffered Path(G, B, s, t)Input: Routing graph G = (V, E), Buffer library BSource node s ∈ V and sink node t ∈ VOutput: Buffered path labeling m1. Q ← {(C, 0, 0, t)}2. while Q ≠ ∅ do3. (c, d, m, u) ← extract min(Q)4. if c = 0, return m5. if u = s,push (0, d + R d · c, m, s) into Q and prunecontinue6. for each (u, v) ∈ E dod ′ ← d + R(u, v) · (c + C(u, v)/2)push (c + C(u, v), d ′ , m, v) into Q and prune7. if p(u) = 1 and m(u) = 08. for each b ∈ B dod ′ ← d + R(b) · c + K(b)m(u) = bpush (C(b), d ′ , m, u) into Q and pruneFigure 28.4: Pseudo code of the dynamic programming based buffered path algorithm.3.2 Graph Based ApproachThe graph based approaches [3,4] first transform the routing graph G = (V, E) into a buffer graphG B = (V B , E B ) and then obtain the minimum delay buffered path by the Dijkstra’s shortest pathalgorithm.driverbufferbufferbufferb1ub2b1vb2buffersink(a)(b)(c)Figure 28.5: Buffer graph.The node set V B of the buffer graph is composed by the source node, sink nodes and a set of5
Page 6 and 7: uffer nodes. A buffer node always h
Page 8 and 9: 4.1 Tree Adjustment TechniqueAs a r
Page 10 and 11: the root of the subtree. The label
Page 12 and 13: s1s2s1s2Figure 28.11: A graph for g
Page 14 and 15: 5.2 Plate Based Tree AdjustmentWhen
Page 16 and 17: During the bottom-up process, each
Page 18 and 19: v1v1v5v6v2v4v3v4(a)(b)Figure 28.14:
Page 20 and 21: For Critical Nets, the cost impact
Page 22 and 23: candidates, which is chosen by the
Page 24 and 25: [10] S. Dechu, Z. C. Shen, and C. C

Buffering in the Layout Environment - Computer Engineering ...

Create successful ePaper yourself

Delete template?

Save as template?