An efficient algorithm for finding double-vertex dominators in circuit graphs

1. An efficient algorithm for finding double-vertex dominators in circuit graphs Maxim Teslenko, Elena Dubrova DATE’05

2. Double-Vertex dominators A

3. Outline • Introduction • Preliminaries • Dominator chain • Experimental results • Conclusion

4. Introduction • A vertex v is said to dominate another vertex u if every path from u to the outputs of the circuit contains v • Dominators provide a general mechanism for identifying re-converging paths in circuits • Single-vertex dominators are rare in circuits

5. Introduction • Knowing re-convergent region is useful in a number of applications including • Computation of signal probabilities for test generation • Switching activities for power and noise analysis • Cut point selection in equivalence checking • Statistical timing analysis

6. Introduction • Introduce a data structure, called dominator chain, which allows representing all possible double-vertex dominators of a given vertex • Present an efficient algorithm for finding double-vertex dominators

7. Outline • Introduction • Preliminaries • Dominator chain • Experimental results • Conclusion

8. Preliminaries • C=(V, E, root) : a single-output directed acyclic circuit graph • Vertex v is the immediate dominator of u, if v dominates u and every other dominator of u dominates v. idom(v)

9. ∩ ∩ ∩ ∩ Multiple-vertex dominator • Definition 1: A set of vertices {v1,…,vk} is a common multiple-vertex dominator of size k for a set of vertices {u1,…,ul}V – {v1,…,vk}, if • Every path from any uj, j {1,…,l} to root contains some vi, i {1,…,k} • For every vi, there exists at least one path from some uj, j {1,…,l}, to root which contains vi and does not contain any other vj, i, j, {1,…,k}, i≠j ∩

10. Example j e n f {j, n} is not a double-vertex dominator of e

11. Immediate common k-vertex dominator • Definition 2: The set W={vi,…,vk} is an immediate common k-vertex dominator of {u1,…,ul}, if W is a common k-vertex dominator of {u1,…, ul} and there is no other common k-vertex dominator of {u1,…ul}, W’, such that each vertex of W’ is either dominated by W or belongs to W

12. Example j e k b l h m b has two immediate 3-vertex dominator: {e, l, m} and {h, j, k}

13. Outline • Introduction • Preliminaries • Dominator chain • Experimental results • Conclusion

14. ∩ ∩ ∩ Dominator Chain • Definition 3: For any u V, the dominator chainD(u) is a vector of type 〈 {V11,V21},{V12,V22},…,{V1m,V2m}〉 whose element Vij, i {1,2}, j {1,…,m}, 0≤m<|V|/2, are vectors of vertices of V. Every pair {V1j,V2j} satisfies the following properties:

15. Example a e h m k u f c d l n g b All double-vertex dominators for u is {a, b}, {a, c}, {a, d}, {e, c}, {e, d}, {h, c}, {h, d}, {h, g}, {k, l}, {m, l}, {k, n}, {m, n} 〈{V11,V21},{V12,V22}〉=〈{〈a, e, h〉,〈b, c, d, g〉} , {〈k, m〉,〈l, n〉}〉

16. ∩ ∩ ∩ ∩ ∩ Dominator Chain • Property 1: For every vVij, there exists a matching vector W=〈w1,…,ws〉, which is a sub-vector (i.e. linear interval) of Vkj, k {1,2}, k≠i, such that • For all r {1,…,s}, {v, wr} is a double-vertex dominator of u; • No other pair of vertices {v, v’}, v’V-W, is a double-vertex dominator of u • The order of the elements of the matching vector W is given by: if {v, wr} is a double-vertex dominator of wt, then t< r, for all r, t {1,…,s}, r≠t

17. ∩ Dominator Chain • Property 2: The immediate double-vertex dominator of u is the first element of V11 and V21. For all j {2,…,m}, the immediate common double-vertex dominator of the last elements of V1j-1 and V2j-1 is the first elements of V1j and V2j. There is no common double-vertex dominator of the last elements of V1m and V2m

18. ∩ Dominator Chain • Property 3: No pair {V1j, V2j}, j {1,…,m}, can be partitioned into two pairs {V1j1,V2j1} and {V1j2,V2j2}, where V1j1∪ V1j2=V1j, V1j1∩ V1j2=Ø, V2j1∪ V2j2=V2j, V2j1∩ V2j2=Ø, which satisfy properties 1 and 2

19. ∩ Dominator Chain • The dominator chain exists for any u V. It contains all possible double-vertex dominators of u Matching vector Double-vertex dominator Dominator Chain

20. Outline • Introduction • Preliminaries • Dominator chain • Experimental results • Conclusion

21. Experimental results • Compare the performance of the presented algorithm to the algorithm [11]

22. Outline • Introduction • Preliminaries • Dominator chain • Experimental results • Conclusion

23. Conclusion • Contributions • Dominator chain represents double-vertex dominators, which has a linear size and can be efficiently manipulated • Design and algorithm for finding double-vertex dominators • Future work • New application : statistical timing analysis