Department of Computer Science and Engineering Bangladesh University of Engineering and Technology Md. Emran Chowdhury Department of CSE Northern University Bangladesh Muhammad Jawaherul Alam Md. Saidur Rahman 6 th International Conference on Electrical & Computer Engineering (ICECE) 2010

2 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding ▒ Conclusion and Future Works

3 Point-Set Embedding a c b d f e S a c b d f e G Each vertex is placed at a distinct point Input

4 f c b d a e a c b d f e S G Point-Set Embedding Each vertex is placed at a distinct point Each edge is drawn by straight or poly line Output Bend

Upward Point-Set Embedding f c b d a e a c b d f e S G Each edge is drawn upward Input Each vertex is placed at a distinct point Output

6 Each edge is drawn upward Upward Point-Set Embedding a c b d f e G’ S f c b d a e a c b d f e G a c b d f e G S f c b d a e f c b d a e S G’ has no upward point-set embedding on S Not every graph has upward point-set embedding on a fixed point-set

a c d b Upward Point-Set Embedding with mapping a c d b S G φ

8 a c d b S G a c d b φ a c b d S φ’φ’ No upward point-set embedding with this mapping Upward Point-Set Embedding with mapping Finding upward point-set embedding with mapping is a real challenge

9 ▒ Problem Definition Contents ▒ Motivation

10 visual analysis of self-modifiable code, based on computing a sequence of drawings whose edges are defined at run-time [Hal91] Motivation Upward Point-set Embedding with mapping That alters its own instructions while it is executing-usually to reduce the instruction path length and improve performance.

11 Motivation The graphs are specified one at a time The vertex locations for the output graphs are determined by the first graph

12 Motivation In VLSI layout, we often want to find point-set embeddings of planar graphs with fewer bends.

13 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results

14 Previous Results and Our Results Problem Graph class Authors Results Giordano et. al. ’07 upward point-set embedding Upward planar digraphs at most two bends per edge Giordano, Liotta, and Whiteside ’09 upward point-set embedding with mapping Upward planar digraphs at most 2n-3 bends per edge This Paper upward point-set embedding with mapping Upward planar digraphs at most n-3 bends per edge upper bound on total number of bends Upward Point-Set Embedding

15 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding

16 Upward Point-Set Embedding SG Input Upward Topological Book Embedding v1v1 v3v3 v4v4 v2v2 v5v v1v1 v3v3 v4v4 v2v2 v5v Upward Point-set Embedding

17 a c b d Upward Topological Book Embedding a c d b S G Spine Left Page Right Page The vertices on the spine The edges on the pages Digraph Upward Topological Book Embedding

18 G contains directed hamiltonian path A directed path containing all the vertices Upward Topological Book Embedding

19 G contains directed hamiltonian path Upward Topological Book Embedding

Upward Topological Book Embedding

Upward Topological Book Embedding

Upward Topological Book Embedding The drawing ….. has no edge crossings since it has the same embeddingas the original graph has no spine crossing has 1 bend per edge

23 G does not contain directed Hamiltonian path a b c d e Upward Topological Book Embedding

24 a b c d e Upward Topological Book Embedding G does not contain directed Hamiltonian path

25 a b c d e Upward Topological Book Embedding G does not contain directed Hamiltonian path

26 a b c d e a b c d e Upward Topological Book Embedding

a b c d e a b c d e Upward Topological Book Embedding Input digraph Each spine crossing corresponds to a dummy vertex

28 Calculation of number of Bends i i+1 i+2 j-2 j-1 j Spine crossing from i to j is at most j-i-2

29 Calculation of number of Bends Spine crossing from i to j is at most j-i-2 Spine Crossings per edge is at most (n-1)-1-2 = n-4 or n-2-2 = n n-3 n-2 n-1 n The edge (1, n) has no crossings

30 Calculation of number of Bends Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Bends per edge is at most n-3 Spine Crossings per edge is at most (n-1)-1-2 = n-4 or n-2-2 = n-4

31 Calculation of number of Bends Spine crossing from i to j is at most j-i-2 The edge (1, n) has no crossings Bends per edge is at most n-3 n-4 spine crossings edge (1, n-1) edge (2, n) n-5 spine crossings edge (1, n-2) edge (2, n-1) edge (3, n) Spine Crossings per edge is at most (n-1)-1-2 = n-4 or n-2-2 = n-4 Total number of spine crossings =2(n-4)+3(n-5)+... +k(n-2-k)+p(n-3-k) where p, k are integers Number of edges which crosses the spine={k(k+1)/2}-1+p

32 ▒ Problem Definition Contents ▒ Motivation ▒ Previous Results and Our Results ▒ Upward Point-Set Embedding ▒ Conclusion and Future Works

33 Conclusion upward planar digraph n-3 bends per edge Quadratic

34 Design a fast algorithm for checking upward point-set embedding Minimize the number of bends in upward point-set embedding Future Works

35 ThankYou

36 Pseudo-code example repeat N times { if STATE is 1 increase A by 1 else decrease A by 1 do something with A } repeat N times { increase A by 1 do something with A } when STATE has to switch { replace the op-code increase above with the op-code to decrease }

37 Assembly style self-modifying code 1.Optimization of a state dependant loop. 2. Runtime code generation, or specialization of an algorithm in runtime or load time (which is popular, for example in the domain of real-time graphics). 3. Altering of in lined state of an object, or simulating the high level construction of closures. 4. Patching of subroutine address calling, as done usually at load time of dynamic libraries. Whether this is regarded 'self-modifying code' or not is a case of terminology.