Chapter 5 Guillotine Cut (2) Portals Ding-Zhu Du Rectilinear Steiner Tree Given a set of points in the rectilinear plane, find a minimum length tree interconnecting them. Those given points are called terminals.
Initially Edge length < RSMT Initially L n = # of terminals 2
2 n x n grid Total moving Length: L L n 2
n n If PTAS exists for grid points, then it exists for general case. (1/3-2/3)-cut 1/3 Shorter edge Longer edge
> 1/3 2/3 Longer edge Cut line position L
2 2 n x n grid Cut line always passes through the center of a cell.
1 ( assume) Depth of (1/3-2/3)-cut Note that every two parallel cut lines has distance at least one. Therefore, the smallest rectangle has area 1. After one cut, each resulting rectangle has area Within a factor of 2/3 from the original one. Hence, depth of cuts < (4 log n)/(log (3/2)) = O(log n)
since depth 4 (2/3) n >1 (1/3-2/3)-Partition O(log n) Portals
m portals divide a cut segment equally. Restriction A rectilinear Steiner tree T is restricted if there exists a (1/3-2/3)-partition such that If a segment of T passes through a cut Line, it passes at a portal. Minimum Restricted RST can be
26 O(m) computed in time n 2 by dynamic programming 2 Choices of each cut line = O(n ) 24 O(m) # of subproblems = n 2
# of subproblem Each subproblems can be described by three facts: 1. Position of for edges of a rectangle. 2. Position of portals at each edge. 3. Set of using portals. O(n 8 ) 4 O(n ) 2
O(m) 4. Partition of using portals on the boundary. (In each part of the partition, all portals are connected and every terminal inside of the rectangle is connected to some tree containing a portal. ) 2 O(m)
Position of portals O(n2 ) O(n 2 ) # of partitions N(k) = # of partitions
. 2x Since f (0) 1, we have 1/ 2 1 1 4x ( 4 x) k /( 2 x) f ( x)
2x k 1 k 1/ 2 k 1 ( 4) / 2 N (k ) k 1 0.5(0.5 1)(0.5 2) (0.5 k ) k
2 k 1 ( 1) 2 (k 1) 2O ( k ) Analysis (idea) Consider a MRST T. Choose a (1/3-2/3)-partition. Modify it into a restricted RST by moving
cross-points to portals. Estimate the total cost of moving crosspoints. Choice of (1/3-2/3)-partition 1/3 2/3 Each cut is chosen to minimize # of cross-points. (# of cross-points) x (1/3 longer edge length)
< (length of T lying in rectangle). Moving cross-points to portals Cost = (# of cross-points) x ( edge length/(m+1)) < (3/(m+1)) x (length of T lying in rectangle) Moving cost at each level of (1/3-2/3)-Partition < (3/(m+1)) x (length of T ) O(log n)
Total cost < O(log n)(3 / (m+1)) x (length of T) Choose m = (1/) O(log n). Then 2 O(m) O(1/) =n .
RSMT has (1+)-approximation with running O(1/) Time n . Thanks, End
Chief Ophthalmology Nemours Children's Clinic. ... Robison D Harley MD Endowed Chair of Pediatric Ophthalmology. 12 month ex 23-week premature infant with tracheostomy, h/o retinopathy of prematurity who has been recently diagnosed with cerebral palsy. ... Provide support for child's...
Treaty seen increasingly as unfair The will to make it work Idealism after WW1 soon disappeared Britain and France not best suited to lead the League. Acted against the League (Hoare-Laval Pact) Depression allowed aggressive individuals to gain support at...
Morgan Kaufmann Publishers. 28 January, 2013. Chapter 4 — The Processor. Performance penalty of stalling on branch: 17% of instructions executed in the SPECint2006 benchmark are branch instructions. If we always stalled for 1 clock cycle on . a branch
RCOT Adaptations without Delay (updated report launched in Cardiff on 19 June) Later living survey findings with Shakespeare Martineau (report launched in Manchester on 25 June) National housing design awards (ceremony on 11 July) 4 schemes celebrated HAPPI recognition
Strengths-Based Self-Reflection Activities and Faculty Mentoring Sessions. Four required self-reflection activities and faculty mentoring sessions during the academic year: Exploring My Signature Themes of Talent (early Fall semester) Setting My Academic Course (late Fall semester) Developing My Talents Into Strengths...
A. dissection of the flowers of both tall and short African violet plants . B. microscopic observation of the nuclei of fruit fly cells . C. biochemical analysis of DNA produced in the F2 generations of roan cattle . D....
Ready to download the document? Go ahead and hit continue!