1. Introduction to the Design and Analysis of Algorithms
ElSayed Badr
Benha University
September 20, 2016

2. Definitions Distance Diameter Radio Labeling Span Radio Number
Gear Graph

3. Distance
Distance: dist(u,v) is the length of a shortest path between u and v in a graph G. u v

4. Diameter
Diameter: d(G) is the longest distance in a graph G u v

5. Radio Labeling A Radio Labeling is a one-to-one mapping
c: V(G) N satisfying the condition
for any distinct vertices (u,v).
C: V(G)- N means when you put in a vertices you get out a number. (The vertices are the input to the mapping c, the output consists of natural numbers.)
2 +
≥ 1+3 1 4 8 12
9≥4 u v

6. Span of a labeling c
Span of a labeling c: the max integer that c maps to a vertex of graph G. 1 4 8 12

7. Radio Number
The Radio Number is the lowest span among all radio labelings of a given graph G.
Notation: rn(G) = min {rn(c)} 3 4 1 6 1 4 8 12

8. Radio Number for Path (chain )Graph
The labeling from 1 to radio number

9. Radio Number for Cyclic Graph
The labeling from 1 to radio number

10. Special Graphs for the project
1- Binomial tree
2- Fibonacci tree 3- 4- 5- 6- 7- 8-

11. Gear Graph
A gear graph is a planar connected graph with 2n+1 vertices and 3n edges. The center vertex is adjacent to n vertices which are of degree- three. Between two degree-three vertices is a degree-two vertex. When n≥5 the diameter is 4. G7

12. Theorem: , when n ≥ 7.

13. Standard labeling for , n oddW2 V3 V2 W3 W1 V4 Z V1 W4
Positions of the the W1 and V1 change W7 V5 V7 W5 W6 V6

14. Standard labeling for , n evenW2 V2 V1 W3 W1 V3 Z V7 W4 W7 V4 V6 W5 W6 V5

15. Prove 1. Define a labeling c 2. Show c is a radio labeling
3. Show span(c) = 4n + 2

16. Lower Bound d(u,v)+ | c(u)-c(v) | ≥ 5 Z 2 3 V W 4 1 Vertex type
Strategy: consider placing labels in a manner that omits the fewest values possible.
(vertex distance)
(label diff)
Vertex
type
Max dist
Min diff Z 2 3 V W 4 1 Z W V
the chart adds up to 5 exactly what we need to meet our condition

17. Lower Bound Z 3 2* 1 W n V 2 1** V’s 2(n-1) n-1 Total 2n + 1 4n + 2
Vertices
Min label diff
Min. # of values omitted
Values used Z 3 2* 1 W n V 2
1**
V’s
2(n-1)
n-1
*Best case: use an extreme value (1 or the span) for Z, otherwise more than two values must be omitted.
one
**Use the remaining extreme value for one of the V vertices, otherwise more than 1 value must be omitted.
other
Explain the way the 1 or the span omits less than when the it is labeled in the center.
Total
2n + 1
4n + 2
when n ≥ 7.

18. The Order Of The PatternW2 X5
X10 V3 V2 X9 W3 X2 W1 X1 X0 Z
X11 V4 X8 V1 W4 X6 W7 X4
X12 V5
X14 V7 X7 W6 W5 X3 V6
X13

19. V W Re-labeling For any given let n =2k or n = 2k+ 1 W2i-1 Xi,
Va Xn+a
W2i Xi,
i= 1,…,k
Examples: G7
V X7+5 =X12
W2i Xn+k+i
i= 1,2,…,k
W5 = W2(3) X3
Where i= (a+1)/2
W6 = W2(3) X7+3+3 = X13

20. { 3+i 1 ≤ i ≤ n; 2+n+3(i-n) n+1 ≤ i ≤ 2n. X5 8 18 X10 X9 15 Example:4 X1
X n+3(i-n)
2+(7)+3( 11 – 7 )= 21 X0 1
X11 21 12 X8
Let’s put the labels on in order from smallest to largest and refer to the formula, e.g. when I is between i and n, each label is just 1 greater than the previous. X6 9 X4 7 24
X12 30
X14 X3 6 X7 10
X13 27

21. Claim: c is a radio labeling for
*Note diam(G) = 4 for all when n ≥ 6
WTS: d(u,v) + | c(u) - c(v)| ≥ 1+ diam(G) = 5 18 V2 15 21 1 V1
Case1: u = C (center), v = {V1, …,Vn}
* Know c(u) = 1
the possible labels for c(v) = { n+5, n+8,…, 4n+2}
Then, d(u,v) = 1 12 Z V1 V2 24 30 27
so, d(u,v) + | c(u) – c(v)|
≥ 1 + |1 - (n +5)|
Example: u = Center v = V1
c(u) = 1 c(v) = 12
We will be using v1 since it is the closest radio labeling number to the center.
= 1 + n + 4
1 + | |
= n +5 ≥ 5  =
= 12 ≥ 5

22. { Upper Bound Our goal is to show: when n ≥ 7. 3+i 1≤ i ≤ n
2+n+3(i-n) n+1≤ i ≤ 2n
2+n+3( i
- n)
n+1≤ i ≤ 2n
2 + n + 3n
4n + 2
when n ≥ 7.

23. Conclusion
Lower Bound
Upper Bound
*When n ≥ 7

24. References
[1] Chartrand, Erwin, and Zhang, A graph labeling problem suggest by FM channel restrictions, manuscript, 2001.
[2] Liu and Zhu, Multi-level distance labeling for paths and cycles, SIAM J. Disc. Math, 2002(revised 2003).

26. Lower Bound Z 1 2 W n V n-1 2(n-1) Total 2n + 1 4n + 2 when n ≥ 7.
Values used
Values omitted V
Vertices Z 1 2 W n V
n-1
2(n-1)
d(u,v)+ | c(u)-c(v) | ≥ 5
The center has a distance of one to all V vertices and a distance of two with W vertices.
Last
other
Every other W vertex has a distance of four.
Total
2n + 1
The V vertices have a distance of two between each other.
4n + 2
when n ≥ 7.

27. G1 1 4 2

28. G2 1 2 5 6 3

29. G3 1 3 6 9 4 7 10

30. G4 1 8 4 5 14 10 20 17 12

31. G5 1 4 24 7 18 10 21 15 5 8 12

32. G6 1 8 12 26 6 23 11 20 5 17 10 14 4