Mathematics for Computer Science

8.13. References 313 (a)Explain why (8.51) follows very simply from Euler’s theorem whenmisrel- atively prime ton. All the rest ...

Chapter 8 Number Theory314 cryptosystem efficiently, then they also have the ability to factor numbers that are products of two ...

8.13. References 315 But what if you don’t knowp;q? Let’s assume that the evil message interceptor claims to have a program that ...

Chapter 8 Number Theory316 In particular, the number 1 has four square roots modulon. The two trivial ones are 1 andn 1 (which i ...

9 Directed graphs & Partial Orders Directed graphs, calleddigraphsfor short, provide a handy way to represent how things are ...

Chapter 9 Directed graphs & Partial Orders318 in 0 in 1 in 2 out 0 out 1 out 2 Figure 9.2 A 6-switch packet routing digraph. ...

9.1. Vertex Degrees 319 u v tail e head Figure 9.4 A directed edgeeDhu!vi. The edgeestarts at the tail vertex,u, and ends at the ...

Chapter 9 Directed graphs & Partial Orders320 12 6 1 4 2 8 10 5 7 3 9 11 Figure 9.5 The Digraph for Divisibility onf1;2;:::; ...

9.2. Walks and Paths 321 So a walk,v, is a sequence of the form vWWDv 0 hv 0 !v 1 iv 1 hv 1 !v 2 iv 2 :::hvk 1 !vkivk wherehvi!v ...

Chapter 9 Directed graphs & Partial Orders322 from 1 to 4, and the rest of the walk 4 h 4! 12 i 12 h 12! 12 i 12 h 12! 12 i1 ...

9.3. Adjacency Matrices 323 As would be expected, this definition of distance satisfies: Lemma 9.2.5.[The Triangle Inequality] d ...

Chapter 9 Directed graphs & Partial Orders324 For example, letHbe the 4-node graph shown in Figure 9.1. Its adjacency matrix ...

9.3. Adjacency Matrices 325 According to this theorem, the square.AG/^2 of the adjacency matrix is the length two walk counting ...

Chapter 9 Directed graphs & Partial Orders326 or costs and the objective is to find least weight, cheapest paths. These refi ...

9.5. Directed Acyclic Graphs & Scheduling 327 This even works fornD 0 , with the usual convention thatG^0 is theidentity rel ...

Chapter 9 Directed graphs & Partial Orders328 New6-3: SB in Computer Science and Engineering 6.UAT All subjects are 12 units ...

9.5. Directed Acyclic Graphs & Scheduling 329 underwear shirt jacket left shoe right shoe belt left sock right sock pants ti ...

Chapter 9 Directed graphs & Partial Orders330 underwear left sock shirt shirt pants tie belt underwear tie right sock jacket ...

9.5. Directed Acyclic Graphs & Scheduling 331 Theorem 9.5.4.Every finite DAG has a topological sort. There are many other wa ...

Chapter 9 Directed graphs & Partial Orders332 underwear shirt jacket left shoe right shoe belt left sock right sock pants ti ...