markov chains and random walks

... which randomness comes h' hear on computer science: random ized algorithms and the probabilistic analysis of 1l�orithms Randomized algorithms: Randomized algorithms are algorithms that make random ... Case 6.10 Exercises :\Iarkov Chains and Random Walks 53 15 156 159 63 166 167 173 74 176 177 182 7.2 Markov Chains: Definitions and Representations 7.1.1 Application: A Randomized Algorithm for 2-Satisfiability ... Random Variables and Expectation 20 2.1 20 22 23 25 26 30 32 34 38 Random Variables and Expectation 2.1.1 Linearity of Expectations 2.1.2 Jensen's Inequality 2.2 The Bernoulli and Binomial Random...

Ngày tải lên: 14/07/2016, 15:42

... pivot point randomly, we will have a randomized version of QuickSort © 2010, Quoc Le & Van Nguyen Probability for Computing 10 Analysis Xij be a random variable that   Takes value if yi and yj are ... problems in the design and analysis of algorithms © 2010, Quoc Le & Van Nguyen Probability for Computing 20 The maximum load When n balls are thrown independently and uniformly at random into n bins, ... balls that are thrown into n bins, with the location of each ball chosen independently and uniformly at random from n possibilities What does the distribution of the balls into the bins look like...

Ngày tải lên: 12/07/2014, 18:20

... problems in the design and analysis of algorithms © 2010, Van Nguyen Probability for Computing The maximum load When n balls are thrown independently and uniformly at random into n bins, the ... balls that are thrown into n bins, with the location of each ball chosen independently and uniformly at random from n possibilities What does the distribution of the balls into the bins look like ... Gather: Visit the buckets in order and put all elements back into the original array © 2010, Van Nguyen Probability for Computing A set of n =2m integers, randomly chosen from [0,2k),km, can...

Ngày tải lên: 12/07/2014, 18:20

... and “consistency” in vertex-signed graphs, and generalizable work on two-graphs (the set of unbalanced triangles of a signed complete graph) and on even and odd polygons and paths in graphs and ... G Brown and Richard D McBride 1984a Solving generalized networks Management Sci 30 (1984), 1497–1523 Zbl 554.90032 (GN: M(bases)) Kenneth S Brown and Persi Diaconis 1998a Random walks and hyperplane ... m ) [See Poljak and Turz´ (1982a), Sol´ and Zaslavsky (1994a) for more on D(Γ); Brown and Spencer e (1971a), Gordon and Witsenhausen (1972a) for D(Kt,t ); Harary, Lindstr¨m, o and Zetterstr¨m...

Ngày tải lên: 07/08/2014, 05:23

... ring and let A and A be two n × n matrices Then rankR (A + A ) ≤ rankR (A) + rankR (A ) Proof: Let A = BC and A = B C , where B is an n × r and C is an r × n matrix, while B is an n × r and C ... of the columns of B and B , and C is an (r + r ) × n matrix, formed from the union of rows of C and C The following theorem shows, that for any prime p, the co-triangle (and, consequently, the ... result of Ramsey [7] and Erd˝s and Szekeres [4], every n-vertex graph has o either a clique on k, or an anti-clique on vertices, if n≥ If we set k = log n log log n k+ −2 k−1 , and = log2 n , then...

Ngày tải lên: 07/08/2014, 06:20

... Alg Universalis (1977), 313–314 a [3] B Bollob´s, Random Graphs, Academic Press 1985 a [4] B Bollob´s, G Brightwell and J Neˇetˇil, Random Graphs and Covering Graphs a s r of Posets, Order (1986) ... Solution March 1948 [6] P.Erd˝s and A.R´nyi, On the evolution of random graphs, Publ Math Inst o e Hungar Acad Sci (1960) 17-61 [7] T.Luczak, The chromatic number of random graphs, Combinatorica ... was introduced by Bollob´s, a Brightwell, and Neˇetˇil [4] s r First consider dense random graphs It is shown in [4] that for arbitrary integer l ≥ and p = p(n) = o n(l−2)/(l−1) , c(Gn,p ) ≤...

Ngày tải lên: 07/08/2014, 06:20

... · · , Cr / and C1 , C2 , · · · , Cr are the cycles encountered in the edge deletion procedures for T and T Since (i1 , i2 ) ∈ E(T ), we may assume that (i1 , i2 ) ∈ E(Cj ) and i1 and i2 are ... formulas (1.3) and (1.4) for Nn,k and dn,k and the following identity 2n−1 (−1)m−n−k m=n 3n n+1+m m−1 n−1 m−n k = n−1+k n−1 2n − k , n+1 (3.2) which can be verified by using the Vandermonde convolution ... [2] E Deutsch, S Feretic and M Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math 256 (2002) 645-654 [3] E Deutsch and M Noy, Statistics on...

Ngày tải lên: 07/08/2014, 13:21

... Theory (1980) 93–100 [4] A Brandst¨dt, V B Le, and J P Spinrad, Graph Classes: A Survey, Society for a Industrial and Applied Mathematics, Philadelphia, 1999 [5] M Burlet and J.-P Uhry, Parity graphs, ... that C3 is the only order-3 hamiltonian graph and that every nonbipartite graph contains an induced cycle Ck with k odd.) References [1] H.-J Bandelt and H M Mulder Distance-hereditary graphs J ... Theory (Ser B) 41 (1986) 182–208 [2] H.-J Bandelt and H M Mulder Metric characterization of parity graphs Discrete Math 91 (1991) 221–230 [3] C A Barefoot and R C Entringer, Extremal maximal uniquely...

Ngày tải lên: 07/08/2014, 15:22

... a clique U with α vertices, where, for any u, w ∈ Γ, ¯ Γ ¯ ¯ ¯ ¯ u ∈ U and w ∈ W , u and w are adjacent if and only if u and w are adjacent ¯ ¯ Lemma 3.1 Let Γ be an amply regular Terwilliger ... w in Γi−1 (u) and bi neighbors of w in Γi+1 (u) (we assume that Γ−1 (u) and Γd+1 (u) are empty sets) In particular, a distance-regular graph Γ is regular with degree b0 , c1 = and c2 = µ(Γ) For ... a Moore strongly regular graph ∆ Then α = and one of the following holds: (1) ∆ is the pentagon and Γ is the icosahedron; (2) ∆ is the Petersen graph and Γ is the Doro graph or the Conway–Smith...

Ngày tải lên: 08/08/2014, 12:22

... was conjectured by Fallat and Kirkland [5] Theorem 1.2 ([7, 8]) Let G be a connected graph of order n with girth g (1) α(G) Then α(Cn,g ), and the equality holds if and only if G ∼ Cn,g = (2) ... u1 u2 and v1 v2 such that the resulting graph is in Un3 and does not isomorphic to one of graphs U1 , U2 , U3 , U5 , U6 , U7 Thus the result follows Similarly, if n1 = and n2 = (or n3 = and n4 ... u1 u2 and e2 = v1 v2 be the edges on the = cycles C1 and C2 of G′ , respectively, such that the degrees of u1 , u2, v1 and v2 in G′ are all Then in G = G′ + e, at least one of u1 , u2, v1 and v2...

Ngày tải lên: 08/08/2014, 12:23

... trees As GCS is vertex transitive and thus regular, the only possible clique trees are single cliques Since rows −t and and rows and t are adjacent, but rows −t and t are not, this is a contradiction ... Proof Suppose rows and are not adjacent Then using Lemmas 3.2 and 3.3, we can permute columns such that rowi (A) = [i, i + r) for i ∈ [3] and {2, r + 1} ⊂ row2 (B) Since rows and are adjacent in ... exactly one case in (9) is true Take some row j such that rows i and j are adjacent in GA and j ≤ ℓ (this is possible since ℓ ≥ and rows and are adjacent) Let {a} = {i, i + r − 1} ∩ rowj (A) = rowi...

Ngày tải lên: 08/08/2014, 12:23

... for both men and women were manufacturing, which had about 300 000 women and 650 000 men in both surveyed years, and the public sector (non-defence), which employed 650 000 women and 850 000 men ... also made gains in both the finance/banking industries and in the defence-related public sector Whereas some 125 000 women worked in finance and banking institutions in 1975, the number increased ... decades between 1975 and 1995 brought significant changes in the representation of women in Freedonia's work force, according to the graphs In 1975, for example, some 300 000 men and 250 000 women...

Ngày tải lên: 04/10/2012, 10:02

... with graphs and rables ,and since they require specific interpretation and writing conyentions,we will first focus on these A Note on terminology: The IELIS Handbook mentions diagrams and tables ... tonnes.Bangladesh,Viet Nam and Thailand had similar leuelsofproduction, i.e between20 and 30 million tonneseach, while Malaysia, Japan and the Philippines ranged between16 and l0 million tonne* The ... understand what to write, as well as ltotl to write it The first section of to this book therefore beginswith a discussionabout the main purpose and featuresof to graphs, tablesand other visuals,and...

Ngày tải lên: 04/10/2012, 10:02

... with graphs and rables ,and since they require specific interpretation and writing conyentions,we will first focus on these A Note on terminology: The IELIS Handbook mentions diagrams and tables ... tonnes.Bangladesh,Viet Nam and Thailand had similar leuelsofproduction, i.e between20 and 30 million tonneseach, while Malaysia, Japan and the Philippines ranged between16 and l0 million tonne* The ... understand what to write, as well as ltotl to write it The first section of to this book therefore beginswith a discussionabout the main purpose and featuresof to graphs, tablesand other visuals,and...

Ngày tải lên: 04/09/2013, 15:20

... differences between directed and undirected edges and the way in which they are modeled Figure 12-4 shows two graphs: I is undirected and J is directed Figure 12-4 Directed and undirected graphs have ... routinely applied to trees and hierarchies, but I’ve personally found that it can get confusing trying to figure out which node is the cousin of another, and so have abandoned most of this extended ... contains only two columns, X and Y: CREATE TABLE Edges ( X int NOT NULL, Y int NOT NULL, PRIMARY KEY (X, Y) ); GO The combination of columns X and Y constitutes the primary key, and each row in the table...

Ngày tải lên: 05/10/2013, 08:48

... tonnes.Bangladesh,Viet Nam and Thailand had similar leuelsofproduction, i.e between20 and 30 million tonneseach, while Malaysia, Japan and the Philippines ranged between16 and l0 million tonne* The ... meat excePt that in 1950 production ofpoulny and sheepand goat meat d) The graplt shouts was less than million tons, uthileproduction ofpork and beefand buffalo uas around 20 million tons 1n€At ... details of what is being shown -the units of measurement and the time frame - and decide how much you need to include in Steps and involve understanding the visual This was discussed Part the Step...

Ngày tải lên: 25/10/2013, 07:20

... leuel of a) and b) a) and c) GDP; unemplolment; literacy; income; expenditure a) and c) b) a) and d) 7.b) and c) and d) 4.^) Part3 'per 1000 12 a) and b) and d) are all too vague, and in a) the ... literacy and the leuel of countries population growth in the uorld The graph shows f.guresfom selected 1998 and 2000 The bighestpercentageoffemale literacy wasfound in betuueen Thailand and Colombia, ... top 10 spenders andthose utho ouere top 10 earnersin tourism in 1998 Allthe top spenders the excePt JaPan and the Netherlands uere also arnong the top 10 earners,abhough UK Germarytand Canada earned...

Ngày tải lên: 25/10/2013, 07:20

... Delegates Committed to Each Candidate Candidate Candidate Candidate Candidate 200 400 600 800 Delegates Candidate has approximately how many more delegates committed than does Candidate 2? A 150 B 200 ... two years was the greatest rise in average test scores? A 1978 and 1979 B 1979 and 1980 C 1980 and 1981 D 1981 and 1982 E 1982 and 1983 E The most efficient way to compute greatest rise is to ... contains Charts and Tables Charts and tables are often used to give an organized picture of information, or data Make sure that you understand the information that is given Column headings and line...

Ngày tải lên: 01/11/2013, 14:20

... any two elements x and y of L have a greatest lower bound, called the infimum of x and y and denoted by inf(x, y), anda smallest greater bound, called the supremum of x and y and denoted by sup(x, ... M.A Kiwi, “Games on line graphes and sand piles”, Theoret Comput Sci., 115 (1993) 321 [7] E Goles, M Latapy, C Magnien, M Morvan, H.D Phan, “Sandpile models and lattices: a comprehensive survey”, ... em So by using Theorem 1, we have g ≥ c and g ≥ d On the other hand, let h be a configuration of CCFG(T,n) satisfying h ≥ c,d We have eV(i, h) ≤ eV(i, c) and eV(i, h) ≤ eV(i, d) for all nodes i...

Ngày tải lên: 12/02/2014, 17:20