Notation Index 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coef

Transcription

1 Notation 9 (there exists) Fn-4 8 (for all) Fn-4 3 (such that) Fn-4 B n (Bell numbers) CL-25 s ο t (equivalence relation) GT-4 n k (binomial coefficient) CL-14 (multinomial coefficient) CL-18 n m 1 ;m 2 ;::: BFE(T ) (breadth first vertex sequence) DT-7, GT-29 BFV(T ) (breadth first vertex sequence) DT-7, GT-29 C(n; k) (binomial coefficient) Cov(X; Y ) (covariance) CL-14 DFV(T ) (depth first vertex sequence) GT-28 xjy (x divides y) GT-25 DFE(T ) (depth first edge sequence) GT-28 μ X (expectation or mean) Fn-22 E(X) (expectation) f ffi g (composition) Fn-22 Fn-6 F n (Fibonacci numbers) (V; E) (simple graph) (V; E; ffi) (graph) N (natural numbers) n (first n integers) GT-2 GT-2 CL-13 DT-18 O( ) (Big oh notation) GT-37 o( ) (little oh notation) GT-38 P k (A) (k-subsets of A) S(A) (permutations of A) PER(A) (permutations of A) CL-14, Fn-7 Fn-7 DT-7, DT-7, Probability notation μ X (expectation, or mean) Fn-22 ρ(x; Y ) (correlation) ff X (standard deviation) E(X) (expectation) Fn-22 Cov(X; Y ) (covariance) Var(X) (variance) P (AjB) (conditional probability) POSV(T ) (postorder sequence of vertices) DT-8 PREV(T ) (preorder sequence of vertices) DT-8 Q (rational numbers) R (real numbers) ρ(x; Y ) (correlation) CL-26, DT-36 Set notation οa (complement) CL-13, A 0 (complement) CL-13, A B (difference) CL-13, A B (intersection) CL-13, A [ B (union) CL-13, AΦB (symmetric difference) A n B (difference) CL-13, A B (Cartesian product) CL-3, A c (complement) CL-13, P k (A) (k-subsets of A) CL-14, jaj (cardinality) CL-2 ff X (standard deviation) S(n; k) (Stirling numbers) ( ) (rate of growth) GT-37 Var(X) (variance) Z (integers) CL-13, CL-23-1

2 Subject Adjacent vertices GT-3 Algorithm backtracking DT-7 Kruskal's (minimum weight spanning tree) GT-31 lineal (= depth-first) spanning tree GT-31 partial GT-45 polynomial time (tractable) GT-43 Prim's (minimum weight spanning tree) GT-30 which is faster? GT-41 Antisymmetric binary relation GT-25 Asymptotic GT-38 Average running time GT-40 Backtracking DT-7 Base (simplest) cases for induction DT-15 Bayes' Theorem DT-37, DT-41 Bell numbers CL-25 Bicomponents GT-21 Biconnected components GT-21 Bijection Binary relation GT-5 antisymmetric GT-25 covering GT-25 equivalence relation GT-4 order relation GT-25 reflexive GT-5 symmetric GT-5 transitive GT-5 Binary tree GT-34 full GT-34 Binomial coefficients CL-14 recursion CL-21 Binomial distribution 2 Binomial theorem CL-16 Bipartite graph GT-23 cycle lengths of GT-33 Blocks of a partition CL-18, CL-23, 4 Boolean variables DT-44 Breadth first vertex (edge) sequence DT-7, GT-29 Card hands and multinomial coefficients CL-21 full house CL-17 straight CL-24 two pairs CL-17 Cartesian product CL-3, Characteristic equation DT-18 Chebyshev's inequality Fn-25 Child vertex DT-2, GT-27 Chromatic number GT-40, GT-45 Circuit in a graph GT-17 Eulerian GT-20 Clique GT-44 Clique problem GT-44 Codomain (range) of a function Fn-2 Coimage of a function 3 Coloring a graph GT-40, GT-45 Coloring problem GT-44 Comparing algorithms GT-41 Complement of a set Complete simple graph GT-15 Component connected GT-18 Composition of an integer CL-8 Composition of functions Fn-6 Conditional probability DT-36 Conjunctive normal form DT-44 Connected components GT-18 Correlation Covariance Covering relation GT-25-2

3 Cumulative distribution (of normal) 4 Cycle in a graph Hamiltonian GT-17 GT-20 Cycle in a permutation Fn-8 Decision tree DT-1 see also Rooted tree ordered tree is equivalent GT-27 probabilistic DT-39 RP-tree is equivalent GT-27 Towers of Hanoi DT-28 traversals DT-7, GT-28 Decreasing (strictly) function or list 6 Decreasing (weakly) function or list Degree of a vertex DT-2, GT-4 Degree sequence of a graph Density function Fn-21 GT-4 Depth first vertex (edge) sequence GT-28 Derangement recursion Deviation standard Dictionary order 1 DT-20 CL-4 Digraph GT-14 functional GT-22 Direct (Cartesian) product CL-3, Direct insertion order for permutations DT-5 Directed graph Directed loop GT-14 GT-14 Disjunctive normal form 6 DT-7, Distribution Fn-21 binomial 2 cumulative 4 hypergeometric CL-29 joint 1 marginal 1 normal 3 Poisson 3 uniform CL-25 Distribution function see Distribution Divide and conquer Domain of a function Domino covering DT-21, GT-42 DT-10 Down degree of a vertex Edge DT-2, GT-2 directed GT-14 incident onvertex loop GT-10 parallel GT-10 Fn-2 DT-2 GT-3 Edge sequence breadth first DT-7, GT-29 depth first DT-7, GT-28 Envelope game Equation characteristic Equivalence class Fn-2 Equivalence relation Error percentage CL-9 relative CL-9 DT-18 GT-4 Eulerian circuit or trail Event CL-25, 9 independent pair GT-4 GT-20 Fn-26, DT-37 Expectation of a random variable Fn-22 Factorial estimate (Stirling's formula) CL-9 Fibonacci recursion First Moment Method Full binary tree GT-34 DT-18-3

4 Functiion probability 9 Function bijection codomain (range) of Fn-2 coimage of 3 composition of Fn-6 decreasing: decision tree DT-13 density Fn-21 distribution, see Distribution domain of Fn-2 generating CL-14 image of 2 image of and Stirling numbers (set partitions) 4 injective (one-to-one) inverse inverse image of 2 monotone 6 one-line notation Fn-2 partial DT-3 range of Fn-2 restricted growth and set partitions 9 strictly decreasing 6 strictly increasing 6 surjective (onto) two-line notation Fn-5 weakly decreasing 6 weakly increasing 6 Functional relation Fn-4 Gambler's ruin problem Generating function Geometric probability Geometric series DT-23 CL-14 DT-48 CL-31 Graph GT-2 see also specific topic biconnected GT-21 bipartite GT-23 bipartite and cycle lengths GT-33 complete simple GT-15 connected GT-18, GT-18 directed GT-14 incidence function GT-3 induced subgraph (by edges or vertices) GT-17 isomorphism GT-6 oriented simple GT-24 random GT-7 rooted GT-27 simple GT-2 subgraph of GT-16 Gray code for subsets Growth rate of, see Hamiltonian cycle Hasse diagram Height of a tree Height ofavertex DT-32 Rate of growth GT-25 GT-20 GT-34 DT-2 Hypergeometric probability CL-29 Image of a function 2 Stirling numbers (set partitions) and 4 Incidence function of a graph Inclusion and exclusion CL-35 CL-27, GT-3 Increasing (strictly) function or list Increasing (weakly) function or list Independent events Fn-26, DT-37 Independent random variables Induced subgraph (by edges or vertices) GT-17 Induction Fn-7, DT-14 base (simplest) cases DT-15 induction hypothesis DT-15 inductive step DT-15 Fn

5 Inequality Tchebycheff Injection Internal vertex Intersection of sets Fn-25 DT-2, GT-27 Inverse image of a function Involution Isolated vertex Fn-9 Isomorph rejection Isomorphic graphs GT-10 DT-13 GT-6 Joint distribution function 2 1 Kruskal's algorithm for minimum weight spanning tree GT-31 Leaf vertex rank of DT-2, GT-27 DT-5 Lexicographic order (lex order) CL-4 List CL-2 circular CL-10 strictly decreasing 6 strictly increasing 6 weakly decreasing 6 weakly increasing 5 with repetition CL-3 without repetition CL-2, CL-9 without repetition are injections Little oh notation GT-38 Local description DT-25 Gray code for subsets DT-33 merge sorting DT-24 permutations in lex order DT-27 Towers of Hanoi DT-29 Loop GT-10 directed GT-14 Matrix permutation Merge sorting Merging sorted lists Monotone function 0 Multinomial coefficient DT-24, GT-43 DT-24 6 Multiset CL-2 and monotone function CL-18 Nondecreasing function or list Nonincreasing function or list Normal distribution Normal form conjunctive disjunctive DT-44 NP-complete problem NP-easy problem NP-hard problem 3 GT-44 GT-44 GT Numbers Bell CL-25 binomial coefficients CL-14 Fibonacci DT-18 Stirling (set paritions) 4 Stirling (set partitions) CL-23 Odds CL-29 One-line notation Fn-2 One-to-one function (injection) Onto function (surjection) Order direct insertion for permutations DT-5 lexicographic (lex) CL-4 Order relation GT-25 Oriented simple graph GT-24 Machine independence Marginal distribution GT-36 1 Parallel edges GT-10 Parent vertex DT-2, GT-27 Partial function DT-3-5

6 Partition set CL-23, 3 set (ordered) CL-18 set and restricted growth function 9 Path in a (directed) graph GT-15 Permutation CL-3,, Fn-6 cycle Fn-8 cycle form Fn-8 cycle length Fn-8 derangement 1 direct insertion order DT-5 involution Fn-9 is a bijection matrix 0 powers of Fn-7 Poisson distribution 3 Polynomial time algorithm (tractable) Postorder sequence of vertices Preorder sequence of vertices DT-8 DT-8 Prim's algorithm for minimum weight spanning tree GT-30 Prime factorization DT-15 Probabilistic decision tree DT-39 Probability conditional DT-36 conditional and decision trees function CL-26 probability space CL-26 Probability distribution function see Distribution Probability function CL-26, 9 see also Distribution Probability space CL-26, 9 see also Distribution Random graphs GT-7 Random variable Fn-20 binomial 2 correlation of two covariance of two independent pair Fn-26 standard deviation of variance of DT-39 GT-43 Range of a function Rank (of a leaf) DT-5 Fn-2 Rate of growth Big oh notation GT-37 comparing GT-41 exponential GT-43 little oh notation GT-38 polynomial GT-38, GT-43 Theta notation GT-37 Rearranging words Recurrence see Recursion Recursion DT-16 CL-18 see also Recursive procedure binomial coefficients CL-21 derangements DT-20 Fibonacci DT-18 guessing solutions DT-21 inductive proofs and DT-14 set partitions (Bell numbers) CL-25 set partitions (Stirling numbers) CL-23 sum of first n integers DT-16 Recursive equation see Recursion Recursive procedure see also Recursion 0-1 sequences DT-24 Gray code for subsets DT-33 merge sorting DT-24 permutations in lex order DT-27 Towers of Hanoi DT-29 Reflexive relation Relation Fn-4 GT-5 see perhaps Binary relation Relative error CL-9 Restricted growth function and set partitions 9 Root DT-2 Rooted graph GT-27-6

7 Rooted tree child DT-2, GT-27 down degree of a vertex DT-2 height ofavertex DT-2 internal vertex DT-2, GT-27 leaf DT-2, GT-27 parent DT-2, GT-27 path to a vertex DT-2 siblings GT-27 RP-tree (rooted plane tree) see Decision tree Rule of Product Rule of Sum Sample space SAT problem CL-5 CL-3 CL-25, 9 Satisfiability problem Sequence Series geometric CL-2 DT-23 Set CL-2 and monotone function 6 Cartesian product CL-13 complement CL-13 complement of difference CL-13 intersection CL-13 intersection of two partition, see Set partition subsets of size k CL-14 symmetric diference of two symmetric difference CL-13 union CL-13 union of two with repetition (multiset) CL-2 Set partition CL-23, 3 ordered CL-18 recursion (Bell numbers) CL-25 recursion (Stirling numbers) CL-23 restricted growth function 9 Simple graph GT-2 Simplest (base) cases for induction Sorting (merge sort) DT-24, GT-43 DT-15 Space probability CL-26 Spanning tree GT-29 lineal (= depth first) GT-32 minimum weight GT-29 Stacks and recursion Standard deviation DT-31 Stirling numbers (set partitions) image of a function 4 Stirling's approximation for n! Strictly decreasing function or list Strictly increasing (or decreasing) function or list 6 Strictly increasing function or list String see List Subgraph GT-16 cycle GT-17 induced by edges or vertices Surjection Symmetric difference of sets Symmetric relation GT-5 Tchebycheff's inequality Fn-25 CL-23 CL GT-17-7

8 Theorem Bayes' DT-37, DT-41 binomial coefficients CL-14 binomial theorem CL-16 bipartite and cycle lengths GT-33 conditional probability DT-37 correlation bounds Fn-25 covariance when independent Fn-28 cycles and multiple paths GT-18 equivalence relations GT-5 expectation is linear expectation of a product Fn-28 induction DT-14 lists with repetition CL-3 lists without repetition CL-9 minimum weight spanning tree GT-30 monotone functions and (multi)sets 6 permutations of set to fixed power Fn-9 Prim's algorithm GT-30 properties of and O GT-37 Rule of Product CL-3 Rule of Sum CL-5 Stirling's formula CL-9 systematic tree traversal DT-8 Tchebycheff's inequality Fn-25 variance of sum Fn-28 walk, trail and path GT-16 Towers of Hanoi DT-28 four pole version DT-35 Tractable algorithm GT-43 Trail in a (directed) graph Transitive relation GT-5 Traveling salesman problem Traversal decision tree DT-7, GT-28 GT-15 GT-44 Tree see also specific topic binary GT-34 decision, see Decision tree height GT-34 ordered tree, see Decision tree rooted, see Rooted tree RP-tree (rooted plane tree), see Decision tree spanning GT-29 spanning, lineal (= depth first) GT-32 spanning, minimum weight GT-29 Two-line notation Fn-5 Uniformly at random CL-25 Union of sets Variance Venn diagram CL-28 Vertex DT-2 adajcent pair GT-3 child DT-2, GT-27 degree of DT-2, GT-4 down degree of DT-2 height of DT-2 internal DT-2, GT-27 isolated GT-10 leaf DT-2, GT-27 parent DT-2, GT-27 Vertex sequence GT-15 breadth first DT-7, GT-29 depth first DT-7, GT-28 Walk in a graph GT-15 Weakly decreasing function or list Weakly increasing function or list Words CL-11, CL