We summarize the performance characteristics of classic algorithms anddata structures for sorting, priority queues, symbol tables, and graph processing. Amadeus pro 2 4 2 build 1968 download free.
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We also summarize some of the mathematics useful in the analysis of algorithms, including commonly encountered functions;useful formulas and appoximations; properties of logarithms;asymptotic notations; and solutions to divide-and-conquer recurrences.
Sorting.
The table below summarizes the number of compares for a variety of sortingalgorithms, as implemented in this textbook.It includes leading constants but ignores lower-order terms.ALGORITHM | CODE | STABLE | BEST | AVERAGE | WORST | REMARKS | |
---|---|---|---|---|---|---|---|
selection sort | Selection.java | ✔ | ½ n 2 | ½ n 2 | ½ n 2 | n exchanges; quadratic in best case | |
insertion sort | Insertion.java | ✔ | ✔ | n | ¼ n 2 | ½ n 2 | use for small or partially-sorted arrays |
bubble sort | Bubble.java | ✔ | ✔ | n | ½ n 2 | ½ n 2 | rarely useful; use insertion sort instead |
shellsort | Shell.java | ✔ | n log3n | unknown | c n 3/2 | tight code; subquadratic | |
mergesort | Merge.java | ✔ | ½ n lg n | n lg n | n lg n | n log n guarantee; stable | |
quicksort | Quick.java | ✔ | n lg n | 2 n ln n | ½ n 2 | n log n probabilistic guarantee; fastest in practice | |
heapsort | Heap.java | ✔ | n† | 2 n lg n | 2 n lg n | n log n guarantee; in place | |
†n lg n if all keys are distinct |
Priority queues.
The table below summarizes the order of growth of the running time ofoperations for a variety of priority queues, as implemented in this textbook.It ignores leading constants and lower-order terms.Except as noted, all running times are worst-case running times.DATA STRUCTURE | CODE | INSERT | MIN | DELETE | MERGE | ||
---|---|---|---|---|---|---|---|
array | BruteIndexMinPQ.java | 1 | n | n | 1 | 1 | n |
binary heap | IndexMinPQ.java | log n | log n | 1 | log n | log n | n |
d-way heap | IndexMultiwayMinPQ.java | logdn | d logdn | 1 | logdn | d logdn | n |
binomial heap | IndexBinomialMinPQ.java | 1 | log n | 1 | log n | log n | log n |
Fibonacci heap | IndexFibonacciMinPQ.java | 1 | log n† | 1 | 1 † | log n† | 1 |
† amortized guarantee |
Symbol tables.
The table below summarizes the order of growth of the running time ofoperations for a variety of symbol tables, as implemented in this textbook.It ignores leading constants and lower-order terms.worst case | average case | ||||||
---|---|---|---|---|---|---|---|
DATA STRUCTURE | CODE | SEARCH | INSERT | DELETE | SEARCH | INSERT | DELETE |
sequential search (in an unordered list) | SequentialSearchST.java | n | n | n | n | n | n |
binary search (in a sorted array) | BinarySearchST.java | log n | n | n | log n | n | n |
binary search tree (unbalanced) | BST.java | n | n | n | log n | log n | sqrt(n) |
red-black BST (left-leaning) | RedBlackBST.java | log n | log n | log n | log n | log n | log n |
AVL | AVLTreeST.java | log n | log n | log n | log n | log n | log n |
hash table (separate-chaining) | SeparateChainingHashST.java | n | n | n | 1 † | 1 † | 1 † |
hash table (linear-probing) | LinearProbingHashST.java | n | n | n | 1 † | 1 † | 1 † |
† uniform hashing assumption |
Graph processing.
The table below summarizes the order of growth of the worst-case running time and memory usage (beyond the memory for the graph itself)for a variety of graph-processing problems, as implemented in this textbook.It ignores leading constants and lower-order terms.All running times are worst-case running times.PROBLEM | ALGORITHM | CODE | TIME | SPACE |
---|---|---|---|---|
path | DFS | DepthFirstPaths.java | E + V | V |
shortest path (fewest edges) | BFS | BreadthFirstPaths.java | E + V | V |
cycle | DFS | Cycle.java | E + V | V |
directed path | DFS | DepthFirstDirectedPaths.java | E + V | V |
shortest directed path (fewest edges) | BFS | BreadthFirstDirectedPaths.java | E + V | V |
directed cycle | DFS | DirectedCycle.java | E + V | V |
topological sort | DFS | Topological.java | E + V | V |
bipartiteness / odd cycle | DFS | Bipartite.java | E + V | V |
connected components | DFS | CC.java | E + V | V |
strong components | Kosaraju–Sharir | KosarajuSharirSCC.java | E + V | V |
strong components | Tarjan | TarjanSCC.java | E + V | V |
strong components | Gabow | GabowSCC.java | E + V | V |
Eulerian cycle | DFS | EulerianCycle.java | E + V | E + V |
directed Eulerian cycle | DFS | DirectedEulerianCycle.java | E + V | V |
transitive closure | DFS | TransitiveClosure.java | V (E + V) | V 2 |
minimum spanning tree | Kruskal | KruskalMST.java | E log E | E + V |
minimum spanning tree | Prim | PrimMST.java | E log V | V |
minimum spanning tree | Boruvka | BoruvkaMST.java | E log V | V |
shortest paths (nonnegative weights) | Dijkstra | DijkstraSP.java | E log V | V |
shortest paths (no negative cycles) | Bellman–Ford | BellmanFordSP.java | V (V + E) | V |
shortest paths (no cycles) | topological sort | AcyclicSP.java | V + E | V |
all-pairs shortest paths | Floyd–Warshall | FloydWarshall.java | V 3 | V 2 |
maxflow–mincut | Ford–Fulkerson | FordFulkerson.java | EV (E + V) | V |
bipartite matching | Hopcroft–Karp | HopcroftKarp.java | V ½ (E + V) | V |
assignment problem | successive shortest paths | AssignmentProblem.java | n 3 log n | n 2 |
Commonly encountered functions.
Here are some functions that are commonly encounteredwhen analyzing algorithms.FUNCTION | NOTATION | DEFINITION |
---|---|---|
floor | ( lfloor x rfloor ) | greatest integer (; le ; x) |
ceiling | ( lceil x rceil ) | smallest integer (; ge ; x) |
binary logarithm | ( lg x) or (log_2 x) | (y) such that (2^{,y} = x) |
natural logarithm | ( ln x) or (log_e x ) | (y) such that (e^{,y} = x) |
common logarithm | ( log_{10} x ) | (y) such that (10^{,y} = x) |
iterated binary logarithm | ( lg^* x ) | (0) if (x le 1;; 1 + lg^*(lg x)) otherwise |
harmonic number | ( H_n ) | (1 + 1/2 + 1/3 + ldots + 1/n) |
factorial | ( n! ) | (1 times 2 times 3 times ldots times n) |
binomial coefficient | ( n choose k ) | ( frac{n!}{k! ; (n-k)!}) |
Useful formulas and approximations.
Here are some useful formulas for approximations that are widely used in the analysis of algorithms.- Harmonic sum: (1 + 1/2 + 1/3 + ldots + 1/n sim ln n)
- Triangular sum: (1 + 2 + 3 + ldots + n = n , (n+1) , / , 2 sim n^2 ,/, 2)
- Sum of squares: (1^2 + 2^2 + 3^2 + ldots + n^2 sim n^3 , / , 3)
- Geometric sum: If (r neq 1), then(1 + r + r^2 + r^3 + ldots + r^n = (r^{n+1} - 1) ; /; (r - 1))
- (r = 1/2): (1 + 1/2 + 1/4 + 1/8 + ldots + 1/2^n sim 2)
- (r = 2): (1 + 2 + 4 + 8 + ldots + n/2 + n = 2n - 1 sim 2n), when (n) is a power of 2
- Stirling's approximation: (lg (n!) = lg 1 + lg 2 + lg 3 + ldots + lg n sim n lg n)
- Exponential: ((1 + 1/n)^n sim e; ;;(1 - 1/n)^n sim 1 / e)
- Binomial coefficients: ({n choose k} sim n^k , / , k!) when (k) is a small constant
- Approximate sum by integral: If (f(x)) is a monotonically increasing function, then( displaystyle int_0^n f(x) ; dx ; le ; sum_{i=1}^n ; f(i) ; le ; int_1^{n+1} f(x) ; dx)
Properties of logarithms.
- Definition: (log_b a = c) means (b^c = a).We refer to (b) as the base of the logarithm.
- Special cases: (log_b b = 1,; log_b 1 = 0 )
- Inverse of exponential: (b^{log_b x} = x)
- Product: (log_b (x times y) = log_b x + log_b y )
- Division: (log_b (x div y) = log_b x - log_b y )
- Finite product: (log_b ( x_1 times x_2 times ldots times x_n) ; = ; log_b x_1 + log_b x_2 + ldots + log_b x_n)
- Changing bases: (log_b x = log_c x ; / ; log_c b )
- Rearranging exponents: (x^{log_b y} = y^{log_b x})
- Exponentiation: (log_b (x^y) = y log_b x )
Aymptotic notations: definitions.
NAME | NOTATION | DESCRIPTION | DEFINITION |
---|---|---|---|
Tilde | (f(n) sim g(n); ) | (f(n)) is equal to (g(n)) asymptotically (including constant factors) | ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 1) |
Big Oh | (f(n)) is (O(g(n))) | (f(n)) is bounded above by (g(n)) asymptotically (ignoring constant factors) | there exist constants (c > 0) and (n_0 ge 0) such that (0 le f(n) le c cdot g(n)) forall (n ge n_0) |
Big Omega | (f(n)) is (Omega(g(n))) | (f(n)) is bounded below by (g(n)) asymptotically (ignoring constant factors) | ( g(n) ) is (O(f(n))) |
Big Theta | (f(n)) is (Theta(g(n))) | (f(n)) is bounded above and below by (g(n)) asymptotically (ignoring constant factors) | ( f(n) ) is both (O(g(n))) and (Omega(g(n))) |
Little oh | (f(n)) is (o(g(n))) | (f(n)) is dominated by (g(n)) asymptotically (ignoring constant factors) | ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 0) |
Little omega | (f(n)) is (omega(g(n))) | (f(n)) dominates (g(n)) asymptotically (ignoring constant factors) | ( g(n) ) is (o(f(n))) |
Common orders of growth.
NAME | NOTATION | EXAMPLE | CODE FRAGMENT |
---|---|---|---|
Constant | (O(1)) | array access arithmetic operation function call | |
Logarithmic | (O(log n)) | binary search in a sorted array insert in a binary heap search in a red–black tree | |
Linear | (O(n)) | sequential search grade-school addition BFPRT median finding | |
Linearithmic | (O(n log n)) | mergesort heapsort fast Fourier transform | |
Quadratic | (O(n^2)) | enumerate all pairs insertion sort grade-school multiplication | |
Cubic | (O(n^3)) | enumerate all triples Floyd–Warshall grade-school matrix multiplication | |
Polynomial | (O(n^c)) | ellipsoid algorithm for LP AKS primality algorithm Edmond's matching algorithm | |
Exponential | (2^{O(n^c)}) | enumerating all subsets enumerating all permutations backtracing search |
Asymptotic notations: properties.
- Reflexivity: (f(n)) is (O(f(n))).
- Constants: If (f(n)) is (O(g(n))) and ( c > 0 ),then (c cdot f(n)) is (O(g(n)))).
- Products: If (f_1(n)) is (O(g_1(n))) and ( f_2(n) ) is (O(g_2(n)))),then (f_1(n) cdot f_2(n)) is (O(g_1(n) cdot g_2(n)))).
- Sums: If (f_1(n)) is (O(g_1(n))) and ( f_2(n) ) is (O(g_2(n)))),then (f_1(n) + f_2(n)) is (O(max { g_1(n) , g_2(n) })).
- Transitivity: If (f(n)) is (O(g(n))) and ( g(n) ) is (O(h(n))),then ( f(n) ) is (O(h(n))).
- Polynomials: Let (f(n) = a_0 + a_1 n + ldots + a_d n^d) with(a_d > 0). Then, ( f(n) ) is (Theta(n^d)).
- Logarithms and polynomials: ( log_b n ) is (O(n^d)) for every ( b > 0) and every ( d > 0 ).
- Exponentials and polynomials: ( n^d ) is (O(r^n)) for every ( r > 0) and every ( d > 0 ).
- Factorials: ( n! ) is ( 2^{Theta(n log n)} ).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = c)for some constant ( 0 < c < infty), then(f(n)) is (Theta(g(n))).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 0),then (f(n)) is (O(g(n))) but not (Theta(g(n))).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = infty),then (f(n)) is (Omega(g(n))) but not (O(g(n))).
Here are some examples.
FUNCTION | (o(n^2)) | (O(n^2)) | (Theta(n^2)) | (Omega(n^2)) | (omega(n^2)) | (sim 2 n^2) | (sim 4 n^2) |
---|---|---|---|---|---|---|---|
(log_2 n) | ✔ | ✔ | |||||
(10n + 45) | ✔ | ✔ | |||||
(2n^2 + 45n + 12) | ✔ | ✔ | ✔ | ✔ | |||
(4n^2 - 2 sqrt{n}) | ✔ | ✔ | ✔ | ✔ | |||
(3n^3) | ✔ | ✔ | |||||
(2^n) | ✔ | ✔ |
Divide-and-conquer recurrences.
For each of the following recurrences we assume (T(1) = 0)and that (n,/,2) means either (lfloor n,/,2 rfloor) or(lceil n,/,2 rceil).RECURRENCE | (T(n)) | EXAMPLE |
---|---|---|
(T(n) = T(n,/,2) + 1) | (sim lg n) | binary search |
(T(n) = 2 T(n,/,2) + n) | (sim n lg n) | mergesort |
(T(n) = T(n-1) + n) | (sim frac{1}{2} n^2) | insertion sort |
(T(n) = 2 T(n,/,2) + 1) | (sim n) | tree traversal |
(T(n) = 2 T(n-1) + 1) | (sim 2^n) | towers of Hanoi |
(T(n) = 3 T(n,/,2) + Theta(n)) | (Theta(n^{log_2 3}) = Theta(n^{1.58..})) | Karatsuba multiplication |
(T(n) = 7 T(n,/,2) + Theta(n^2)) | (Theta(n^{log_2 7}) = Theta(n^{2.81..})) | Strassen multiplication |
(T(n) = 2 T(n,/,2) + Theta(n log n)) | (Theta(n log^2 n)) | closest pair |
Master theorem.
Let (a ge 1), (b ge 2), and (c > 0) and suppose that(T(n)) is a function on the non-negative integers that satisfiesthe divide-and-conquer recurrence$$T(n) = a ; T(n,/,b) + Theta(n^c)$$with (T(0) = 0) and (T(1) = Theta(1)), where (n,/,b) meanseither (lfloor n,/,b rfloor) or either (lceil n,/,b rceil).- If (c < log_b a), then (T(n) = Theta(n^{log_{,b} a}))
- If (c = log_b a), then (T(n) = Theta(n^c log n))
- If (c > log_b a), then (T(n) = Theta(n^c))
Last modified on September 12, 2020.
Copyright © 2000–2019Robert SedgewickandKevin Wayne.All rights reserved.
Copyright © 2000–2019Robert SedgewickandKevin Wayne.All rights reserved.
Learning some algebraic rules for various exponents, radicals, laws, binomials, formulas, and equations will help you successfully study and solve problems in an Algebra II course. You should also be able to recognize formulas to find slope, slope-intercept, distance, and midpoint (which are formulas from geometry) to help you through Algebra II.
Algebra: Rules of Exponents
Exponents are shorthand for repeated multiplication. The rules for performing operations involving exponents allow you to change multiplication and division expressions with the same base into something simpler to work with. Remember that in xa, the x is the base and the a is the exponent.
Assume x ≠ 0:
Linear Equations: How to Find Slope, y-Intercept, Distance, Midpoint
Cheatsheet 1 3 2 4 Fraction
In algebra, linear equations means you’re dealing with straight lines. When you’re working with the xy-coordinate system, you can use the following formulas to find the slope, y-intercept, distance, and midpoint between two points.
Consider the two points (x1, y1) and (x2, y2):
Slope of the line through the points:
Slope-intercept form of the line with y-intercept b:
![Cheatsheet 1 3 2 4 50 adalah Cheatsheet 1 3 2 4 50 adalah](https://photos.donedeal.ie/ddimg/YjEzODQ2MDU3MWRhYzlmMDhmZjg0MGI1NGE2NTViZGKYCUAQNPACDlqtQktUq3ylaHR0cDovL3MzLWV1LXdlc3QtMS5hbWF6b25hd3MuY29tL2RvbmVkZWFsLmllLXBob3Rvcy9waG90b18xNTk2OTEwOTV8fHx8fHwzMDB4MjI1fHx8fHw=.jpeg)
Point-slope form of the line with slope m:
Distance formula:
Midpoint formula:
Rewrite Absolute Value Equations as Linear Equations
To work with an absolute value equation in algebra, you first need to rewrite it as a linear equation. The same goes for an absolute value inequality, which you rewrite as a linear inequality.
When rewriting absolute value equations or inequalities, you drop the absolute value bars.
|ax + b| = c→ax + b = c or ax + b = –c
|ax + b| > c →ax + b > c or ax + b > –c
|ax + b| < c → –c < ax + b < c
9 Number Systems in Algebra to Know
Cheatsheet 1 3 2 4 50 Adalah
A number system in algebra is a set of numbers — and different number systems are used to solve different types of algebra problems. Mac os x 10.7 0 free. Number systems include real numbers, natural numbers, whole numbers, integers, rational numbers, irrational numbers, even numbers, and odd numbers.
- Real numbers: Real numbers comprise the full spectrum of numbers. They cover the gamut and can take on any form — fractions or whole numbers, decimal points or no decimal points. The full range of real numbers includes decimals that can go on forever. Real numbers are different from imaginary or complex numbers.
- Natural numbers: A natural number is a number that comes naturally. What numbers did you first use? Remember someone asking, “How old are you?” You proudly held up four fingers and said, “Four!” Natural numbers are greater than zero but don’t include fractions: 1, 2, 3, 4, 5, 6, 7, and so on, into infinity. You use natural numbers to count items and to make lists.
- Whole numbers: Whole numbers are just all the natural numbers plus a zero: 0, 1, 2, 3, 4, 5, and so on, into infinity. They act like natural numbers and are used when whole amounts (no fractions) are required. Zero can also indicate none.Algebraic problems often require you to round the answer to the nearest whole number. This makes perfect sense when the problem involves people, cars, animals, houses, or anything that shouldn’t be cut into pieces.
- Integers: Integers incorporate all the whole numbers and their opposites (or additive inverses of the whole numbers). Integers can be described as being positive and negative whole numbers and 0: . . . –3, –2, –1, 0, 1, 2, 3, . . . .Integers are popular in algebra. When you solve a long, complicated problem and come up with an integer, you can be joyous because your answer is probably right. After all, it’s not a fraction! This doesn’t mean that answers in algebra can’t be fractions or decimals. It’s just that most textbooks and reference books try to stick with nice answers to increase the comfort level and avoid confusion.
- Rational numbers: Rational numbers are numbers that act rationally! In this case, acting rationally means that the decimal equivalent of the rational number behaves. The decimal ends somewhere, or it has a repeating pattern to it. That’s what constitutes “behaving.” Some examples of rational numbers with decimals that terminate include 2, 3.4, 5.77623, and –4.5.Some examples of rational numbers with decimals that repeat the same pattern include the following:(The horizontal bar over the 164 and the 6 lets you know that these numbers repeat forever.) In all cases, rational numbers can be written as a fraction. They all have a fraction that they are equal to.
- Irrational numbers: Irrational numbers are real numbers that are not rational numbers. An irrational number cannot be written as a fraction, and decimal values for irrationals never end and never have a nice pattern to them. For example, pi, with its never-ending decimal places, is irrational.
- Imaginary/complex numbers: A number that isn’t real can be imaginary or complex. An imaginary number contains some multiple of i, which is the following:For example, 2 + 3i is a complex number.
- Even numbers: An even number is one that divides evenly by 2, such as 2, 4, 18, and 352.
- Odd numbers: An odd number is one that does not divide evenly by 2, such as 1, 3, 27, and 485.
Algebra II: What Is the Binomial Theorem?
Cheatsheet 1 3 2 4
A binomial is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers to complete computations. The binomial theorem says that if a and b are real numbers and n is a positive integer, then
You can see the rule here, in the second line, in terms of the coefficients that are created using combinations. The powers on a start with n and decrease until the power is zero in the last term. That’s why you don’t see an a in the last term — it’s a0, which is really a 1. The powers on b increase from b0 until the last term, where it’s bn. Notice that the power of b matches k in the combination.
Use the Properties of Proportions to Simplify Fractions
In algebra, the properties of proportions come in handy when solving equations involving fractions. When you can, change an algebraic equation with fractions in it to a proportion for easy solving.
If
then the following are all true:
A proportion is an equation involving two ratios (fractions) set equal to each other. The following equation is a proportion:
Both fractions in that proportion reduce to Ltf reeder 1 2 – google rss reader.
so it’s fairly easy to see how this statement is true.
Proportions have some interesting, helpful, and easy-to-use properties. For example, in the following proportion,
the cross-products are equal: a∙d = b∙c.
Cheatsheet 1 3 2 4 Blackjack Betting System
The reciprocals are equal (you can flip the fractions):
You can reduce the fractions vertically or horizontally: You can divide out factors that are common to both numerators or both denominators or the left fraction or the right fraction. (You can’t, however, divide out a factor from the numerator of one fraction and the denominator of the other.)
Algebra II: Raise Binomials to a Power
A binomial is a mathematical expression that has two terms. In algebra, people frequently raise binomials to powers in order to solve equations. Here are some examples:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5