Design And Analysis Of Algorithms Introduction – Programming Assignments First and second exams (20% each) Closed books, closed notes Final exam (40%) Late homework 12% penalty for each day late, up to 2 days Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest and Clifford Stein
Improve algorithm efficiency: work faster, process more data, do something otherwise impossible Solve significantly large-scale problems Technology only improves everything by a constant factor Compare algorithms Algorithms as a field of work Algorithms Learn about a set of standard algorithms Do New discoveries appear Multiple application areas Learn algorithm design and analysis techniques Introduction
Design And Analysis Of Algorithms Introduction
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7 Analysis of algorithms Predict the amount of required resources: memory: how much space is needed? computation time: how fast does the algorithm run? FACT: runtime grows with input size. Size of input (number of elements in input) Size of array, degree of polynomial, number of elements in matrix, # vertices and edges of bits in binary representation of input in graph Def: Execution time = number of primitive operations (steps) performed before stop Arithmetic operations (+, -, *), data movement, control, decision making (if, then), comparison Introduction
For example: Sort n numbers Friend’s computer = 109 instructions / second Friend’s algorithm = 2n2 instructions Your computer = 107 instructions / second Your algorithm = 50nlgn instructions Your friend = You = Sort 106 numbers! 2000/100 = 20 times better! Introduction
Alg.: MIN (a, …, a[n]) m ← a; for i ← 2 to n if a[i] < m then m ← a[i]; Execution time: number of elementary operations (steps) performed before termination T(n) =1 [first step] + (n) [for loop] + (n-1) [conditions] + (n-1) [ this then becomes defined] = 3n – 1 Order of increase (rate): The leading term of the formula expresses the asymptotic behavior of the algorithm.
1 (constant running time): Instructions are executed one or more times logN (logarithmic) A large problem is solved by dividing the original problem into smaller ones, by a constant fraction at each step N (linear) A small amount of processing. performed on each input element N logN The problem is solved by dividing it into smaller problems, solving them independently and combining the solution.
Pdf) Design And Analysis Of Physical Design Algorithms*
N2 (quadratic) Typical for algorithms processing all even data elements (double nested loops) N3 (cube) Processing ternary data (triply nested loops) NK (polynomial) 2N (exponential) Several exponential algorithms are suitable for practical use Introduction
16 Asymptotic notation A way to describe the behavior of functions in the limit Abstracts low-order terms and constant factors. asymptotic “small and equal”: f(n) “≤” g(n) asymptotic “greater and equal”:f(n) “≥” g(n) asymptotic “equality”: f(n) “=” g( n) Introduction
N2/2 – n/2 (6n3 + 1)lgn/(n + 1) n vs n2 notation n3 vs n2 n vs logn = (n2) = (n2lgn) n ≠ (n2) O notation 2n2 vs . n3 n2 against. n2 n3 against. nlogn n3 = (n2) 2n2 = O(n3) n = (logn) n2 = O(n2) n (n2) n3 O(nlgn) Introduction
18 Recursive Algorithms Binary search: for sorted array A finds whether x is in array A[lo…hi] Alg.:BINARY-SEARCH (A, lo, hi, x) if (lo > hi) returns FALSE mid (lo +hi)/2 if x = A[mid] return TRUE if ( x A[mid] ) BINARY-SEARCH (A, avg+1, hi, x) 12 11 10 9 7 5 3 2 1 4 6 8 avg lo hi Log in
Advanced Algorithms And Data Structures
19 Definition of iterations: Iteration = an equation or inequality that describes a function in terms of its value over smaller inputs and one or more base cases, for example: T(n) = T(n-1) + n Useful for analyzing of iterative algorithms Solve Iteration Methods Iteration Method Substitution Method Recursion Tree Method Master Method Introduction
2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A Divide and conquer Merge sort Quick sort Incomparable methods Enum sort Radix sort Code sort Intro
Provides an upper bound on runtime An absolute guarantee that the algorithm will not run longer, regardless of the inputs. Best case An input is the fastest the algorithm runs Average case Predicts the runtime. random (e.g. cards ordered in reverse order) (e.g
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Levitin, A. (2014). An introduction to the design and analysis of algorithms (3rd ed.). Pearson. Retrieved from https:///book/811582/introduction-to-the-design-and-analysis-of-algorithms-international-edition-pdf (original work published 2014)
What Is An Algorithm? Characteristics, Types And How To Write It
Levitin, A. (2014) Introduction to algorithm design and analysis. 3rd edition. Pearson. Available at: https:///book/811582/introduction-to-the-design-and-analysis-of-algorithms-international-edition-pdf (Accessed: 14 October 2022).
Levitin, Anany. An introduction to the design and analysis of algorithms. 3rd edition. Pearson, 2014. Web. 14 October 2022.
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