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Algorithm

Algorithm. An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time. Features of Algorithm.

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Algorithm

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  1. Algorithm • An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time. Design and Analysis of Algorithms - Chapter 1

  2. Features of Algorithm • “Besides merely being a finite set of rules that gives a sequence of operations for solving a specific type of problem, an algorithm has the five important features” [Knuth1] • finiteness (otherwise: computational method) • termination • definiteness • precise definition of each step • input (zero or more) • output (one or more) • effectiveness • “Its operations must all be sufficiently basic that they can in principle be done exactly and in a finite length of time by someone using pencil and paper” [Knuth1] Design and Analysis of Algorithms - Chapter 1

  3. Historical Perspective • Muhammad ibn Musa al-Khwarizmi – 9th century mathematician www.lib.virginia.edu/science/parshall/khwariz.html • Euclid’s algorithm for finding the greatest common divisor Design and Analysis of Algorithms - Chapter 1

  4. Greatest Common Divisor (GCD) Algorithm 1 • Step 1 Assign the value of min{m,n} to t • Step 2 Divide m by t. If the remainder of this division is 0, go to Step 3; otherwise, to Step 4 • Step 3 Divide n by t. If the remainder of this division is 0, return the value of t as the answer and stop; otherwise, proceed to Step 4 • Step 4 Decrease the value of t by 1. Go to Step 2 • Note: m and n are positive integers Design and Analysis of Algorithms - Chapter 1

  5. Correctness • The algorithm terminates, because t is decreased by 1 each time we go through step 4, 1 divides any integer, and t eventually will reach 1 unless the algorithm stops earlier. • The algorithm is partially correct, because when it returns t as the answer, t is the minimum value that divides both m and n Design and Analysis of Algorithms - Chapter 1

  6. GCD Procedure 2 • Step 1 Find the prime factors of m • Step 2 Find the prime factors of n • Step 3 Identify all the common factors in the two prime expansions found in Steps 1 and 2. If p is a common factor repeated i times in m and j times in n, assign to p the multiplicity min{i,j} • Step 4 Compute the product of all the common factors with their multiplicities and return it as the GCD of m and n • Note: as written, the algorithm requires than m and n be integers greater than 1, since 1 is not a prime Design and Analysis of Algorithms - Chapter 1

  7. Procedure 2 or Algorithm 2? • Procedure 2 is not an algorithm unless we can provide an effective way to find prime factors of a number • The sieve of Eratosthenes is an algorithm that provides such an effective procedure Design and Analysis of Algorithms - Chapter 1

  8. Euclid’s Algorithm • E0 [Ensure m geq n] If m lt n, exchange m with n • E1 [Find Remainder] Divide m by n and let r be the remainder (We will have 0 leq r lt n) • E2 [Is it zero?] If r=0, the algorithm terminates; n is the answer • E3 [Reduce] Set m to n, n to r, and go back to E1 Design and Analysis of Algorithms - Chapter 1

  9. Termination of Euclid’s Algorithm • The second number of the pair gets smaller with each iteration and cannot become negative: indeed, the new value of n is r = m mod n, which is always smaller than n. Eventually, r becomes zero, and the algorithms stops. Design and Analysis of Algorithms - Chapter 1

  10. Correctness of Euclid’s Algorithm • After step E1, we have m = qn + r, for some integer q. If r = 0, then m is a multiple of n, and clearly in such a case n is the GCD of m and n. If r !=0, note that any number that divides both m and n must divide m – qn = r, and any number that divides both n and r must divide qn + r = m; so the set of divisors of {m,n} is the same as the set of divisors of {n,r}. In particular, the GCD of {m,n} is the same as the GCD of {n,r}. Therefore, E3 does not change the answer to the original problem. [Knuth1] Design and Analysis of Algorithms - Chapter 1

  11. Book Variation (p.4) • Step 1 If n=0, return the value of m as the answer and stop; otherwise, proceed to Step 2. • Step 2 Divide m by n and assign the value of the remainder to r. • Step 3 Assign the value of n to m and the value of r to n. Go to Step 1. • Note: m and n are nonnegative, not-both-zero integers Design and Analysis of Algorithms - Chapter 1

  12. Pseudocode for book version while n <> 0 do r <- m mod n m <- n n <- r return m Design and Analysis of Algorithms - Chapter 1

  13. Algorithm Design Techniques • Euclid’s algorithm is an example of a “Decrease and Conquer” algorithm: it works by replacing its instance by a simpler one (in step E3) • It can be shown (exercise 5.6b) that the instance size, when measured by the second parameter, n, is always reduced by at least a factor of 2 after two successive iterations of Euclid’s algorithm • In Euclid’s original, the remainder m mod n is computed by repeated subtraction of n from m • Step E0 is not necessary for correctness; it improves time efficiency Design and Analysis of Algorithms - Chapter 1

  14. Notion of algorithm problem algorithm “computer” input output Algorithmic solution Design and Analysis of Algorithms - Chapter 1

  15. Example of computational problem: sorting • Statement of problem: • Input: A sequence of n numbers <a1, a2, …, an> • Output: A reordering of the input sequence <a´1, a´2, …, a´n> so that a´i≤ a´j whenever i < j • Instance: The sequence <5, 3, 2, 8, 3> • Algorithms: • Selection sort • Insertion sort • Merge sort • (many others) Design and Analysis of Algorithms - Chapter 1

  16. Selection Sort • Input: array a[1],…,a[n] • Output: array a sorted in non-decreasing order • Algorithm: • for i=1 to n swap a[i] with smallest of a[i+1],…a[n] • see also pseudocode, section 3.1 Design and Analysis of Algorithms - Chapter 1

  17. Sorting: Some Terms • Key • Stable sorting • In place sorting Design and Analysis of Algorithms - Chapter 1

  18. Graphs: Some Terms • Graph • Vertices, edges • Undirected, directed • Loops • Complete, dense, sparse • Paths: sequences of vertices or of edges? • Path length: number of edges in path • Walk: repeated vertices and edges allowed • Path: repeated vertices are allowed, repeated edges are not allowed • Simple path: neither repeated vertices nor repeated edges are allowed • Cycle: simple path that starts and ends at the same vertex Design and Analysis of Algorithms - Chapter 1

  19. Some Well-known Computational Problems • Sorting • Searching • Shortest paths in a graph • Minimum spanning tree • Primality testing • Traveling salesman problem • Knapsack problem • Chess • Towers of Hanoi • Program termination Design and Analysis of Algorithms - Chapter 1

  20. Basic Issues Related to Algorithms • How to design algorithms • How to express algorithms • Proving correctness • Efficiency • Theoretical analysis • Empirical analysis • Optimality Design and Analysis of Algorithms - Chapter 1

  21. Brute force Divide and conquer Decrease and conquer Transform and conquer Algorithm design strategies See http://www.aw-bc.com/info/levitin/taxonomy.html This is not a unique taxonomy. • Greedy approach • Dynamic programming • Backtracking and Branch and bound • Space and time tradeoffs Design and Analysis of Algorithms - Chapter 1

  22. Analysis of Algorithms • How good is the algorithm? • Correctness • Time efficiency • Space efficiency • Does there exist a better algorithm? • Lower bounds • Optimality Design and Analysis of Algorithms - Chapter 1

  23. What is an algorithm? • Recipe, process, method, technique, procedure, routine,… with the following requirements: • Finiteness • terminates after a finite number of steps • Definiteness • rigorously and unambiguously specified • Input • valid inputs are clearly specified • Output • can be proved to produce the correct output given a valid input • Effectiveness • steps are sufficiently simple and basic Design and Analysis of Algorithms - Chapter 1

  24. Why study algorithms? • Theoretical importance • the core of computer science • Practical importance • A practitioner’s toolkit of known algorithms • Framework for designing and analyzing algorithms for new problems Design and Analysis of Algorithms - Chapter 1

  25. Correctness • Termination • Well-founded sets: find a quantity that is never negative and that always decreases as the algorithm is executed • Partial Correctness • For recursive algorithms: induction • For iterative algorithms: axiomatic semantics, loop invariants Design and Analysis of Algorithms - Chapter 1

  26. Proving Correctness • Make sure that the algorithm will terminate. • Define a precondition (allowable input and state) and a postcondition (correct output). • Show that, given the precondition, algorithm will fulfill the postcondition. • In many cases, the algorithm is either recursive or iterative. Determine which one applies. Design and Analysis of Algorithms - Chapter 1

  27. Recursive Algorithms Use induction and show correct fulfillment of postcondition in: • basis case (input is of size 0) • case where input is of size n+1, assuming (inductive assumption) that postcondition is fulfilled when input is of size n (simple induction) or input is of all sizes from 0 to n (strong or complete induction) Design and Analysis of Algorithms - Chapter 1

  28. Iterative Algorithms • Determine an appropriate invariant. • Show that the invariant is true before entering the loop for the first time. • Show that, if the invariant is true and the loop condition is false, the postcondition is fulfilled. • Show, by induction, that, if the invariant is true for input of size n and the loop condition is true, the invariant is also true for input of size n+1. Design and Analysis of Algorithms - Chapter 1

  29. Example: Recursive Factorial (* Pre: i GEQ 0 *) fun fact(i) = if i = 0 then 1 else i * fact(i-1); (* Post: fact(i) = i! *) Design and Analysis of Algorithms - Chapter 1

  30. Example: Iterative Factorial (* Pre’: i>= 0*) fun fact(i) int j=1, f=1; (* Pre: i>=0 and j=f=1 *) while j <= i f = f * j; j := j+1; (* Post: f = i! *) return f; (* Post’: fact(i) = i! *) Design and Analysis of Algorithms - Chapter 1

  31. Example: Product Algorithm • To be added Design and Analysis of Algorithms - Chapter 1

  32. Example: Recursive BinarySearch • To be added Design and Analysis of Algorithms - Chapter 1

  33. Example: Iterative Binary Search • To be added Design and Analysis of Algorithms - Chapter 1

  34. Example: Recursively Computing Binomial Coefficients • Multiple calls • To be added Design and Analysis of Algorithms - Chapter 1

  35. Example: RecursiveSelectionSort (* Author: Kingston *)(* Pre: a LTE b + 1 *)(* SelectionSort(A, a, b) *)(* Post: A[a] LTE A[a+1] LTE A[a+2] LTE ... A[b] *)(* and A is a permutation of the input array *)procedure SelectionSort (var A: AType; a,b: integer);var i: integer;begin if a = b + 1 then (* do nothing *) else i := MinIndex(A, a, b); if (i <> a) then Swap(A[i],A[a]); end; SelectionSort(A, a + 1, b); end;end SelectionSort;(* MinIndex returns the index of a minimum element of the non-empty arrayA[i..j], and Swap(A[i],A[j] swaps A[i] and A[j] *) Design and Analysis of Algorithms - Chapter 1

  36. Example: Max Contiguous Subvector • Algorithm:Maximum-Sum Contiguous Subvector • /* Precondition: x[1..n] is an array of floating-point (real) numbers. */ • procedureMaxSumContigVector (x[1..n]: float)// Returns the maximum sum of a continuous subarray drawn from x.MaxSoFar := 0.0MaxEndingHere := 0i := 1while i ≠ n + 1 doMaxEndingHere := max(0.0, MaxEndingHere + x[i])MaxSoFar := max(MaxSoFar, MaxEndingHere)i := i + 1 • endreturn MaxSoFar • /* Postcondition: MaxSoFar is the maximum sum of any contiguous subvector in x[1..n]. */ Design and Analysis of Algorithms - Chapter 1

  37. Complexity • Space complexity • Time complexity • For iterative algorithms: sums • For recursive algorithms: recurrence relations Design and Analysis of Algorithms - Chapter 1

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