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2IL65 Algorithms

2IL65 Algorithms. Fall 2013 Lecture 4: Sorting in linear time. Building a heap. One more time …. Building a heap. Build-Max-Heap (A) heap-size = A.length for i = A.length downto 1 do Max-Heapify(A,i). 14. 3. 8. 11. Building a heap. Build-Max-Heap2 (A) heap-size[A] = 1

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2IL65 Algorithms

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  1. 2IL65 Algorithms Fall 2013Lecture 4: Sorting in linear time

  2. Building a heap One more time …

  3. Building a heap Build-Max-Heap(A) • heap-size = A.length • for i = A.length downto 1 • do Max-Heapify(A,i)

  4. 14 3 8 11 Building a heap Build-Max-Heap2(A) • heap-size[A] = 1 • for i = 2 to A.length doMax-Heap-Insert(A, A[i]) Max-Heap-Insert(A, key) • heap-size[A] = heap-size[A] + 1 • A[heap-size[A] ] = - infinity • Increase-Key(A, heap-size[A], key) Lower bound worst case running time? 14 3 8 11 2 24 35 28 16 5 20 21 A[i] moves up until it reaches the correct position

  5. Building a heap Build-Max-Heap2(A) • heap-size[A] = 1 • for i = 2 to A.length doMax-Heap-Insert(A, A[i]) Running time: Θ(1) + ∑2≤i ≤ n (time for Max-Heap-Insert(A, A[i])) • Max-Heap-Insert (A, A[i] ) takes O(log i) = O(log n) time ➨ worst case running time is O(n log n) • If A is sorted in increasing order, then A[i] is always the largest element when Max-Heap-Insert(A, A[i])) is called and must move all the way up the tree ➨ Max-Heap-Insert (A, A[i]) takes Ω(log i) time. Worst case running time: Θ(1) + ∑2≤i ≤ nΩ(log i) = Ω(1 + ∑2≤i ≤ n log i) = Ω (n log n) since ∑2≤i ≤ n log i ≥ ∑n/2≤i ≤ n log (n/2) = n/2 log (n/2)

  6. log2 n = Θ(log n2) ? √n = Ω(log4 n) ? 2lg n = Ω(n2) ? 2n = Ω(n2) ? log(√n) = Θ(log n) ? no yes no yes yes Quiz

  7. Sorting in linear time

  8. The sorting problem Input: a sequence of n numbers ‹a1, a2, …, an› Output: a permutation of the input such that ‹ai1≤ … ≤ain› Why do we care so much about sorting? • sorting is used by many applications • (first) step of many algorithms • many techniques can be illustrated by studying sorting

  9. Can we sort faster than Θ(n log n) ?? Worst case running time of sorting algorithms: InsertionSort: Θ(n2) MergeSort: Θ(n log n) HeapSort: Θ(n log n) Can we do this faster? Θ(n loglog n) ? Θ(n) ?

  10. Upper and lower bounds Upper bound How do you show that a problem (for example sorting) can be solved in Θ(f(n)) time? ➨ give an algorithm that solves the problem in Θ(f(n)) time. Lower bound How do you show that a problem (for example sorting) can not be solved faster than in Θ(f(n)) time? ➨ prove that every possiblealgorithm that solves the problem needs Ω(f(n)) time.

  11. Lower bounds Lower bound How do you show that a problem (for example sorting) can not be solved faster than in Θ(f(n)) time? ➨ prove that every possible algorithm that solves the problem needs Ω(f(n)) time. Model of computation: which operations is the algorithm allowed to use? Bit-manipulations?Random-access (array indexing) vs. pointer-machines?

  12. Comparison-based sorting InsertionSort(A) • for j = 2 to A.length • do begin 6. 7. end Which steps precisely the algorithm executes — and hence, which element ends up where — only depends on the result of comparisons between the input elements. key = A[ j ] ; i = j -1 while i > 0 and A[ i ] > key do begin A[ i+1] = A[ i ]; i = i -1 end A[ i +1] = key

  13. or ≤, =, >, ≥ Decision tree for comparison-based sorting exchange of elements, assignments, etc. … A[.] < A[.] A[.] < A[.] A[.] < A[.] A[.] < A[.] A[.] < A[.] A[.] < A[.] A[.] < A[.]

  14. Decision tree for comparison-based sorting (InsertionSort for 3 elements)

  15. Comparison-based sorting • every permutation of the input follows a different path in the decision tree ➨ the decision tree has at least n! leaves • the height of a binary tree with n! leaves is at least log(n!) • worst case running time ≥ longest path from root to leaf ≥ log(n!) = Ω(n log n)

  16. Lower bound for comparison-based sorting TheoremAny comparison-based sorting algorithm requires Ω(n log n) comparisons in the worst case. ➨ The worst case running time of MergeSort and HeapSort is optimal.

  17. Sorting in linear time … Three algorithms which are faster: • CountingSort • RadixSort • BucketSort (not comparison-based, make assumptions on the input)

  18. CountingSort Input: array A[1..n] of numbers Assumption: the input elements are integers in the range0 to k, for some k Main idea:count for every A[i] the number of elements less than A[i]➨ position of A[i] in the output array Beware of elements that have the same value! position(i) = number of elements less than A[i] in A[1..n] + number of elements equal to A[i] in A[1..i]

  19. CountingSort position(i) = number of elements less than A[i] in A[1..n] + number of elements equal to A[i] in A[1..i] 5 3 10 5 4 5 7 7 9 3 10 8 5 3 3 8 3 3 3 3 4 5 5 5 5 7 7 8 8 9 10 10 numbers < 5 third 5 from left position: (# less than 5) + 3

  20. CountingSort position(i) = number of elements less than A[i] in A[1..n] + number of elements equal to A[i] in A[1..i] LemmaIf every element A[i] is placed on position(i), then the array is sorted and the sorted order is stable. Numbers with the same value appear in the same order in the output array as they do in the input array.

  21. CountingSort C[i] will contain the number of elements ≤ i CountingSort(A,k) //Input: array A[1..n] of integers in the range 0..k //Output: array B[1..n] which contains the elements of A, sorted • for i = 0 to k do C[i] = 0 • for j = 1 to A.length do C[A[j]] = C[A[j]] + 1 • //C[i] now contains the number of elements equal to i • for i = 1 to k do C[i] = C[i ] + C[i-1] • //C[i] now contains the number of elements less than or equal to i • for j = A.length downto 1 • do B[C[A[ j ] ] ] = A[j]; C[A[ j ]] = C[A[ j ]] – 1

  22. CountingSort CountingSort(A,k) //Input: array A[1..n] of integers in the range 0..k //Output: array B[1..n] which contains the elements of A, sorted • for i = 0 to k do C[i] = 0 • for j = 1 to A.length do C[A[j]] = C[A[j]] + 1 • //C[i] now contains the number of elements equal to i • for i = 1 to k do C[i] = C[i ] + C[i-1] • //C[i] now contains the number of elements less than or equal to i • for j = A.length downto 1 • do B[C[A[ j ] ] ] = A[j]; C[A[ j ]] = C[A[ j ]] – 1 Correctness lines 6/7: Invariant Inv(m): for m ≤ i ≤ n: B[position(i)] contains A[i] for 0 ≤ i ≤ k: C[i] = ( # numbers smaller than i ) + ( # numbers equal to i in A[1..m-1]) Inv(m+1) holds before loop is executed with j =m, Inv(m) holds afterwards

  23. CountingSort: running time CountingSort(A,k) //Input: array A[1..n] of integers in the range 0..k //Output: array B[1..n] which contains the elements of A, sorted • for i = 0 to k do C[i] = 0 • for j = 1 to A.length do C[A[j]] = C[A[j]] + 1 • //C[i] now contains the number of elements equal to i • for i = 1 to k do C[i] = C[i ] + C[i-1] • //C[i] now contains the number of elements less than or equal to i • for j = A.length downto 1 • do B[C[A[ j ] ] ] = A[j]; C[A[ j ]] = C[A[ j ]] – 1 line 1: ∑0≤i≤kΘ(1) = Θ(k) line 2: ∑1≤i≤nΘ(1) = Θ(n) line 4: ∑0≤i≤kΘ(1) = Θ(k) lines 6/7: ∑1≤i≤nΘ(1) = Θ(n) Total: Θ(n+k) ➨ Θ(n) if k = O(n)

  24. CountingSort TheoremCountingSort is a stable sorting algorithm that sorts an array of n integers in the range 0..k in Θ(n+k) time.

  25. RadixSort Input: array A[1..n] of numbers Assumption: the input elements are integers with ddigits example(d = 4): 3288, 1193, 9999, 0654, 7243, 4321 RadixSort(A, d) • for i ← 1 to d • do use a stable sort to sort array A on digit i dth digit 1st digit

  26. 720 355 436 457 657 329 839 720 329 436 839 355 457 657 329 355 436 457 657 720 839 RadixSort: example Correctness: Induction on i 329 457 657 839 436 720 355 sort on 1st digit sort on 2nd digit sort on 3rd digit

  27. RadixSort Running time: If we use CountingSort as stable sorting algorithm ➨ Θ(n + k) per digit TheoremGiven nd-digit numbers in which each digit can take up to k possible values, RadixSort correctly sorts these numbers in Θ(d (n + k)) time. each digit is an integer in the range 0..k

  28. BucketSort Input: array A[1..n] of numbers Assumption: the input elements lie in the interval [0..1) (no integers!) BucketSort is fast if the elements are uniformly distributed in [0..1)

  29. 1 0 0.792 2 1 0.1 0.1 0.15 0.13 0.287 0.287 0.256 0.15 0.346 0.346 0.734 0.5 0.53 0.5 0.13 0.792 0.734 0.256 n-1 n 0.53 BucketSort • Throw input elements in “buckets”, sort buckets, concatenate … input array A[1..n]; numbers in [0..1) auxiliary array B[0..n-1] bucket B[i] contains numbers in [i/n … (i+1)/n]

  30. 1 0.792 2 0.1 0.1 0.13 0.15 0.287 0.256 0.287 0.15 0.346 0.346 0.346 0.734 0.5 0.5 0.53 0.53 0.5 0.13 0.734 0.792 0.256 n 0.53 BucketSort • Throw input elements in “buckets”, sort buckets, concatenate … input array A[1..n]; numbers in [0..1) auxiliary array B[0..n-1] bucket B[i] contains numbers in [i/n … (i+1)/n] 0 1 0.1 0.15 0.13 0.287 0.256 0.792 0.734 n-1

  31. BucketSort BucketSort(A) // Input: array A[1..n] of numbers with 0 ≤ A[i ] < 1 // Output: sorted list, which contains the elements of A • n = A.length • initialize auxiliary array B[0..n-1]; each B[i] is a linked list of numbers • for i = 1 to n • do insert A[i] into list B[ n∙A[i ] ] • for i = 0 to n-1 • do sort list B[i], for example with InsertionSort • concatenate the lists B[0], B[1], …, B[n-1] together in order

  32. BucketSort Running time? Define ni = number of elements in bucket B[i] ➨ running time = Θ(n) + ∑0≤i≤n-1Θ(ni2) • worst case: • best case: • expected running time if the numbers are randomly distributed ? all numbers fall into the same bucket ➨ Θ(n2) all numbers fall into different buckets ➨Θ(n)

  33. BucketSort: expected running time Define ni = number of elements in bucket B[i] ➨ running time = Θ(n) + ∑0≤i≤n-1Θ(ni2) Assumption: Pr { A[j] falls in bucket B[i] } = 1/n for each i E [ running time ] = E [Θ(n) + ∑0≤i≤n-1Θ(ni2) ] = Θ ( n + ∑0≤i≤n-1 E [ni2 ] ) What is E [ni2 ] ? (some math with indicator random variables – see book for details) ➨ E [ni2 ] = 2 - 1/n = Θ(1) ➨ expected running time = Θ(n) but E [ni2 ] ≠ E [ni ]2 We have E [ni] = 1 …

  34. Tutorials this week Thursday, 15:45-17:30 (as usual) Friday, 13:45 -15:30 (as usual)

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