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Analysis of Algorithms CS 477/677

Analysis of Algorithms CS 477/677. Lecture 8 Instructor: Monica Nicolescu. Quick Announcement. Hand imaging collection (hand shape) Reza Amayeh: amayeh@cse.unr.edu Lab address: LME building room 314 Time: Mon, Wed, Fri: 10:00 am until 5:00pm Thu, Thr: 3:00 pm until 5:00pm.

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Analysis of Algorithms CS 477/677

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  1. Analysis of AlgorithmsCS 477/677 Lecture 8 Instructor: Monica Nicolescu

  2. Quick Announcement • Hand imaging collection (hand shape) • Reza Amayeh: amayeh@cse.unr.edu • Lab address: LME building room 314 • Time: • Mon, Wed, Fri: 10:00 am until 5:00pm • Thu, Thr: 3:00 pm until 5:00pm CS 477/677 - Lecture 8

  3. How Fast Can We Sort? • Insertion sort, Bubble Sort, Selection Sort • Merge sort • Quicksort • What is common to all these algorithms? • These algorithms sort by making comparisons between the input elements • To sort n elements, comparison sorts must make (nlgn) comparisons in the worst case (n2) (nlgn) (nlgn) CS 477/677 - Lecture 8

  4. one execution trace node leaf: Decision Tree Model • Represents the comparisons made by a sorting algorithm on an input of a given size: models all possible execution traces • Control, data movement, other operations are ignored • Count only the comparisons • Decision tree for insertion sort on three elements: CS 477/677 - Lecture 8

  5. A 0 1 2 3 4 5 C 2 2 4 7 7 8 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 2 0 5 0 2 3 2 0 2 3 3 3 3 0 5 3 B Counting Sort • Assumption: • The elements to be sorted are integers in the range 0 to k • Idea: • Determine for each input element x, the number of elements smaller than x • Place element x into its correct position in the output array CS 477/677 - Lecture 8

  6. Analysis of Counting Sort Alg.: COUNTING-SORT(A, B, n, k) • for i ← 0to k • do C[ i ] ← 0 • for j ← 1to n • do C[A[ j ]] ← C[A[ j ]] + 1 • C[i] contains the number of elements equal to i • for i ← 1to k • do C[ i ] ← C[ i ] + C[i -1] • C[i] contains the number of elements ≤i • for j ← ndownto 1 • do B[C[A[ j ]]] ← A[ j ] • C[A[ j ]] ← C[A[ j ]] - 1 (k) (n) (k) (n) Overall time: (n + k) CS 477/677 - Lecture 8

  7. Analysis of Counting Sort • Overall time: (n + k) • In practice we use COUNTING sort when k = O(n)  running time is (n) • Counting sort is stable • Numbers with the same value appear in the same order in the output array • Important when satellite data is carried around with the sorted keys CS 477/677 - Lecture 8

  8. Radix Sort • Considers keys as numbers in a base-R number • A d-digit number will occupy a field of d columns • Sorting looks at one column at a time • For a d digit number, sort the least significant digit first • Continue sorting on the next least significant digit, until all digits have been sorted • Requires only d passes through the list CS 477/677 - Lecture 8

  9. RADIX-SORT Alg.: RADIX-SORT(A, d) for i ← 1to d do use a stable sort to sort array A on digit i • 1 is the lowest order digit, d is the highest-order digit CS 477/677 - Lecture 8

  10. Analysis of Radix Sort • Given n numbers of d digits each, where each digit may take up to k possible values, RADIX-SORT correctly sorts the numbers in (d(n+k)) • One pass of sorting per digit takes (n+k) assuming that we use counting sort • There are d passes (for each digit) CS 477/677 - Lecture 8

  11. Correctness of Radix sort • We use induction on number of passes through each digit • Basis: If d = 1, there’s only one digit, trivial • Inductive step: assume digits 1, 2, . . . , d-1 are sorted • Now sort on the d-th digit • If ad < bd, sort will put a before b: correct a < b regardless of the low-order digits • If ad > bd, sort will put a after b: correct a > b regardless of the low-order digits • If ad = bd, sort will leave a and b in the same order and a and b are already sorted on the low-order d-1 digits CS 477/677 - Lecture 8

  12. Bucket Sort • Assumption: • the input is generated by a random process that distributes elements uniformly over [0, 1) • Idea: • Divide [0, 1) into n equal-sized buckets • Distribute the n input values into the buckets • Sort each bucket • Go through the buckets in order, listing elements in each one • Input: A[1 . . n], where 0 ≤ A[i] < 1 for all i • Output: elements ai sorted • Auxiliary array: B[0 . . n - 1] of linked lists, each list initially empty CS 477/677 - Lecture 8

  13. BUCKET-SORT Alg.: BUCKET-SORT(A, n) for i ← 1to n do insert A[i] into list B[nA[i]] for i ← 0to n - 1 do sort list B[i] with insertion sort concatenate lists B[0], B[1], . . . , B[n -1] together in order return the concatenated lists CS 477/677 - Lecture 8

  14. / / .12 / .39 / .26 .68 / .17 .78 .23 / .21 .72 / .94 / / / Example - Bucket Sort 1 0 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 8 10 9 CS 477/677 - Lecture 8

  15. / / .78 .23 .68 .78 / .17 / .72 .26 / .39 .94 / .72 .39 / .21 .12 .17 .12 .23 .94 / .26 .21 .68 / / / Example - Bucket Sort 0 1 2 3 4 5 6 7 Concatenate the lists from 0 to n – 1 together, in order 8 9 CS 477/677 - Lecture 8

  16. Correctness of Bucket Sort • Consider two elements A[i], A[ j] • Assume without loss of generality that A[i] ≤ A[j] • Then nA[i] ≤ nA[j] • A[i] belongs to the same group as A[j] or to a group with a lower index than that of A[j] • If A[i], A[j] belong to the same bucket: • insertion sort puts them in the proper order • If A[i], A[j] are put in different buckets: • concatenation of the lists puts them in the proper order CS 477/677 - Lecture 8

  17. Analysis of Bucket Sort Alg.: BUCKET-SORT(A, n) for i ← 1to n do insert A[i] into list B[nA[i]] for i ← 0to n - 1 do sort list B[i] with insertion sort concatenate lists B[0], B[1], . . . , B[n -1] together in order return the concatenated lists O(n) (n) O(n) (n) CS 477/677 - Lecture 8

  18. Conclusion • Any comparison sort will take at least nlgn to sort an array of n numbers • We can achieve a better running time for sorting if we can make certain assumptions on the input data: • Counting sort: each of the n input elements is an integer in the range 0 to k • Radix sort: the elements in the input are integers represented with d digits • Bucket sort: the numbers in the input are uniformly distributed over the interval [0, 1) CS 477/677 - Lecture 8

  19. A Job Scheduling Application • Job scheduling • The key is the priority of the jobs in the queue • The job with the highest priority needs to be executed next • Operations • Insert, remove maximum • Data structures • Priority queues • Ordered array/list, unordered array/list CS 477/677 - Lecture 8

  20. Example CS 477/677 - Lecture 8

  21. PQ Implementations & Cost Worst-case asymptotic costs for a PQ with N items Insert Remove max ordered array N 1 ordered list N 1 unordered array 1 N unordered list 1 N Can we implement both operations efficiently? CS 477/677 - Lecture 8

  22. Background on Trees • Def:Binary tree = structure composed of a finite set of nodes that either: • Contains no nodes, or • Is composed of three disjoint sets of nodes: a root node, a left subtree and a right subtree root 4 Right subtree Left subtree 1 3 2 16 9 10 14 8 CS 477/677 - Lecture 8

  23. 4 4 1 3 1 3 2 16 9 10 2 16 9 10 14 8 7 12 Complete binary tree Full binary tree Special Types of Trees • Def:Full binary tree = a binary tree in which each node is either a leaf or has degree exactly 2. • Def:Complete binary tree = a binary tree in which all leaves have the same depth and all internal nodes have degree 2. CS 477/677 - Lecture 8

  24. The Heap Data Structure • Def:A heap is a nearly complete binary tree with the following two properties: • Structural property: all levels are full, except possibly the last one, which is filled from left to right • Order (heap) property: for any node x Parent(x) ≥ x 8 It doesn’t matter that 4 in level 1 is smaller than 5 in level 2 7 4 5 2 Heap CS 477/677 - Lecture 8

  25. Definitions • Height of a node = the number of edges on a longest simple path from the node down to a leaf • Depth of a node = the length of a path from the root to the node • Height of tree = height of root node = lgn, for a heap of n elements Height of root = 3 4 1 3 Height of (2)= 1 Depth of (10)= 2 2 16 9 10 14 8 CS 477/677 - Lecture 8

  26. Array Representation of Heaps • A heap can be stored as an array A. • Root of tree is A[1] • Left child of A[i] = A[2i] • Right child of A[i] = A[2i + 1] • Parent of A[i] = A[ i/2 ] • Heapsize[A] ≤ length[A] • The elements in the subarray A[(n/2+1) .. n] are leaves • The root is the maximum element of the heap A heap is a binary tree that is filled in order CS 477/677 - Lecture 8

  27. Heap Types • Max-heaps (largest element at root), have the max-heap property: • for all nodes i, excluding the root: A[PARENT(i)] ≥ A[i] • Min-heaps (smallest element at root), have the min-heap property: • for all nodes i, excluding the root: A[PARENT(i)] ≤ A[i] CS 477/677 - Lecture 8

  28. Operations on Heaps • Maintain the max-heap property • MAX-HEAPIFY • Create a max-heap from an unordered array • BUILD-MAX-HEAP • Sort an array in place • HEAPSORT • Priority queue operations CS 477/677 - Lecture 8

  29. Operations on Priority Queues • Max-priority queues support the following operations: • INSERT(S, x): inserts element x into set S • EXTRACT-MAX(S): removes and returns element of S with largest key • MAXIMUM(S): returns element of S with largest key • INCREASE-KEY(S, x, k): increases value of element x’s key to k (Assume k ≥ x’s current key value) CS 477/677 - Lecture 8

  30. Maintaining the Heap Property • Suppose a node is smaller than a child • Left and Right subtrees of i are max-heaps • Invariant: • the heap condition is violated only at that node • To eliminate the violation: • Exchange with larger child • Move down the tree • Continue until node is not smaller than children CS 477/677 - Lecture 8

  31. Maintaining the Heap Property Alg:MAX-HEAPIFY(A, i, n) • l ← LEFT(i) • r ← RIGHT(i) • ifl ≤ n and A[l] > A[i] • thenlargest ←l • elselargest ←i • ifr ≤ n and A[r] > A[largest] • thenlargest ←r • iflargest  i • then exchange A[i] ↔ A[largest] • MAX-HEAPIFY(A, largest, n) • Assumptions: • Left and Right subtrees of i are max-heaps • A[i] may be smaller than its children CS 477/677 - Lecture 8

  32. A[2] violates the heap property A[4] violates the heap property Heap property restored Example MAX-HEAPIFY(A, 2, 10) A[2]  A[4] A[4]  A[9] CS 477/677 - Lecture 8

  33. MAX-HEAPIFY Running Time • Intuitively: • A heap is an almost complete binary tree  must process O(lgn) levels, with constant work at each level • Running time of MAX-HEAPIFY is O(lgn) • Can be written in terms of the height of the heap, as being O(h) • Since the height of the heap is lgn CS 477/677 - Lecture 8

  34. Alg:BUILD-MAX-HEAP(A) n = length[A] fori ← n/2downto1 do MAX-HEAPIFY(A, i, n) Convert an array A[1 … n] into a max-heap (n = length[A]) The elements in the subarray A[(n/2+1) .. n] are leaves Apply MAX-HEAPIFY on elements between 1 and n/2 1 4 2 3 1 3 4 5 6 7 2 16 9 10 8 9 10 14 8 7 Building a Heap A: CS 477/677 - Lecture 8

  35. Readings • Chapter 8, 6 CS 477/677 - Lecture 8

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