Searching, Sorting & Analysis

Transcription

1 Searching, Sorting & Anaysis Unit 2 Chapter 8 CS 2308 Fa 2018 Ji Seaman 1 Definitions of Search and Sort Search: find a given item in an array, return the index of the item, or -1 if not found. Sort: rearrange the items in an array into some order (smaest to biggest, aphabetica order, etc.). There are various methods (agorithms) for carrying out these common tasks. Which ones are better? Why? 2 Very simpe method. Linear Search Compare first eement to target vaue, if not found then compare second eement to target vaue... Repeat unti: target vaue is found (return its index) or we run out of items (return -1). Linear Search in C++ first attempt int searchlist (int ist[], int size, int target) int position = -1; for (int i=0; i<size; i++) //position of target if (ist[i] == target) //found the target! position = i; //record which item return position; Is this agorithm correct (does it cacuate the right vaue)? 3 Is this agorithm efficient (does it do unnecessary work)? 4

2 Linear Search in C++ second attempt Program that uses inear search int searchlist (int ist[], int size, int target) int position = -1; boo found = fase; //position of target //fag, true when target is found for (int i=0; i < size &&!found; i++) if (ist[i] == target) //found the target! found = true; //set the fag position = i; //record which item return position; Is this agorithm correct (does it cacuate the right vaue)? Is this agorithm efficient (does it do unnecessary work)? 5 #incude <iostream> using namespace std; int searchlist(int[], int, int); int main() const int SIZE=5; int idnums[size] = 871, 750, 988, 100, 822; int resuts, id; cout << Enter the empoyee ID to search for: ; cin >> id; resuts = searchlist(idnums, SIZE, id); if (resuts == -1) cout << That id number is not registered\n ; ese cout << That id number is found at ocation ; cout << resuts+1 << end; 6 Evauating the Agorithm Does it do any unnecessary work? Is it time efficient? How woud we know? We measure time efficiency of agorithms in terms of number of main steps required to finish. For search agorithms, the main step is comparing an array eement to the target vaue. Number of steps depends on: size of input array whether or not vaue is in array Efficiency of Linear Search how many main steps (comparisons to target)? N is the number of eements in the array Best Average Worst N=50,000 In terms of N ,000 N/2 50,000 N Note: if we search for many items that are not in the array, the average case wi be greater than N/2. where the vaue is in the array 7 8

3 Binary Search Works ony for SORTED arrays Divide and conquer stye agorithm Compare target vaue to midde eement in ist. if equa, then return its index if ess than midde eement, repeat the search in the first haf of ist if greater than midde eement, repeat the search in ast haf of ist If current search ist is narrowed down to 0 target is 11 target < 50 target > 7 Binary Search Agorithm exampe We use first and ast to indicate beginning and end of current search ist target == 11 first first first ast ast ast eements, return Binary Search in C++ int binarysearch (int array[], int size, int target) int first = 0, //index of beginning of search ist ast = size 1, //index of end of search ist midde, //index of midpoint of search ist position = -1; //position of target vaue boo found = fase; //fag whie (first <= ast &&!found) midde = (first + ast) /2; if (array[midde] == target) found = true; position = midde; ese if (target < array[midde]) ast = midde 1; ese first = midde + 1; return position; //cacuate midpoint What if first + ast is odd? What if first==ast? //search ist = ower haf //search ist = upper haf 11 Program using Binary Search #incude <iostream> using namespace std; int binarysearch(int[], int, int); How is this program different from the one on side 6? int main() const int SIZE=5; int idnums[size] = 100, 750, 822, 871, 988; int resuts, id; cout << Enter the empoyee ID to search for: ; cin >> id; resuts = binarysearch(idnums, SIZE, id); if (resuts == -1) cout << That id number is not registered\n ; ese cout << That id number is found at ocation ; cout << resuts+1 << end; 12

4 Efficiency of Binary Search Cacuate worst case (target not in ist) for N=1024 # Items eft to search # Comparisons so far = 2 10 <==> og = 10 Goa: cacuate this vaue from N 13 Efficiency of Binary Search If N is the number of eements in the array, how many comparisons (steps)? 1024 = 2 10 <==> og = 10 N = 2 steps Best Worst <==> og 2 N = steps N=50,000 In terms of N og 2 N To what power do I raise 2 to get N? Rounded up to next whoe number 14 Is Log 2 N better than N? Is binary search better than inear search? Compare vaues of N/2, N, and Log 2 N as N increases: N N/2 Log2N ,000 2, ,000 25, N and N/2 are growing much faster than og N! sower growing is more efficient (fewer steps). Is this reay a fair comparison? Sorting Agorithms Sort: rearrange the items in an array into ascending or descending order. Bubbe Sort Seection Sort unsorted 16 sorted

5 The Bubbe Sort Bubbe sort Exampe: first pass On each pass: - Compare first two eements. If the first is bigger, they exchange paces (swap). - Compare second and third eements. If second is bigger, exchange them. - Repeat unti ast two eements of the ist are compared. Repeat this process (keep doing passes) unti a pass competes with no exchanges > 2, swap > 3, swap !(7 > 8), no swap !(8 > 9), no swap > 1, swap finished pass 1, did 3 swaps Note: argest eement is now in ast position Note: This is one compete pass! 18 Bubbe sort Exampe: second and third pass Bubbe sort Exampe: passes 4, 5, and <3<7<8, no swap,!(8<1), swap (8<9) no swap finished pass 2, did one swap <3<7, no swap,!(7<1), swap <8<9, no swap finished pass 3, did one swap 2 argest eements in ast 2 positions 3 argest eements in ast 3 positions <3,!(3<1) swap, 3<7<8< finished pass 4, did one swap !(2<1) swap, 2<3<7<8< finished pass 5, did one swap <2<3<7<8<9, no swaps finished pass 6, no swaps, ist is sorted! 20

6 Bubbe sort how does it work? At the end of the first pass, the argest eement is moved to the end (it s bigger than a its neighbors) At the end of the second pass, the second argest eement is moved to just before the ast eement. The back end (tai) of the ist remains sorted. Each pass increases the size of the sorted portion. No exchanges impies each eement is smaer than its next neighbor (so the ist is sorted). Bubbe Sort in C++ void bubbesort (int array[], int size) boo swap; int temp; do swap = fase; for (int i = 0; i < (size-1); i++) if (array [i] > array[i+1]) temp = array[i]; array[i] = array[i+1]; array[i+1] = temp; swap = true; whie (swap); Program using bubbe sort Seection Sort #incude <iostream> using namespace std; void bubbesort(int [], int); void showarray(int [], int); int main() int vaues[6] = 7, 2, 3, 8, 9, 1; cout << The unsorted vaues are: \n ; showarray (vaues, 6); bubbesort (vaues, 6); cout << The sorted vaues are: \n ; showarray(vaues, 6); void showarray (int array[], int size) for (int i=0; i<size; i++) cout << array[i] << ; cout << end; Output: The unsorted vaues are: The sorted vaues are: There is a pass for each position (0..size-1) On each pass, the smaest (minimum) eement in the rest of the ist is exchanged (swapped) with eement at the current position. The first part of the ist (the part that is aready processed) is aways sorted Each pass increases the size of the sorted portion. 24

7 Seection sort Exampe Seection Sort in C++ My version is the min a[5], swap with a[0] is the min a[1], sef-swap a[1] is the min a[2], sef-swap a[2] is the min a[5], swap with a[3] is the min a[5], swap with a[4] sorted Note: underined portion of ist is sorted. Note: This is five passes 25 // Returns the index of the smaest eement, starting at start int findindexofmin (int array[], int size, int start) int minindex = start; for (int i = start+1; i < size; i++) if (array[i] < array[minindex]) minindex = i; return minindex; // Sorts an array, using findindexofmin void seectionsort (int array[], int size) int temp; int minindex; for (int index = 0; index < (size -1); index++) minindex = findindexofmin(array, size, index); //swap temp = array[minindex]; array[minindex] = array[index]; array[index] = temp; Note: saving the index We need to find the index of the minimum vaue so that we can do the swap 26 void seectionsort(int array[], int size) for(int i=0; i<size; i++) for(j=i+1; j<size; j++) if(array[i]>array[j]) temp=array[i]; array[i]=array[j]; array[j]=temp; Seection Sort in C++ compact version This version might do more swapping than the previous one For each eement, it scans the remainder of the array If the next eement is smaer than the eement at the current position (i), then swap them This finds the smaest eement in the remainder of the ist, and it ends up in position (i) 27 Program using Seection Sort #incude <iostream> using namespace std; int findindexofmin (int [], int, int); void seectionsort(int [], int); void showarray(int [], int); int main() int vaues[6] = 7, 2, 3, 8, 9, 1; cout << The unsorted vaues are: \n ; showarray (vaues, 6); seectionsort (vaues, 6); cout << The sorted vaues are: \n ; showarray(vaues, 6); void showarray (int array[], int size) for (int i=0; i<size; i++) cout << array[i] << ; cout << end; Output: The unsorted vaues are: The sorted vaues are:

8 Anaysis of Agorithms using Big O notation Time Efficiency of Agorithms Which agorithm is better, inear search or binary search? Which agorithm is better, bubbe sort or seection sort? How can we answer these questions? Anaysis of agorithms is the determination of the amount of resources (such as time and storage) necessary to execute them. 29 To cassify the time efficiency of an agorithm: Express time (using number of main steps), as a mathematica function of input size (or n beow). Binary search: f(n) = og 2 (n) Need a way to be abe to compare these math functions to determine which is better. We are mosty concerned with which function has smaer vaues (# of steps) at very arge data sizes. We compare the growth rates of the functions and prefer the one that grows more sowy. 30 Cassifications of (math) functions Comparing growth of functions Constant f(x)=b O(1) Logarithmic f(x)=ogb(x) O(og n) Linear f(x)=ax+b O(n) Linearithmic f(x)=x ogb(x) O(n og n) Time (# of steps) Quadratic f(x)=ax 2 +bx+c O(n 2 ) Exponentia f(x)=2 x O(2 n ) Last coumn is big O notation, used in CS. It ignores a but dominant term, constant factors 31 Data size (N) 3 32

9 Time Efficiency of Agorithms Efficiency of Searches (Assuming the array is aready sorted) To cassify the time efficiency of an agorithm: Linear Search, worst case: Express time (using number of main steps), as a mathematica function of input size. Determine which cassification the function fits into. Linear search: f(n) = n Binary Search, worst case: Binary search: f(n) = og 2 (n) O(N) O(og N) Nearer to the top of the cassification chart (on side 31) is sower growth, and more efficient (constant is better than ogarithmic, etc.) 33 Which is sower growing (and thus fewer steps at arge input sizes)? O(og N) Which search agorithm is more time efficient? Binary search 34 Efficiency of Seection Sort Efficiency of Bubbe Sort N is the number of eements in the ist Outer oop executes N-1 times Inner oop executes N-1, then N-2, then N-3,... then once. One comparison per oop iteration. Tota number of comparisons (in inner oop): f(n) = (N-1) + (N-2) = sum of 1 to N-1 sum of 1..N: Subtract N from each side: N + (N-1) + (N-2) = N(N+1)/2 (N-1) + (N-2) = N(N+1)/2 - N = (N 2 +N)/2-2N/2 = (N 2 +N-2N)/2 = N 2 /2 - N/2 Each pass makes N-1 comparisons There wi be (at most) N passes So worst case it s: f(n) = (N-1)*N = N 2 - N If you change the agorithm to ook at ony the unsorted part of the array in each pass, it s exacty ike the seection sort: (N-1) + (N-2) = N 2 /2 - N/2 sti O(N 2 ) Neither agorithm is more efficient in the worst case. O(N 2 ) O(N 2 )