Algorithms and Data Structures*TopicsLinear SearchLinear SearchLinear SearchBinary SearchBinary SearchQuicksortPartitionQuicksortSwapLibrariesqsort – standard C libraryqsort (int)qsort (strings)Big “Oh”Timingnlog n vs. n2Growing ArraysGrowing Arrays in CGrowing ArraysListsLists in CLists in CAppend Element to Back of ListLookup Element in ListApply Function to Elements in ListPrint Elements of a ListCount Elements of a ListFree Elements in ListDelete Element in ListTreesBinary Search TreeBinary Search Tree LookupHash TablesHash TableHash Table LookupHash FunctionStandard Template LibraryIteratorsperlJava CollectionAlgorithms and Data Structures*Objective: To review the fundamental algorithms and data structures that are commonly used in programs. To see how to use and implement these algorithms and data structures in different languages and to see what language and library support exists for them. Sorting and Searching Arrays and Vectors Lists Hash Tables Generic programming*This material comes from chapter 2 ofBrian Kernighan and Rob Pike,The Practice of ProgrammingTopicsBinary SearchQuicksortBig “Oh”VectorsListsHash TablesC, C++, Java, PerlLinear SearchJust exhaustively examine each elementWorks on any list (sorted or unsorted)Onlysearch for a linked-listRuntime: best: θ(1) worst: θ(n) average: θ(n)Linear Searchtypdef struct Nameval Nameval;struct Nameval {char *name;int value;}/* HTML characters, e.g. AElig is a ligature of A and E. *//* values are Unicode/ISO10646 encoding. */Nameval htmlchars[] = {“Aelig”, 0x00c6,“Aacute”, 0x00c1,“Acirc”, 0x00c2,/* … */“zeta”, 0x03b6};Linear Search/* lookup: sequential search for name in tab;return index */int lookup(char *name, Nameval tab[], int ntab){int i;for (i = 0; i < ntab; i++)if (strcmp(name, tab[i].name) == 0)return i;return –1; /* no match */}Binary SearchOnly works on sortedcollectionsOnly works on indexed collections (arrays, vectors)Start in the middle: Find it?Cut search space in ½Need θ(lg(n)) time, worst (and avg.)Binary Search/* lookup: binary search for name in tab; return index or –1 if not found. */int lookup(char *name, Nameval tab[], int ntab) {int low, high, mid, cmp;low = 0; high = ntab – 1;while (low <= hight) {mid = (low + high)/2;cmp = strcmp(name, tab[mid].name);if (cmp < 0)high = mid – 1;else if (cmp > 0)low = mid + 1;else /* found match */return mid;}return –1; /* no match */}Quicksortpick one element of the array (pivot)partition the other elements into two groups those less than the pivot those that are greater than or equal to the pivotPivot is now in the right placerecursively sort each (strictly smaller) groupPartitionunexaminedplast i n-1unexamined< pp >= p0 1 last i n-1< p p >= p0last n-1Quicksort/* quicksort: sort v[0]..v[n-1] into increasing order. */void quicksort(int v[], int n){int i, last;if (n <= 1) /* nothing to do */return;swap(v, 0, rand()%n); /* move pivot element to v[0] */last = 0;for (i = 1; i < n; i++) /* partition */if (v[i] < v[0])swap(v, ++last, i);swap(v, 0, last); /* restore pivot */quicksort(v, last); /* recursively sort each part. */quicksort(v+last+1, n-last-1);}Swap/* swap: interchange v[i] and v[j]. */void swap(int v[], int i, int j){int temp;temp = v[i];v[i] = v[j];v[j] = temp;}LibrariesC: qsortC++: sort (algorithm library from STL)java.util.collections.sortperl: sortqsort – standard C libraryqsort( void *a, int n, int s,int(*cmp)(void *a, void *b) );Sorts array a, which has n elementsEach element is s bytescmp is a function that you must provide compares 2 single elements, *a & *b qsort must pass pointers, since it doesn't know type (but cmpdoes, since you provide it for a given sort) returns -1 if a<b, 0 if a==b, and 1 if a>b For sorting in descendingorder, return 1 if a < bqsort (int)int arr[N];qsort(arr, N, sizeof(arr[0]), icmp);/* icmp: integer compare of *p1 and *p2 */int icmp(const void *p1, const void *p2){int v1, v2;v1 = *((int*) p1);v2 = *((int*) p2);if( v1 < v2 )return –1;else if( v1 == v2 )return 0;elsereturn 1;}qsort (strings)char *str[N];qsort(str, N, sizeof(str[0]), scmp);/* scmp: string compare of *p1 and *p2 *//* p1 is a ptr to a string, or char*, so is a *//* ptr to a ptr, or a char** */int scmp(const void *p1, const void *p2){char *v1, *v2;v1 = *((char**) p1);v2 = *((char**) p2);return strcmp(v1, v2);}Big “Oh”Notation Name ExampleO(1) constant array indexO(log n) logarithmic binary searchO(n) linear string comparisonO(nlog n) n log n quicksort/mergesortO(n2) quadratic insertion sortO(n3) cubic matrix multiplicationO(2n) exponential set partitioning*It is more precise to use Θ for order classesTimingUnix time command [jjohnson@ws56 lec1]$ time sign < /usr/share/dict/words | sort | squash > temptime 0.11user 0.01system 0:00.27elapsed 43%CPU (0avgtext+0avgdata 0maxresident)k 0inputs+0outputs (91major+13minor)pagefaults 0swapsclock()counting cyclesnlog n vs. n2Growing ArraysArrays provide O(1) access and insertion timeSorted arrays provide O(log n) search time and O(n) insertion time [have to move elements]If the number of elements in an array is not known ahead of time it may be necessary to resize the array.Involves dynamic memory allocation and copyingTo minimize the cost it is best to resize in chunks (some factor of its current size)Growing Arrays in Ctypedef struct Nameval Nameval;struct Nameval {char *name;int value;};struct NvTable {int nval; /* current number of values */int max; /* allocated number of values */Nameval *data; /* array of name-value pairs */};enum { NVINIT = 1, NVGROW = 2 };struct NvTable symList;Growing Arrays/* addname: add new name and value to symList */int addname( Nameval newname ) {Nameval *nvp;if( symList.data == NULL) { /* first time */symList.dat =(Nameval*) malloc(NVINIT * sizeof(Nameval));if( symList.dat == NULL ) /* check for no memory */return –1;symList.max = NVINIT;symList.nval = 0;} else if( symList.nval >= symList.max) { /* grow */nvp = (Nameval*) realloc( symList.data,(NVGROW*symList.max)*sizeof(Nameval));if( nvp == NULL ) /* realloc failed */return –1;symList.max *= NVGROW;symList.data = nvp;}symList.data[symList.nval] = newname;return symList.nval++;}ListsA sequence of elementsSpace is allocate for each new element and consecutive elements are linked together with a pointer.O(1) time to insert at front, O(n) to append unless pointer to last