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IS 2610: Data Structures Priority Queue, Heapsort, Searching March 15, 2004

Priority Queues Applications require that we process records with keys in order Collect records Process one with largest key Maybe collect more records Applications Simulation systems (event times) Scheduling (priorities) Priority queue : A data structure of items with keys that support two basic operations: Insert a new item; Delete the item with the largest key

Priority Queue Build and maintain the following operations Construct the queue Insert a new item Delete the maximum item Change the priority of an arbitrary specified item Delete an arbitrary specified item Join two priority queues into one large one

Priority Queue: Elementary operations void PQinit(int); int PQempty(); void PQinsert(Item); Item PQdelmax(); #include <stdlib.h> #include "Item.h" static Item *pq; static int N; void PQinit(int maxN) { pq = malloc(maxN*sizeof(Item)); N = 0; } int PQempty() { return N == 0; } void PQinsert(Item v) { pq[N++] = v; } Item PQdelmax() { int j, max = 0; for (j = 1; j < N; j++) if (less(pq[max], pq[j])) max = j; exch(pq[max], pq[N-1]); return pq[--N]; }

Heap Data Structure Def 9.2 A tree is heap-ordered if the key in each node is larger than or equal to the keys in all of that node’s children (if any). Equivalently, the key in each node of a heap-ordered tree is smaller than or equal to the key in that node’s parent (if any) Property 9.1 No node in a heap-ordered tree has a key larger than the key at the root. Heap can efficiently support the basic priority-queue operations

Heap Data Structure Def 9.2 A heap is a set of nodes with keys arranged in a complete heap-ordered binary tree, [represented as an array]. A complete tree allows using a compact array representation The parent of node i is in position  i /2  The two children of the node i are in positions 2 i and 2 i + 1 . Disadvantage of using arrays?

Algorithms on Heaps Heapifying Modify heap to violate the heap condition Add new element Change the priority Restructure heap to restore the heap condition Priority of child becomes greater A E T A I M G R N O X S P

Bottom-up heapify First exchange T and R Then exchange T and S fixUp(Item a[], int k) { while (k > 1 && less(a[k/2], a[k])) { exch(a[k], a[k/2]); k = k/2; } } A E R A I M G S N O X T P

Top-down heapify Exchange with the larger child Priority of parent becomes smaller A E R A I M G S N P O T X fixDown(Item a[], int k, int N) { int j; while (2*k <= N) { j = 2*k; if (j < N && less(a[j], a[j+1])) j++; if (!less(a[k], a[j])) break; exch(a[k], a[j]); k = j; } } I M N O X P

Heap-based priority Queue Property 9.2 Insert requires no more than lg n one comparison at each level Delete maximum requires no more than 2 lg n two comparisons at each level #include <stdlib.h> #include "Item.h" static Item *pq; static int N; void PQinit(int maxN) { pq = malloc((maxN+1)*sizeof(Item)); N = 0; } int PQempty() { return N == 0; } void PQinsert(Item v) { pq[++N] = v; fixUp(pq, N); } Item PQdelmax() { exch(pq[1], pq[N]); fixDown(pq, 1, N-1); return pq[N--]; }

Sorting with a priority Queue Use PQinsert to put all the elements on the priority queue Use PQdelmax to remove them in decreasing order Heap construction takes < n lg n void PQsort(Item a[], int l, int r) { int k; PQinit(); for (k = l; k <= r; k++) PQinsert(a[k]); for (k = r; k >= l; k--) a[k] = PQdelmax(); }

Bottom-up Heap Right to left Bottom-up G E X A M P R T I A S O L E N T A X M I P void heapsort(Item a[], int l, int r) { int k, N = r-l+1; Item* pq = a + l -1; for (k = N/2; k >= 1; k--) fixDown(pq, k, N); while (N > 1) { exch(pq[1], pq[N]); fixDown(pq, 1, --N); } }

Bottom-up Heap Note: most nodes are at the bottom Bottom up heap construction takes linear time G E T A M I R X P A S O L E N G E S A M I R T O A X P L E N

Radix Sort Decompose keys into pieces Binary numbers are sequence of bytes Strings are sequence of characters Decimal number are sequence of digits Radix sort: Sorting methods built on processing numbers one piece at a time Treat keys as numbers represented in base R and work with individual digits R = 10 in many applications where keys are 5- to 10-digit decimal numbers Example: postal code, telephone numbers, SSN

Radix Sort If keys are integers Can use R = 2, or a multiple of 2 If keys are strings of characters Can use R = 128 or 256 (aligns with a byte) Radix sort is based on the abstract operation Extract the i th digit from a key Two approaches to radix sort Most-significant-digit (MSD) radix sorts (left-to-right) Least-significant-digit (LSD) radix sorts (right-to-left)

Bits, Bytes and Word The key to understanding Radix sorts is to recognize that Computers generally are built to process bits in groups called machine words (groups of bytes) Sort keys are commonly organized as byte sequences Small byte sequence can also serve as array indices or machine addresses Hence an abstraction can be used

Bits, Bytes and Word Def: A byte is a fixed-length sequence of bits; a string is a variable-length sequence of bytes; a word is a fixed-length sequence of bytes digit(A, 2)?? #define bitsword 32 #define bitsbyte 8 #define bytesword 4 #define R (1 << bitsbyte) #define digit(A, B) (((A) >> (bitsword-((B)+1)*bitsbyte)) & (R-1)) // Another possibility #define digit(A, B) A[B]

Binary Quicksort Partition a file based on leading bits Sort the sub-files recursively quicksortB(int a[], int l, int r, int w) { int i = l, j = r; if (r <= l || w > bitsword) return; while (j != i) { while (digit(a[i], w) == 0 && (i < j)) i++; while (digit(a[j], w) == 1 && (j > i)) j--; exch(a[i], a[j]); } if (digit(a[r], w) == 0) j++; quicksortB(a, l, j-1, w+1); quicksortB(a, j, r, w+1); } void sort(Item a[], int l, int r) { quicksortB(a, l, r, 0); }

Binary Quicksort 0 01 1 11 0 11 1 00 1 01 0 10 001 010 011 100 101 111 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1

MSD Radix Sort Binary Quicksort is a MSD with R = 2; For general R, we will partition the array into R different bins/buckets

LSD Radix Sort Examine bytes Right to left now so b c ab ace for no b w ab ago tip ca b t ag and

Symbol Table A symbol table is a data structure of items with keys that supports two basic operations: insert a new item, and return an item with a given key Examples: Account information in banks Airline reservations

Symbol Table ADT Key operations Insert a new item Search for an item with a given key Delete a specified item Select the k th smallest item Sort the symbol table Join two symbol tables void STinit(int); int STcount(); void STinsert(Item); Item STsearch(Key); void STdelete(Item); Item STselect(int); void STsort(void (*visit)(Item));

Key-indexed ST Simplest search algorithm is based on storing items in an array, indexed by the keys static Item *st; static int M = maxKey; void STinit(int maxN) { int i; st = malloc((M+1)*sizeof(Item)); for (i = 0; i <= M; i++) st[i] = NULLitem; } int STcount() { int i, N = 0; for (i = 0; i < M; i++) if (st[i] != NULLitem) N++; return N; } void STinsert(Item item) { st[key(item)] = item; } Item STsearch(Key v) { return st[v]; } void STdelete(Item item) { st[key(item)] = NULLitem; } Item STselect(int k) { int i; for (i = 0; i < M; i++) if (st[i] != NULLitem) if (k-- == 0) return st[i]; } void STsort(void (*visit)(Item)) { int i; for (i = 0; i < M; i++) if (st[i] != NULLitem) visit(st[i]); }

Sequential Search based ST When a new item is inserted, we put it into the array by moving the larger elements over one position (as in insertion sort) To search for an element Look through the array sequentially If we encounter a key larger than the search key – we report an error

Binary Search Divide and conquer methodology Divide the items into two parts Determine which part the search key belongs to and concentrate on that part Keep the items sorted Use the indices to delimit the part searched. Item search(int l, int r, Key v) { int m = (l+r)/2; if (l > r) return NULLitem; if eq(v, key(st[m])) return st[m]; if (l == r) return NULLitem; if less(v, key(st[m])) return search(l, m-1, v); else return search(m+1, r, v); } Item STsearch(Key v) { return search(0, N-1, v); }

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