DataStruct 8. Heaps and Priority Queues

Transcription

1 DataStruct 8. Heaps and Priority Queues Michael T. Goodrich, et. al, Data Structures and Algorithms in C++, 2 nd Ed., John Wiley & Sons, Inc., November 13, 2013 Advanced Networking Technology Lab. (YU-ANTL) Dept. of Information & Comm. Eng, College of Engineering, Yeungnam University, KOREA (Tel : ; Fax : ytkim@yu.ac.kr)

2 Priority Queue ADT Outline Implementing a Priority Queue with a List Heaps Implementing a Priority Queue with Heap based on Vector Complete Tree Adaptable Priority Queues DS 8-2

3 8.1 Priority Queue ADT A priority queue stores a collection of entries Typically, an entry is a pair (key, value), where the key indicates the priority Main methods of the Priority Queue ADT insert(e) : inserts an entry e removemin() : removes the entry with smallest key Additional methods min() : returns, but does not remove, an entry with smallest key size(), empty() Applications: Standby flyers Auctions Stock market DS 8-3

4 Total Order Relations Keys in a priority queue can be arbitrary objects on which an order is defined Two distinct entries in a priority queue can have the same key Mathematical concept of total order relation Reflexive property: x x Anti-symmetric property: x y y x x = y Transitive property: x y y z x z DS 8-4

5 Comparator ADT Implements the boolean function isless(p,q), which tests whether p < q Can derive other relations from this: (p == q) is equivalent to (!isless(p, q) &&!isless(q, p)) Can implement in C++ by overloading () Two ways to compare 2D points: class LeftRight { // left-right comparator public: bool operator()(const Point2D& p, const Point2D& q) const { return p.getx() < q.getx(); } }; class BottomTop { // bottom-top public: bool operator()(const Point2D& p, const Point2D& q) const { return p.gety() < q.gety(); } }; DS 8-5

6 Priority Queue ADT Functions supported in Priority Queue ADT size() empty() insert(e) min() removemin() Operation Output Priority Queue insert(5) - {5} insert(9) - {5, 9} insert(2) - {2, 5, 9} insert(7) - {2, 5, 7, 9} min() [2] {2, 5, 7, 9} removemin() - {5, 7, 9} size() 3 {5, 7, 9} min() [5] {5, 7, 9} removemin() - {7, 9} removemin() - {9} removemin() - {} empty() true {} removemin() error {} DS 8-6

7 C++ Priority Queue Interface informal PriorityQueue interface template <typename E, typename C> // element and comparator class PriorityQueue { // priority-queue interface public: int size() const; // number of elements bool isempty() const; // is the queue empty? void insert(const E& e); // insert element const E& min() const throw(queueempty); // minimum element void removemin() throw(queueempty); // remove minimum }; DS 8-7

8 Sorting with Priority Queue We can use a priority queue to sort a set of comparable elements 1. Insert the elements one by one with a series of insert operations 2. Remove the elements in sorted order with a series of removemin operations The running time of this sorting method depends on the priority queue implementation Algorithm PriorityQueueSort(L, P) Input STL list, L, of n elements and a priority queue, P, that compares elements using a total order relation Output The sorted list L while!l.empty () e L.front(); L.popFront() P.insert (e) while!p.empty() e P.min() P.removeMin() L.push_back(e) DS 8-8

9 STL Priority Queue Class Using STL Priority Queue Class #include <queue> using namespace std; priority_queue<int> priq_1; priority_queue<point2d, vector<point2d>, LeftRight> priq_2; Operations size() empty() push(e) top() pop() DS 8-9

10 8.2 Implementing a Priority Queue with a List class ListPriorityQueue (1) template <typename E, typename C> class ListPriorityQueue { public: int size() const; // number of elements bool empty() const; // is the queue empty? void insert(const E& e); // insert element const E& min() const; // minimum element void removemin(); // remove minimum private: std::list<e> L; // priority queue contents C isless; // less-than comparator }; DS 8-10

11 class ListPriorityQueue (2) template <typename E, typename C> // number of elements int ListPriorityQueue<E,C>::size() const { return L.size(); } template <typename E, typename C> // is the queue empty? bool ListPriorityQueue<E,C>::empty() const { return L.empty(); } template <typename E, typename C> // insert element void ListPriorityQueue<E,C>::insert(const E& e) { typename std::list<e>::iterator p; p = L.begin(); while (p!= L.end() &&!isless(e, *p)) ++p; // find element that is larger than e L.insert(p, e); // insert e before p } template <typename E, typename C> // minimum element const E& ListPriorityQueue<E,C>::min() const { return L.front(); } // minimum is at the front template <typename E, typename C> // remove minimum void ListPriorityQueue<E,C>::removeMin() { L.pop_front(); } DS 8-11

12 Selection-Sort Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence Running time of Selection-sort: 1. Inserting the elements into the priority queue with n insert operations takes O(n) time 2. Removing the elements in sorted order from the priority queue with n removemin() operations takes time proportional to n + (n 1) Selection-sort runs in O(n 2 ) time DS 8-12

13 Selection-Sort Example List L Priority Queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (7,4) (g) () (7,4,8,2,5,3,9) Phase 2 (a) (2) (7,4,8,5,3,9) (b) (2,3) (7,4,8,5,9) (c) (2,3,4) (7,8,5,9) (d) (2,3,4,5) (7,8,9) (e) (2,3,4,5,7) (8,9) (f) (2,3,4,5,7,8) (9) (g) (2,3,4,5,7,8,9) () DS 8-13

14 Insertion-Sort Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence Running time of Insertion-sort: 1. Inserting the elements into the priority queue with n insert operations takes time proportional to n 2. Removing the elements in sorted order from the priority queue with a series of n removemin() operations takes O(n) time Insertion-sort runs in O(n 2 ) time DS 8-14

15 Insertion-Sort Example List L Priority queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (4,7) (c) (2,5,3,9) (4,7,8) (d) (5,3,9) (2,4,7,8) (e) (3,9) (2,4,5,7,8) (f) (9) (2,3,4,5,7,8) (g) () (2,3,4,5,7,8,9) Phase 2 (a) (2) (3,4,5,7,8,9) (b) (2,3) (4,5,7,8,9) (g) (2,3,4,5,7,8,9) () DS 8-15

16 In-place Insertion-Sort Instead of using an external data structure, we can implement selectionsort and insertion-sort inplace A portion of the input sequence itself serves as the priority queue For in-place insertion-sort We keep sorted the initial portion of the sequence We can use swaps instead of modifying the sequence DS 8-16

17 Recall Priority Queue ADT A priority queue stores a collection of entries Typically, an entry is a pair (key, value), where the key indicates the priority Main methods of the Priority Queue ADT insert(e) : inserts an entry e removemin() : removes the entry with smallest key Additional methods min() : returns, but does not remove, an entry with smallest key size(), empty() Applications: Standby flyers Auctions Stock market DS 8-17

18 Recall PQ Sorting We use a priority queue Insert the elements with a series of insert operations Remove the elements in sorted order with a series of removemin operations The running time depends on the priority queue implementation: Unsorted sequence gives selection-sort: O(n 2 ) time Sorted sequence gives insertion-sort: O(n 2 ) time Can we do better? DS 8-18

19 8.3 Heaps A heap is a binary tree storing keys at its nodes and satisfying the following properties: Heap-Order: for every internal node v other than the root, key(v) key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0,, h 1, there are 2 i nodes of depth i at depth h 1, the internal nodes are to the left of the external nodes The last node of a heap is the rightmost node of maximum depth last node DS 8-19

20 Complete Binary Tree Complete Binary Tree Property a heap T with height h is a complete binary tree, that is, levels 0, 1, 2,, h-1 of T have the maximum number of nodes possible (namely, level i has 2 i nodes, for 0 i h-1 ) and nodes at level h fill this level from left to right DS 8-20

21 Vector Representation of a Complete Binary Tree Vector representation of a complete binary tree if v is the root of T, then f(v) = 1 if v is the left child of node u, then f(v) = 2 f(u) if v is the right child of node u, then f(v) = 2 f(u) + 1 DS 8-21

22 Vector-based Heap Implementation We can represent a heap with n keys by means of a vector of length n + 1 For the node at rank i the left child is at rank 2i the right child is at rank 2i + 1 Links between nodes are not explicitly stored The cell of at rank 0 is not used Operation insert() corresponds to inserting at rank n + 1 Operation removemin() corresponds to removing at rank n Yields in-place heap-sort DS 8-22

23 Height of a Heap Theorem: A heap storing n keys has height O(log n) Proof: (we apply the complete binary tree property) Let h be the height of a heap storing n keys Since there are 2 i keys at depth i = 0,, h 1 and at least one key at depth h, we have n h Thus, n 2 h, i.e., h log n depth 0 1 h 1 h keys h 1 1 DS 8-23

24 C++ implementation of a Complete Binary Tree class VectorCompleteTree (1) template <typename E> class VectorCompleteTree { // private member data and protected utilities private: // member data std::vector<e> V; // tree contents public: // publicly accessible types typedef typename std::vector<e>::iterator Position; // a position in the tree protected: // protected utility functions Position pos(int i) // map an index to a position { return V.begin() + i; } int idx(const Position& p) const // map a position to an index { return p - V.begin(); } DS 8-24

25 class VectorCompleteTree (2) public: VectorCompleteTree() : V(1) {} // constructor, insert dummy data at index 0 int size() const { return V.size() - 1; } Position left(const Position& p) { return pos(2*idx(p)); } Position right(const Position& p) { return pos(2*idx(p) + 1); } Position parent(const Position& p) { return pos(idx(p)/2); } bool hasleft(const Position& p) const { return 2*idx(p) <= size(); } bool hasright(const Position& p) const { return 2*idx(p) + 1 <= size(); } bool isroot(const Position& p) const { return idx(p) == 1; } Position root() { return pos(1); } Position last() { return pos(size()); } void addlast(const E& e) { V.push_back(e); } void removelast() { V.pop_back(); } void swap(const Position& p, const Position& q) { E e = *q; *q = *p; *p = e; } }; DS 8-25

26 Heaps and Priority Queues We can use a heap to implement a priority queue We store a (key, element) item at each internal node We keep track of the position of the last node (2, Sue) (5, Pat) (6, Mark) (9, Jeff) (7, Anna) last node DS 8-26

27 Insertion into a Heap Method insertitem of the priority queue ADT corresponds to the insertion of a key k to the heap The insertion algorithm consists of three steps Find the insertion node z (the new last node) Store k at z Restore the heap-order property (discussed next) z 6 insertion node z 6 DS 8-27

28 Upheap After the insertion of a new key k, the heap-order property may be violated Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k Since a heap has height O(log n), upheap runs in O(log n) time z 1 5 z DS 8-28

29 Insertion with up-heap bubbling (1) DS 8-29

30 Insertion with up-heap bubbling (2) DS 8-30

31 Insertion with up-heap bubbling (3) DS 8-31

32 Removal from a Heap Method removemin of the priority queue ADT corresponds to the removal of the root key from the heap w 2 6 The removal algorithm consists of three steps last node Replace the root key with the key of the last node w 7 Remove w Restore the heap-order property (discussed next) 9 5 w 6 new last node DS 8-32

33 Downheap After replacing the root key with the key k of the last node, the heap-order property may be violated Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root Downheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k Since a heap has height O(log n), downheap runs in O(log n) time w w 6 DS 8-33

34 Down-heap Bubbling after a Removal (1) DS 8-34

35 Down-heap Bubbling after a Removal (2) DS 8-35

36 C++ implementation of HeapPriorityQueue HeapPriorityQueue for OS Task Scheduling (1) /** Task.h */ #ifndef TASK_H #define TASK_H #include <iostream> #include <string> using namespace std; class Task { friend ostream& operator<<(ostream&, const Task&); public: Task(); // constructor Task(string tname, int pri, int dur); string gettaskname() {return taskname;} int getpriority() { return priority;} int getduration() { return duration;} void settaskname(string n) { taskname = n;} void setpriority(int); void setduration(int); bool operator>(task& t) { return (priority > t.priority);} bool operator<(task& t) { return (priority < t.priority);} private: string taskname; // name of task int priority; // priority of task in scheduling: 0 ~ 49 int duration; // duration of processing time, [ms] }; #endif DS 8-36

37 HeapPriorityQueue for OS Task Scheduling (2) /** Task.cpp */ #include <iostream> #include "Task.h" using namespace std; Task::Task() { } Task::Task(string tname, int pri, int dur) { taskname = tname; setpriority(pri); setduration(dur); } void Task::setPriority(int pri) { if (pri >= 0 && pri <50) priority = pri; else priority = 49; } DS 8-37

38 HeapPriorityQueue for OS Task Scheduling (3) /** Task.cpp (2) */ void Task::setDuration(int dur) { if (dur > 0 && dur <100) duration = dur; else duration = 1; } ostream& operator<<(ostream& fout, const Task& t) { fout <<"Task - name (" << t.taskname; fout << "), pri (" << t.priority << "), duration (" << t.duration; fout << ")"; } return fout; DS 8-38

39 HeapPriorityQueue for OS Task Scheduling (4) /** VectorCompleteTree.h (1) */ #ifndef VECTORCOMPLETETREE_H #define VECTORCOMPLETETREE_H #include <iostream> #include <vector> #include "Task.h" using namespace std; typedef std::vector<task>::iterator Position; class VectorCompleteTree { private: std::vector<task> V; public: protected: Position pos(int i) {return V.begin() + i; } int idx(const Position& p) const { return p - V.begin(); } DS 8-39

40 HeapPriorityQueue for OS Task Scheduling (5) /** VectorCompleteTree.h (2) */ public: private: }; VectorCompleteTree() : V(1) {} int size() const {return V.size() - 1;} Position left(const Position& p) { return pos(2 * idx(p));} Position right(const Position& p) { return pos(2 * idx(p)) + 1;} Position parent(const Position& p) { return pos(idx(p) / 2); } bool hasleft(const Position& p) const { return (2*idx(p) <= size()); } bool hasright(const Position& p) const { return (2*idx(p) + 1 <= size()); } bool isroot(const Position& p) const { return idx(p) == 1; } Position getroot() {return pos(1); } Position getlast() {return pos(size());} void addlast(const Task& enode) { V.push_back(eNode); numtn++;} void removelast() {V.pop_back();numTN--;} void swap(const Position& p, const Position& q) { Task e = *q; *q = *p; *p = e;} void printvectorcompletetreebylevel(position p, int level); int numtn; #endif DS 8-40

41 HeapPriorityQueue for OS Task Scheduling (6) /** VectorCompleteTree.cpp */ #include "VectorCompleteTree.h" #include "Task.h" using namespace std; void VectorCompleteTree::printVectorCompleteTreeByLevel(Position p, int level) { Position poschild; if (level == 0) cout << "\ nroot (data: "; cout <<*p; cout << ")" << endl; for (int i=0; i<level; i++) cout << " "; if (hasleft(p)) { poschild = left(p); cout << "L (data: "; printvectorcompletetreebylevel(poschild, level+1); } else { cout << "L (NULL)" << endl; } DS 8-41

42 HeapPriorityQueue for OS Task Scheduling (7) /** VectorCompleteTree.cpp (2) */ for (int i=0; i<level; i++) cout << " "; if (hasright(p)) { poschild = right(p); } else { } cout << "R (data: "; printvectorcompletetreebylevel(poschild, level+1); cout << "R (NULL)" << endl; } DS 8-42

43 HeapPriorityQueue for OS Task Scheduling (8) /** HeapPriorityQueue.h */ #ifndef HEAPPRIORITYQUEUE_H #define HEAPPRIORITYQUEUE_H #include <iostream> #include "VectorCompleteTree.h" #include "Task.h" using namespace std; class HeapPriorityQueue { public: //HeapPriorityQueue(); int size() const; bool empty() const; void insert(const Task& ele); const Task& min(); void removemin(); VectorCompleteTree* getvctree() {return &vctree;} private: VectorCompleteTree vctree; }; #endif DS 8-43

44 HeapPriorityQueue for OS Task Scheduling (9) /** HeapPriorityQueue.cpp (1) */ #include "HeapPriorityQueue.h" #include "Task.h" int HeapPriorityQueue::size() const { return vctree.size(); } bool HeapPriorityQueue::empty() const { return size() == 0; } const Task& HeapPriorityQueue::min() { return *(vctree.getroot()); } void HeapPriorityQueue::insert(const Task& e) { vctree.addlast(e); Position v = vctree.getlast(); while (!vctree.isroot(v)) { Position u = vctree.parent(v); if (*v > *u) break; vctree.swap(v, u); v = u; } } DS 8-44

45 HeapPriorityQueue for OS Task Scheduling (10) /** HeapPriorityQueue.cpp (2) */ void HeapPriorityQueue::removeMin() { if (size() == 1) vctree.removelast(); else { Position u = vctree.getroot(); Position z = vctree.getlast(); vctree.swap(u, z); vctree.removelast(); while (vctree.hasleft(u)) { Position v = vctree.left(u); Position w = vctree.right(u); if (vctree.hasright(u) && (*v > *w)) v = w; if (*v < *u) { vctree.swap(u, v); u = v; } else break; } } } DS 8-45

46 HeapPriorityQueue for OS Task Scheduling (11) /** main.cpp (1) */ #include <iostream> #include <string> #include <stdlib.h> #include "Task.h" #include "VectorCompleteTree.h" #include "HeapPriorityQueue.h" #include "Task.h" using namespace std; void main() { string tname = ""; char tmp[10]; int priority = -1; int duration = 0; Task* ptask; HeapPriorityQueue taskheappriq; VectorCompleteTree* pvct = taskheappriq.getvctree(); DS 8-46

47 HeapPriorityQueue for OS Task Scheduling (12) /** main.cpp (2) */ for (int i=10; i>0; i--) { _itoa_s(i, tmp, 10); tname = "task_" + string(tmp); priority = i; duration = i; ptask = new Task(tName, priority, duration); taskheappriq.insert(*ptask); cout << "\ nsize of task Heap Priority Queue (after insertion of task ( ; cout << tname << ", " << priority << ", " << duration <<") : ; cout << taskheappriq.size() << endl; cout << "Heap Priority Queue Architecture: " << endl; pvct->printvectorcompletetreebylevel(pvct->getroot(), 0); cout << endl; } for (int i=0; i<10; i++) { cout << "Top priority task in Heap Priority Queue : " << taskheappriq.min(); cout << endl; cout << " removemin()... "; taskheappriq.removemin(); cout << ", size after removemin(): " << taskheappriq.size() << endl; } } // end main(); DS 8-47

48 HeapPriorityQueue for OS Task Scheduling (13) DS 8-48

49 HeapPriorityQueue for OS Task Scheduling (14) DS 8-49

50 Heap-Sort Consider a priority queue with n items implemented by means of a heap the space used is O(n) methods insert and removemin take O(log n) time methods size, empty, and min take time O(1) time Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time The resulting algorithm is called heap-sort Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort DS 8-50

51 8.4 Adaptable Priority Queues Methods of the Adaptable Priority Queue ADT insert(e): Insert the entry e into PQ and return a position referring to this entry remove(p): Remove from PQ the entry referenced by position p replace(p, e): Replace with e the element associated with the entry referenced by p and return the position of the altered entry Locating Entries In order to implement the operations remove(p) and replace(p, e), and we need fast ways of locating an entry p in a priority queue (PQ) DS 8-51

52 List-based Implementation A location-aware list entry is an object storing key value position (or rank) of the item (entry, e) in the list In turn, the position (or array cell) stores the entry Back pointers (or ranks) are updated during swaps header nodes/positions trailer 2 B 4 D 5 E 8 H entries DS 8-52

53 Adaptable Priority Queue (1) template <typename E, typename C> class AdaptPriorityQueue { // adaptable priority queue protected: typedef std::list<e> ElementList; // list of elements public: // Position class definition class Position { // a position in the queue private: typename ElementList::iterator q; // a position in the list public: const E& operator*() { return *q; } // the element at this position friend class AdaptPriorityQueue; // grant access }; public: int size() const; // number of elements bool empty() const; // is the queue empty? const E& min() const; // minimum element Position insert(const E& e); // insert element void removemin(); // remove minimum void remove(const Position& p); // remove at position p Position replace(const Position& p, const E& e); // replace at position p DS 8-53

54 Adaptable Priority Queue (2) // class AdaptPriorityQueue (continued...) private: ElementList L; C isless; }; // priority queue contents // less-than comparator template <typename E, typename C> typename AdaptPriorityQueue<E,C>::Position AdaptPriorityQueue<E,C>::insert(const E& e) { typename ElementList::iterator p = L.begin(); while (p!= L.end() &&!isless(e, *p)) ++p; L.insert(p, e); Position pos; pos.q = --p; return pos; } // insert element // find larger element // insert before p // inserted position DS 8-54

55 Adaptable Priority Queue (3) template <typename E, typename C> // remove at position p void AdaptPriorityQueue<E,C>::remove(const Position& p) { L.erase(p.q); } template <typename E, typename C> // replace at position p typename AdaptPriorityQueue<E,C>::Position AdaptPriorityQueue<E,C>::replace(const Position& p, const E& e) { L.erase(p.q); // remove the old entry return insert(e); // insert replacement } DS 8-55

56 Location-Aware Entries A locator-aware entry identifies and tracks the location of its (key, value) object within a data structure Intuitive notion: Coat claim check Valet claim ticket Reservation number Main idea: Since entries are created and returned from the data structure itself, it can return location-aware entries, thereby making future updates easier DS 8-56

57 Homework DS-8 DS-8.1 Projects P-8.3 (Priority queue based on heap) DS-8.2 Projects P-8.4 (in-place heap-sort algorithm) DS-8.3 Projects P-8.7 (Priority queue for job scheduling) DS 8-57