Algorithms. Algorithms 2.4 PRIORITY QUEUES. API and elementary implementations binary heaps heapsort event-driven simulation

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2 lgorithms B SDWICK KVI WY 2.4 IIY QUUS lgorithms F U H D I I I and elementary implementations binary heaps heapsort event-driven simulation B SDWICK KVI WY

3 2.4 IIY QUUS lgorithms I and elementary implementations binary heaps heapsort event-driven simulation B SDWICK KVI WY

4 riority queue Collections. Insert and delete items. Which item to delete? Stack. emove the item most recently added. Queue. emove the item least recently added. andomized queue. emove a random item. riority queue. emove the largest (or smallest) item. operation argument insert insert insert remove max insert insert insert remove max insert insert insert remove max Q X M L return value Q X 4

5 riority queue I equirement. eneric items are Comparable. Key must be Comparable (bounded type parameter) public class MaxQ<Key extends Comparable<Key>> MaxQ() MaxQ(Key[] a) create an empty priority queue create a priority queue with given keys void insert(key v) insert a key into the priority queue Key delmax() return and remove the largest key boolean ismpty() is the priority queue empty? Key max() return the largest key int size() number of entries in the priority queue 5

6 riority queue client example Challenge. Find the largest M items in a stream of items. Fraud detection: isolate $$ transactions. S monitoring: flag most suspicious documents. Constraint. ot enough memory to store items. huge, M large % more tinybatch.txt uring 6/17/ voneumann 3/26/ Dijkstra 8/22/ voneumann 1/11/ Dijkstra 11/18/ Hoare 5/10/ voneumann 2/12/ Hoare 8/18/ uring 1/11/ hompson 2/27/ uring 2/11/ Hoare 8/12/ voneumann 10/13/ Dijkstra 9/10/ uring 10/12/ Hoare 2/10/ % java opm 5 < tinybatch.txt hompson 2/27/ voneumann 2/12/ voneumann 1/11/ Hoare 8/18/ voneumann 3/26/ sort key 7

7 riority queue client example Challenge. Find the largest M items in a stream of items. Fraud detection: isolate $$ transactions. S monitoring: flag most suspicious documents. huge, M large Constraint. ot enough memory to store items. MinQ<ransaction> pq = new MinQ<ransaction>(); use a min-oriented pq while (StdIn.hasextLine()) { String line = StdIn.readLine(); ransaction item = new ransaction(line); } pq.insert(item); if (pq.size() > M) pq.delmin(); pq contains largest M items ransaction data type is Comparable (ordered by $$) 8

8 riority queue client example Challenge. Find the largest M items in a stream of items. implementation time space sort log elementary Q M M binary heap log M M best in theory M order of growth of finding the largest M in a stream of items 9

9 riority queue elementary implementations Challenge. Implement all operations efficiently. implementation insert del max max unordered array 1 ordered array 1 1 goal log log log order of growth of running time for priority queue with items 12

10 2.4 IIY QUUS lgorithms I and elementary implementations binary heaps heapsort event-driven simulation B SDWICK KVI WY

11 Complete binary tree Binary tree. mpty or node with links to left and right binary trees. Complete tree. erfectly balanced, except for bottom level. complete tree with = 16 nodes (height = 4) roperty. Height of complete tree with nodes is!lg ". f. Height increases only when is a power of 2. 14

12 Binary heap representations Binary heap. rray representation of a heap-ordered complete binary tree. Heap-ordered binary tree. Keys in nodes. arent's key no smaller than children's keys. rray representation. Indices start at 1. ake nodes in level order. o explicit links needed! i a[i] - S I H S 1 I H 2 S I 10 H 11 Heap representations 16

13 Binary heap properties roposition. Largest key is a[1], which is root of binary tree. roposition. Can use array indices to move through tree. arent of node at k is at k/2. Children of node at k are at 2k and 2k+1. i a[i] - S I H S 1 I H 2 S I 10 H 11 Heap representations 17

14 Binary heap demo Insert. dd node at end, then swim it up. emove the maximum. xchange root with node at end, then sink it down. heap ordered H I H I 18

15 Binary heap demo Insert. dd node at end, then swim it up. emove the maximum. xchange root with node at end, then sink it down. heap ordered S I H S I H 19

16 romotion in a heap Scenario. Child's key becomes larger key than its parent's key. o eliminate the violation: xchange key in child with key in parent. epeat until heap order restored. S private void swim(int k) { while (k > 1 && less(k/2, k)) { exch(k, k/2); k = k/2; } parent of node at k is at k/2 } I 2 5 H S 5 1 violates heap order (larger key than parent) I H eter principle. ode promoted to level of incompetence. 20

17 Insertion in a heap Insert. dd node at end, then swim it up. Cost. t most 1 + lg compares. insert H public void insert(key x) { pq[++] = x; swim(); } I H S key to insert I S add key to heap violates heap order swim up S I H 21

18 Demotion in a heap Scenario. arent's key becomes smaller than one (or both) of its children's. o eliminate the violation: xchange key in parent with key in larger child. epeat until heap order restored. why not smaller child? private void sink(int k) { while (2*k <= ) { int j = 2*k; } } children of node at k are 2k and 2k+1 if (j < && less(j, j+1)) j++; if (!less(k, j)) break; exch(k, j); k = j; violates heap order (smaller than a child) 2 H I 2 S 5 S 10 I H 5 op-down reheapify (sink) ower struggle. Better subordinate promoted. 22

19 Delete the maximum in a heap Delete max. xchange root with node at end, then sink it down. Cost. t most 2 lg compares. remove the maximum key to remove S public Key delmax() { Key max = pq[1]; exch(1, --); sink(1); pq[+1] = null; return max; } prevent loitering I I S H H exchange key with root violates heap order remove node from heap S sink down H I 23

20 Binary heap: Java implementation public class MaxQ<Key extends Comparable<Key>> { private Key[] pq; private int ; public MaxQ(int capacity) { pq = (Key[]) new Comparable[capacity+1]; } fixed capacity (for simplicity) public boolean ismpty() { return == 0; } public void insert(key key) public Key delmax() { /* see previous code */ } Q ops private void swim(int k) private void sink(int k) { /* see previous code */ } heap helper functions } private boolean less(int i, int j) { return pq[i].compareo(pq[j]) < 0; } private void exch(int i, int j) { Key t = pq[i]; pq[i] = pq[j]; pq[j] = t; } array helper functions 24