Priority QueuesPriority queueA priority queue ADTEvaluating implementationsArray implementationsLinked list implementationsBinary tree implementationsHeap implementationArray representation of a heapUsing the heapCommentsJava 5 java.util.PriorityQueueThe EndPriority Queues2Priority queueA stack is first in, last outA queue is first in, first outA priority queue is least-first-outThe “smallest” element is the first one removed(You could also define a largest-first-out priority queue)The definition of “smallest” is up to the programmer (for example, you might define it by implementing Comparator or Comparable)If there are several “smallest” elements, the implementer must decide which to remove firstRemove any “smallest” element (don’t care which)Remove the first one added3A priority queue ADTHere is one possible ADT:PriorityQueue(): a constructorvoid add(Comparable o): inserts o into the priority queueComparable removeLeast(): removes and returns the least elementComparable getLeast(): returns (but does not remove) the least elementboolean isEmpty(): returns true iff emptyint size(): returns the number of elementsvoid clear(): discards all elements4Evaluating implementationsWhen we choose a data structure, it is important to look at usage patternsIf we load an array once and do thousands of searches on it, we want to make searching fast—so we would probably sort the arrayIf we load a huge array and expect to do only a few searches, we probably don’t want to spend time sorting the arrayFor almost all uses of a queue (including a priority queue), we eventually remove everything that we addHence, when we analyze a priority queue, neither “add” nor “remove” is more important—we need to look at the timing for “add + remove”5Array implementationsA priority queue could be implemented as an unsorted array (with a count of elements)Adding an element would take O(1) time (why?)Removing an element would take O(n) time (why?)Hence, adding and removing an element takes O(n) timeThis is an inefficient representationA priority queue could be implemented as a sorted array (again, with a count of elements)Adding an element would take O(n) time (why?)Removing an element would take O(1) time (why?)Hence, adding and removing an element takes O(n) timeAgain, this is inefficient6Linked list implementationsA priority queue could be implemented as an unsorted linked listAdding an element would take O(1) time (why?)Removing an element would take O(n) time (why?)A priority queue could be implemented as a sorted linked listAdding an element would take O(n) time (why?)Removing an element would take O(1) time (why?)As with array representations, adding and removing an element takes O(n) timeAgain, these are inefficient implementations7Binary tree implementationsA priority queue could be represented as a (not necessarily balanced) binary search treeInsertion times would range from O(log n) to O(n) (why?)Removal times would range from O(log n) to O(n) (why?)A priority queue could be represented as a balanced binary search treeInsertion and removal could destroy the balanceWe need an algorithm to rebalance the binary treeGood rebalancing algorithms require only O(log n) time, but are complicated8Heap implementationA priority queue can be implemented as a heapIn order to do this, we have to define the heap propertyIn Heapsort, a node has the heap property if it is at least as large as its childrenFor a priority queue, we will define a node to have the heap property if it is as least as small as its children (since we are using smaller numbers to represent higher priorities)128 3Heapsort: Blue node has the heap property38 12Priority queue: Blue node has the heap property9Array representation of a heapLeft child of node i is 2*i + 1, right child is 2*i + 2Unless the computation yields a value larger than lastIndex, in which case there is no such childParent of node i is (i – 1)/2Unless i == 01214186833 12 6 18 14 8 0 1 2 3 4 5 6 7 8 9 10 11 12lastIndex = 510Using the heapTo add an element:Increase lastIndex and put the new value thereReheap the newly added nodeThis is called up-heap bubbling or percolating upUp-heap bubbling requires O(log n) timeTo remove an element:Remove the element at location 0Move the element at location lastIndex to location 0, and decrement lastIndex Reheap the new root node (the one now at location 0)This is called down-heap bubbling or percolating downDown-heap bubbling requires O(log n) timeThus, it requires O(log n) time to add and remove an element11CommentsA priority queue is a data structure that is designed to return elements in order of priorityEfficiency is usually measured as the sum of the time it takes to add and to remove an elementSimple implementations take O(n) timeHeap implementations take O(log n) timeBalanced binary tree implementations take O(log n) timeBinary tree implementations, without regard to balance, can take O(n) (linear) timeThus, for any sort of heavy-duty use, heap or balanced binary tree implementations are better12Java 5 java.util.PriorityQueueJava 5 finally has a PriorityQueue class, based on heapsPriorityQueue<E> queue = new PriorityQueue<E>();boolean add(E o)boolean remove(Object o)boolean offer(E o)E peek()boolean poll()void clear()int size()13The
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