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CS 315 Lecture 18 Nov 15 Goals Complete heap operations review insert, deletemin decreaseKey, increaseKey heap building Heap sorting Some applications of heap

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Code for percolateDown void percolateDown( int hole ) { int child; Comparable tmp = array[ hole ]; for( ; hole * 2 <= currentSize; hole = child ) {child = hole * 2; if( child != currentSize && array[child +1] < array[ child ] ) child++; if( array[ child ] < tmp ) array[ hole ] = array[ child ]; else break; } array[ hole ] = tmp; }

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Code for percolateDown void percolateDown( int hole ) { int child; Comparable tmp = array[ hole ]; for( ; hole * 2 <= currentSize; hole = child ) { child = hole * 2; if( child != currentSize && array[ child + 1 ] < array[ child ] ) child++; if( array[ child ] < tmp ) array[ hole ] = array[ child ]; else break; } array[ hole ] = tmp; } Summary: percolateDown(j) can be called when the heap property is violated at node j in that array[j] could be > array[2j] or array[2j+1]. (All the nodes in the subtree rooted at j satisfy the heap property.) After the call, the heap property is satisfied at all the nodes in the subtree rooted at j.

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increaseKey operation Suppose the priority of a key stored in position p changes. Specifically, we want to increase the key in A[p] by quantity d. Example: 21 24 6526 32 3119 16 68 13 p = 3, d = 6 New key is 22 After changing the key to 22, call percolateDown(3) Change this to 22

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increaseKey operation Example: 21 24 6526 32 3119 22 68 13 percolateDown(3) 21 24 6526 32 3122 19 68 13

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increaseKey operation void increaseKey(int j, int d) { // change array[j] to array[j]+d and restore heap // d >= 0 is a pre-condition array[j] += d; percolateDown(j); }

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decreaseKey operation In this case, we need to percolate the changed key up the heap. Example: j = 4 and d = 12. Now, it is possible that array[j] is < array[j / 2] so we need to move the key up until the correct place is found. 21 24 6526 32 3119 16 68 13 Change to 12 13 21 6526 32 3119 16 68 12

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decreaseKey operation void decreaseKey(int j, int d) { // change array[j] to array[j] – d and restore heap // d >= 0 is a pre-condition int temp = array[j]- d; while (j > 1 && array[j/2] > temp){ array[j] = array[j/2]; j/= 2; }; array[j] = temp; } Note that this procedure is very similar to insert (shown below): void insert(const Comparble & x) { if (currentSize == array.size() – 1) array.reSize(array.size() * 2); //percolate up int hole = ++currentSize; for (; hole > 1 && x < array[ hole / 2]; hole/= 2) array[ hole ] = array[ hole / 2 ]; array[ hole ] = x;}

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Removing a key at index p in a heap Example: Remove key at index 5 21 24 6526 32 3119 22 68 13 Remove key One way to do this: apply decreaseKey to make the key the smallest. (Ex: d = 32) perform deleteMin What is the resulting heap?

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Converting an array into a heap We pass a vector items containing n (arbitrary) keys to the heap class constructor whose task it is to store these keys as a heap. Read these n keys into the heap array. call a procedure buildHeap to convert array into a heap. binaryHeap(const vector & items) { int n = items.size(); for (int i = 0; i < n; ++i) array[i+1] = items[i]; buildHeap(n); }

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Converting an array into a heap Starting with an arbitrary array, how to convert it to a heap? Solution 1: We can view the array as a heap of size 1. Successively insert keys array[2], array[3], etc. into the heap until all the keys are inserted. void buildHeap(int n) { currentSize = 1; for (int j= 2; j <= n; ++j) { insert(array[j]); } }; This takes O(n log n) time. (n operations each taking O(log n) time.)

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Alternate (faster) way to buildHeap Idea: perform percolateDown(j) starting from j = currentSize/2 to j = 1. Why does it work? Before percolateDown(j) is called, we have already performed percolateDown at 2j and 2j+1, and so in the subtree rooted at j, the only node where the heap property may not hold is j. Thus, after percolateDown(j) is called, the subtree rooted at j is a heap. The last call is percolateDown(1) so the tree rooted at 1 is a heap at the end. Code for solution 2: void buildHeap2() { n = array.size(); currentSize = n; for (int j = n/2; j > 0; --j) percolateDown(j); }

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BuildHeap using percolateDown is faster Solution 2 builds a heap in O(n) time. Analysis of this faster algorithm is presented in the text. We will skip it since it is complicated. (Involves summation of a series called the arithmetic- geometric series.)

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Heap-sorting To sort n numbers: read the numbers into an array A. call heap constructor to build a heap from A. perform n deleteMin operations. Store the successive outputs of deleteMin in the array A. A is now sorted. Total number of operations = O(n) (for builing heap) + O(n log n) (for n deleteMin operations) = O(n log n)

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Heap-sorting void sort(vector & out) { // the sorted array is output in array out out.resize(currentSize); int j=0; while (!isEmpty()) out[j++] = deleteMin(); } It is possible to collect the output in the heap array itself. As the heap size is shrinking, we can use the unused slots of the array to collect the delete min keys. The resulting heap will be sorted in descending order.

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Heap-sorting animation

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17 Application of Heaps: Interval Partitioning Interval partitioning. n Lecture j starts at s j and finishes at f j. n Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room. Ex: This schedule uses 4 classrooms to schedule 10 lectures. Time 99:301010:301111:301212:3011:3022:30 h c b a e d g f i j 33:3044:30

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18 Interval Partitioning Interval partitioning. n Lecture j starts at s j and finishes at f j. n Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room. Ex: This schedule uses only 3. Time 99:301010:301111:301212:3011:3022:30 h c a e f g i j 33:3044:30 d b

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19 Interval Partitioning: Lower Bound on Optimal Solution Def. The depth of a set of open intervals is the maximum number that contain any given time. Key observation. Number of classrooms needed depth. Ex: Depth of schedule below = 3 schedule below is optimal. Q. Does there always exist a schedule equal to depth of intervals? Time 99:301010:301111:301212:3011:3022:30 h c a e f g i j 33:3044:30 d b a, b, c all contain 9:30

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20 Interval Partitioning: Greedy Algorithm Greedy algorithm. Consider lectures in increasing order of start time: assign lecture to any compatible classroom. Implementation. O(n log n). n For each classroom k, maintain the finish time of the last job added. n Keep the classrooms in a priority queue. Sort intervals by starting time so that s 1 s 2 ... s n. d 0 for j = 1 to n { if (lecture j is compatible with some classroom k) schedule lecture j in classroom k else allocate a new classroom d + 1 schedule lecture j in classroom d + 1 d d + 1 } number of allocated classrooms

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Machine Scheduling m identical machines (drill press, cutter, sander, etc.) n jobs/tasks to be performed assign jobs to machines so that the time at which the last job completes is minimum

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Another formulation of the same problem A set of n items should be distributed to m people (or parties) so that each one gets nearly the same share as the others. Each item has a weight (value).

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Machine Scheduling Example 3 machines and 7 jobs job times are [6, 2, 3, 5, 10, 7, 14] possible schedule A B C time -----------> 6 2 3 7 13 21

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Machine Scheduling Example Finish time = 21 Objective: Find schedules with minimum finish time. A B C time -----------> 6 2 3 7 13 21

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LPT Schedules Longest Processing Time first. Jobs are scheduled in the order 14, 10, 7, 6, 5, 3, 2 Each job is scheduled on the machine on which it finishes earliest. For each machine, maintain information about when it is ready. Choice of data structure: priority queue

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LPT Schedule [14, 10, 7, 6, 5, 3, 2] A B C 14 10 713 15 16 Finish time is 16!

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LPT Schedule LPT rule does not guarantee minimum finish time schedules. Usually LPT finish time is very close to minimum finish time.

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LPT implementation using a minheap Minheap has the finish times of the m machines. Initial finish times are all 0. To schedule a job remove the machine with minimum finish time from the priority queue. Update the finish time of the selected machine and put the machine back into the priority queue.

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LPT implementation using a minheap ListPtrs LPTschedule(vector jobs, int m) { // jobs is a sorted array of job durations, m is the // number of machines ListPtrs L(m); // array of m null pointers BinaryHeap H; for (int j=0; j < m ; ++j) { pair p = new pair(j,0); H.insert(p); } for (int j=0; j < jobs.size(); ++j) { pair p = H.deleteMin(); int id = p.getId(); int ft = p.getFinishTime(); L[id].insert(j); ft+=jobs[j]; p = new pair(id, ft); H.insert(p); } }

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LPT implementation using a min-heap Sorting n jobs takes O(n log n) steps. Initialization of priority queue with m 0’s O(m) time One delete-min and one insert to schedule each job each put and remove min operation takes O(log m) time time to schedule is O(m + n log m) overall time is O(n log n + n log m + m)

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Summary of heap binary heap is implemented as an array. (Tree representation is conceptual – for understanding, not explicit.) heap data members are: array (a vector) and currentSize. efficient heap operations: insert, findmin, deletemin, decreaseKey, increaseKey – all of which can be performed in O(log n), findmin can be done in O(1) in the worst-case. an important helper function is percolateDown. It also takes O(log n) time.

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Summary of heap build-heap is the process of converting a set of keys into a heap-ordered array. build-heap can be performed in O(n log n) time by a series of n – 1 insertions. faster build-heap can be performed in O(n) time using n/2 percolate-down operations. heap-sorting takes O(n log n) in the worst-case Step 1: build a heap Step 2: perform n – 1 deleteMin operations and put the key at the end of the heap-array.

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Summary of heap Some heap operations are expensive (i.e., can take n steps in the worst-case): search(x) – search for a key x. delete(x) – delete the key x (if present) in a heap Priority queue has many applications Example: scheduling problem, shortest-path problem (Dijkstra’s algorithm) etc.

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