1 CSE 326 Data Structures Part 6: Priority Queues, AKA Heaps Henry Kautz Autumn 2002

2 Not Quite Queues Consider applications –ordering CPU jobs –searching for the exit in a maze –emergency room admission processing Problems? –short jobs should go first –most promising nodes should be searched first –most urgent cases should go first

3 Priority Queue ADT Priority Queue operations –create –destroy –insert –deleteMin –is_empty Priority Queue property: for two elements in the queue, x and y, if x has a lower priority value than y, x will be deleted before y F(7) E(5) D(100) A(4) B(6) insert deleteMin G(9)C(3)

4 Applications of the Priority Q Hold jobs for a printer in order of length Store packets on network routers in order of urgency Simulate events Anything greedy

5 Discrete Event Simulation An event is a pair (x,t) where x describes the event and t is time it should occur A discrete event simulator (DES) maintains a set S of events which it intends to simulate in time order repeat { Find and remove (x 0,t 0 ) from S such that t 0 is minimum; Do whatever x 0 says to do, in the process new events (x 2,t 2 )…(x k,t k ) may be generated; Insert the new events into S; }

6 Emergency Room Simulation Patient arrive at time t with injury of criticality C –If no patients waiting and a free doctor, assign them to doctor and create a future departure event; else put patient in the Criticality priority queue Patient departs at time t –If someone in Criticality queue, pull out most critical and assign to doctor; create a future departure event time queue arrive(t,c) criticality (triage) queue assign patient to doctor patient generator depart(t) arrive(t,c)

7 Naïve Priority Queue Data Structures Unsorted list: –insert: –deleteMin: Sorted list: –insert: –deleteMin:

8 BST Tree Priority Queue Data Structure Regular BST: –insert: –deleteMin: AVL Tree: –insert: –deleteMin: Can we do better?

9 Binary Heap Priority Q Data Structure Heap-order property –parent’s key is less than children’s keys –result: minimum is always at the top Structure property –complete tree with fringe nodes packed to the left –result: depth is always O(log n); next open location always known How do we find the minimum?

Nifty Storage Trick Calculations: –child: –parent: –root: –next free: 0

Nifty Storage Trick Calculations: –child: left = 2*node right=2*node+1 –parent: floor(node/2) –root: 1 –next free: length+1 0

12 DeleteMin pqueue.deleteMin()

13 Percolate Down

14 DeleteMin Code Comparable deleteMin(){ x = A[1]; A[1]=A[size--]; percolateDown(1); return x; } percolateDown(int hole) { tmp=A[hole]; while (2*hole <= size) { left = 2*hole; right = left + 1; if (right <= size && A[right] < A[left]) target = right; else target = left; if (A[target] < tmp) { A[hole] = A[target]; hole = target; } else break; } A[hole] = tmp; } runtime: Trick to avoid repeatedly copying the value at A[1] Move down

15 Insert pqueue.insert(3) 3

16 Percolate Up

17 Insert Code void insert(Comparable x) { // Efficiency hack: we won’t actually put x // into the heap until we’ve located the position // it goes in. This avoids having to copy it // repeatedly during the percolate up. int hole = ++size; // Percolate up for( ; hole>1 && x < A[hole/2] ; hole = hole/2) A[hole] = A[hole/2]; A[hole] = x; } runtime:

18 Performance of Binary Heap In practice: binary heaps much simpler to code, lower constant factor overhead Binary heap worst case Binary heap avg case AVL tree worst case BST tree avg case Insert O(log n)O(1) percolates 1.6 levels O(log n) Delete Min O(log n)

19 Changing Priorities In many applications the priority of an object in a priority queue may change over time –if a job has been sitting in the printer queue for a long time increase its priority –unix “renice” Must have some (separate) way of find the position in the queue of the object to change (e.g. a hash table)

20 Other Priority Queue Operations decreaseKey –Given the position of an object in the queue, increase its priority (lower its key). Fix heap property by: increaseKey –given the position of an an object in the queue, decrease its priority (increase its key). Fix heap property by: remove –given the position of an an object in the queue, remove it. Do increaseKey to infinity then …

21 BuildHeap Task: Given a set of n keys, build a heap all at once Approach 1: Repeatedly perform Insert(key) Complexity:

22 BuildHeap Floyd’s Method pretend it’s a heap and fix the heap-order property! buildHeap(){ for (i=size/2; i>0; i--) percolateDown(i); }

23 Build(this)Heap

24 Finally…

25 Complexity of Build Heap Note: size of a perfect binary tree doubles (+1) with each additional layer At most n/4 percolate down 1 level at most n/8 percolate down 2 levels at most n/16 percolate down 3 levels… O(n)

26 Heap Sort Input: unordered array A[1..N] 1.Build a max heap (largest element is A[1]) 2.For i = 1 to N-1: A[N-i+1] = Delete_Max()

27 Properties of Heap Sort Worst case time complexity O(n log n) –Build_heap O(n) –n Delete_Max’s for O(n log n) In-place sort – only constant storage beyond the array is needed

28 Thinking about Heaps Observations –finding a child/parent index is a multiply/divide by two –operations jump widely through the heap –each operation looks at only two new nodes –inserts are at least as common as deleteMins Realities –division and multiplication by powers of two are fast –looking at one new piece of data terrible in a cache line –with huge data sets, disk accesses dominate

Solution: d-Heaps Each node has d children Still representable by array Good choices for d: –optimize performance based on # of inserts/removes –choose a power of two for efficiency –fit one set of children in a cache line –fit one set of children on a memory page/disk block

30 Coming Up Mergeable heaps –Leftist heaps –Skew heaps –Binomial queues Read Weiss Ch. 6 Midterm results

31 New Operation: Merge Merge(H1,H2): Merge two heaps H1 and H2 of size O(N). –E.g. Combine queues from two different sources to run on one CPU. 1.Can do O(N) Insert operations: O(N log N) time 2.Better: Copy H2 at the end of H1 (assuming array implementation) and use Floyd’s Method for BuildHeap. Running Time: O(N) Can we do even better? (i.e. Merge in O(log N) time?)

32 Binomial Queues Binomial queues support all three priority queue operations Merge, Insert and DeleteMin in O(log N) time Idea: Maintain a collection of heap-ordered trees –Forest of binomial trees Recursive Definition of Binomial Tree (based on height k): –Only one binomial tree for a given height –Binomial tree of height 0 = single root node –Binomial tree of height k = B k = Attach B k-1 to root of another B k-1

33 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

34 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

35 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

36 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

37 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

38 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

39 Building a Binomial Tree To construct a binomial tree B k of height k: 1.Take the binomial tree B k-1 of height k-1 2.Place another copy of B k-1 one level below the first 3.Attach the root nodes Binomial tree of height k has exactly 2 k nodes (by induction) B 0 B 1 B 2 B 3

40 Why Binomial? Why are these trees called binomial? –Hint: how many nodes at depth d? B 0 B 1 B 2 B 3

41 Why Binomial? Why are these trees called binomial? –Hint: how many nodes at depth d? Number of nodes at different depths d for B k = [1], [1 1], [1 2 1], [ ], … Binomial coefficients of (a + b) k = k!/((k-d)!d!) B 0 B 1 B 2 B 3

42 Definition of Binomial Queues 3 Binomial Queue = “forest” of heap-ordered binomial trees B 0 B 2 B 0 B 1 B 3 Binomial queue H1 5 elements = 101 base 2  B 2 B 0 Binomial queue H2 11 elements = 1011 base 2  B 3 B 1 B 0

43 Binomial Queue Properties Suppose you are given a binomial queue of N nodes 1.There is a unique set of binomial trees for N nodes 2.What is the maximum number of trees that can be in an N-node queue? –1 node  1 tree B 0 ; 2 nodes  1 tree B 1 ; 3 nodes  2 trees B 0 and B 1 ; 7 nodes  3 trees B 0, B 1 and B 2 … –Trees B 0, B 1, …, B k can store up to … + 2 k = 2 k+1 – 1 nodes = N. –Maximum is when all trees are used. So, solve for (k+1). –Number of trees is  log(N+1) = O(log N)

44 Binomial Queues: Merge Main Idea: Merge two binomial queues by merging individual binomial trees –Since B k+1 is just two B k ’s attached together, merging trees is easy Steps for creating new queue by merging: 1.Start with B k for smallest k in either queue. 2.If only one B k, add B k to new queue and go to next k. 3.Merge two B k ’s to get new B k+1 by making larger root the child of smaller root. Go to step 2 with k = k + 1.

45 Example: Binomial Queue Merge H1: H2:

46 Example: Binomial Queue Merge H1: H2:

47 Example: Binomial Queue Merge H1: H2:

48 Example: Binomial Queue Merge H1: H2:

49 Example: Binomial Queue Merge H1: H2:

50 Example: Binomial Queue Merge H1: H2:

51 Binomial Queues: Merge and Insert What is the run time for Merge of two O(N) queues? How would you insert a new item into the queue?

52 Binomial Queues: Merge and Insert What is the run time for Merge of two O(N) queues? –O(number of trees) = O(log N) How would you insert a new item into the queue? –Create a single node queue B 0 with new item and merge with existing queue –Again, O(log N) time Example: Insert 1, 2, 3, …,7 into an empty binomial queue

53 Insert 1,2,…,7 1

54 Insert 1,2,…,7 1 2

55 Insert 1,2,…,

56 Insert 1,2,…,

57 Insert 1,2,…,

58 Insert 1,2,…,

59 Insert 1,2,…,

60 Insert 1,2,…,

61 Binomial Queues: DeleteMin Steps: 1.Find tree B k with the smallest root 2.Remove B k from the queue 3.Delete root of B k (return this value); You now have a new queue made up of the forest B 0, B 1, …, B k-1 4.Merge this queue with remainder of the original (from step 2) Run time analysis: Step 1 is O(log N), step 2 and 3 are O(1), and step 4 is O(log N). Total time = O(log N) Example: Insert 1, 2, …, 7 into empty queue and DeleteMin

62 Insert 1,2,…,

63 DeleteMin

64 Merge

65 Merge

66 Merge

67 Merge DONE!

68 Implementation of Binomial Queues Need to be able to scan through all trees, and given two binomial queues find trees that are same size –Use array of pointers to root nodes, sorted by size –Since is only of length log(N), don’t have to worry about cost of copying this array –At each node, keep track of the size of the (sub) tree rooted at that node Want to merge by just setting pointers –Need pointer-based implementation of heaps DeleteMin requires fast access to all subtrees of root –Use First-Child/Next-Sibling representation of trees

69 Efficient BuildHeap for Binomial Queues Insert one at a time - O(n log n) Better algorithm: –Start with each element as a singleton tree –Merge trees of size 1 –Merge trees of size 2 –Merge trees of size 4 Complexity:

70 Other Mergeable Priority Queues: Leftist and Skew Heaps Leftist Heaps: Binary heap-ordered trees with left subtrees always “longer” than right subtrees –Main idea: Recursively work on right path for Merge/Insert/DeleteMin –Right path is always short  has O(log N) nodes –Merge, Insert, DeleteMin all have O(log N) running time (see text) Skew Heaps: Self-adjusting version of leftist heaps (a la splay trees) –Do not actually keep track of path lengths –Adjust tree by swapping children during each merge –O(log N) amortized time per operation for a sequence of M operations See Weiss for details…

71 Coming Up For Wednesday: Read Weiss on Leftist Heaps and Skew Heaps How do they work? How does their complexity compare to Binomial Queues? Which seems easiest to implement? Which is likely to have lowest overhead?

72 Leftist Heaps An alternative heap structure that also enables fast merges Based on binary trees rather than k-ary trees

73 Idea: Hang a New Tree = ? 10 Now, just percolate down!

74 Idea: Hang a New Tree = ? Now, just percolate down!

75 Idea: Hang a New Tree = ? Now, just percolate down!

76 Idea: Hang a New Tree = Now, just percolate down! Note: we just gave up the nice structural property on binary heaps!

77 Problem? Need some other kind of balance condition… = 2 ?

78 Leftist Heaps Idea: make it so that all the work you have to do in maintaining a heap is in one small part Leftist heap: –almost all nodes are on the left –all the merging work is on the right

79 the null path length (npl) of a node is the number of nodes between it and a null in the tree Random Definition: Null Path Length npl(null) = -1 npl(leaf) = 0 npl(single-child node) = another way of looking at it: npl is the height of complete subtree rooted at this node 0

80 Leftist Heap Properties Heap-order property –parent’s priority value is  to childrens’ priority values –result: minimum element is at the root Leftist property –null path length of left subtree is  npl of right subtree –result: tree is at least as “heavy” on the left as the right Are leftist trees complete? Balanced?

81 Leftist tree examples NOT leftistleftist every subtree of a leftist tree is leftist, comrade!

82 Right Path in a Leftist Tree is Short If the right path has length at least r, the tree has at least 2 r - 1 nodes Proof by induction Basis: r = 1. Tree has at least one node: = 1 Inductive step: assume true for r’ < r. The right subtree has a right path of at least r - 1 nodes, so it has at least 2 r nodes. The left subtree must also have a right path of at least r - 1 (otherwise, there is a null path of r - 3, less than the right subtree). Again, the left has 2 r nodes. All told then, there are at least: 2 r r = 2 r - 1 So, a leftist tree with at least n nodes has a right path of at most log n nodes

83 Merging Two Leftist Heaps merge(T 1,T 2 ) returns one leftist heap containing all elements of the two (distinct) leftist heaps T 1 and T 2 a L1L1 R1R1 b L2L2 R2R2 merge T1T1 T2T2 a < b a L1L1 recursive merge b L2L2 R2R2 R1R1

84 Merge Continued a L1L1 R’R’ R ’ = Merge(R 1, T 2 ) a R’R’ L1L1 npl(R ’ ) > npl(L 1 ) constant work at each merge; recursively traverse RIGHT right path of each tree; total work = O(log n)

85 Operations on Leftist Heaps merge with two trees of total size n: O(log n) insert with heap size n: O(log n) –pretend node is a size 1 leftist heap –insert by merging original heap with one node heap deleteMin with heap size n: O(log n) –remove and return root –merge left and right subtrees merge

86 Example merge ? merge 10 5 ? 0 merge

87 Sewing Up the Example ? ? ? Done?

88 Finally…

89 Skew Heaps Problems with leftist heaps –extra storage for npl –two pass merge (with stack!) –extra complexity/logic to maintain and check npl Solution: skew heaps –blind adjusting version of leftist heaps –amortized time for merge, insert, and deleteMin is O(log n) –worst case time for all three is O(n) –merge always switches children when fixing right path –iterative method has only one pass What do skew heaps remind us of?

90 Merging Two Skew Heaps a L1L1 R1R1 b L2L2 R2R2 merge T1T1 T2T2 a < b a L1L1 merge b L2L2 R2R2 R1R1 Notice the old switcheroo!

91 Example merge merge merge

92 Skew Heap Code void merge(heap1, heap2) { case { heap1 == NULL: return heap2; heap2 == NULL: return heap1; heap1.findMin() < heap2.findMin(): temp = heap1.right; heap1.right = heap1.left; heap1.left = merge(heap2, temp); return heap1; otherwise: return merge(heap2, heap1); }

93 Comparing Heaps Binary Heaps d-Heaps Binomial Queues Leftist Heaps Skew Heaps

94 Summary of Heap ADT Analysis Consider a heap of N nodes Space needed: O(N) –Actually, O(MaxSize) where MaxSize is the size of the array –Pointer-based implementation: pointers for children and parent Total space = 3N + 1 (3 pointers per node + 1 for size) FindMin: O(1) time; DeleteMin and Insert: O(log N) time BuildHeap from N inputs: What is the run time? –N Insert operations = O(N log N) –O(N): Treat input array as a heap and fix it using percolate down Thanks, Floyd! Mergable Heaps: Binomial Queues, Leftist Heaps, Skew Heaps