Data Structures & Algorithms Binomial Queues

Data Structures & Algorithms Binomial Queues

Priority Queue None of the Priority Queue implementations supports insert, remove_max, andjoin that are all fast in the worst case. ??? lg N lg N lg N lg N lg N lg N

Binomial Queues • None of the Priority Queue implementations supports insert, remove_max, andjoin that are all fast in the worst case. Fast = log time or better • Fast join eliminates arrays • Fast remove_max requires some order Observation: • Need to relax heap restrictions to get fast join • Yet keep enough order to make others operations fast

Binomial Queues Developed by Vuillemin (1978). Defn. 9.4: A binary tree comprising nodes with keys is left heap ordered if the key in the node is >= all the keys in that node's left subtree. Big ?1 Med ?2 <Big <?1 <Med Left heap ordered complete binary tree

Binomial Queues Defn. 9.5: A power-of-2 heap is a left-heap-ordered tree consisting of a root node with an empty right subtree and a complete left subtree. The tree corresponding to a power-of-2 heap by the left-child right-sibling correspondence is a binomial tree Biggest Biggest X X Left- heap- ordered complete Z Y Y binomial tree < X binomial tree < Y binomial tree < Z Z

Power-of-2 Heap empty T 23-1 node left heap T I R O O L G H I G R L H A A Power-of-2 heap Corresponding Binomial Tree 23 nodes

Binomial Queues • Defn. 9.6: A binomial queue is a set of power-of-2 heaps, no two of the same size. • The structure of a binomial queue is determined by that queue's size, by correspondence with the binary representation of integers.

Binomial Queues Recall that a positive integer N has a binary representation, so N is an an-1 … a2 a1 a0 in binary, and N = an2n + an-12n-1 … + a222 + a12 + a0 where ai is either 0 or 1. A binomial queue with N elements has a power-of-2 heap of size 2i whenever ai = 1.

Binomial Queue 9 nodes: T M 8-node Power-of-2 heap 1-node Power-of-2 heap R O L H G I A 9 = 1001 = 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20

Binomial Queue 10 nodes: T M S 8-node Power-of-2 heap R 2-node Power-of-2 heap O L H G I A 10 = 1010 = 1 x 23 + 0 x 22 + 1 x 21 + 0 x 20

Binomial Queues Now consider joining two binomial queues, one of size N and one of size M, where N = anan-1…a1a0 and M = bnbn-1…b1b0. If at most one of ai and bi is 1 for all i,then just combine the collections. Otherwise, must combine two power-of-2 heaps to make a power-of-2 heap twice as large – like having to “carry a 1” So how do we join two power-of-2 heaps?

Binomial Queues Joining two power-of-2 heaps... larger root 23-1 node left-heap 23-1 node left-heap T U O P R O L O I H N D A G C M

Binomial Queues Joining two power-of-2 heaps... 24-1 node left-heap U T O P R O L O I H N D A G C M

Binomial Queues Do we have to sift down the root of the left sub-tree? What would that entail? How much would it cost? Can we afford to do it?

Binomial Queues Consider “fixing” left subtree as heap.... Broken 24-1 node heap P H G O F N C K D E L M A B I J May have to siftDown smaller of two roots

Binomial Queues Consider “fixing” left subtree as heap.... P H G O F N C K D E L M A B I J May have to siftDown smaller of two roots

Binomial Queues Consider “fixing” left subtree as heap.... P O G H F N C K D E L M A B I J May have to siftDown smaller of two roots

Binomial Queues Consider “fixing” left subtree as heap.... P O G N F H C K D E L M A B I J May have to siftDown smaller of two roots

Binomial Queues Consider “fixing” left subtree as heap.... P O G N F M C K D E L H A B I J May have to siftDown smaller of two roots

Binomial Queues Consider “fixing” left subtree as heap.... 24-1 node heap P O G N F M C K D E L H A B I J Great! We now have a heap on the left.

Binomial Queues But do we have to sift down the root of the left sub-tree? What would that entail? Sifting down the smaller root each time How much would it cost? k for a tree of size 2k May do for k = 1, 2, 3, 4, …, n-1 Where final tree is of size 2n Can we afford to do it? Uh … no – this is O(n2), or O(lg2 N) That's better than N, but not “fast”

Binomial Queues Great News! Left sub-tree only has to be left-heap-ordered, not heap-ordered! No need to sift down (make the power-of-2 heap with the smaller tree the left subtree) But is insert still fast? Can't be worse! Fewer constraints! Still need to give details.... How about remove-max?

Binomial Queues First Insert: Make new node a unit power-of-2 heap. If there is no “units” heap already, done! But if there is already a “units” heap, merge with it and create a “carry-one” heap of size 2. If there is already a size 2 power-of-2 heap, merge with it and make “carry-one” heap of size 4, etc. until a 0 is found (a power of 2 for which there is no heap).

Binomial Queues Insert M That was easy! H G O F C K D E A B I J 13-node binomial queue 12-node binomial queue

Binomial Queues Insert O H G F N C D E L M A B 11-node binomial queue

Binomial Queues Insert H G F N O C D E L M A B 12-node binomial queue

Binomial Queues Remove-max: Max has to be root of one of the lg N power-of-2 heaps – check them all, and remove the largest root. Now break up that left-heap ordered tree of size 2k-1 into k power-of-2 heaps. Then just join these with the remaining original power-of-2 heaps.

Binomial Queues Remove_max P H G O F N C K D E L M A B I J 16-node power-of-2 heap

Binomial Queues Remove_max P H G O F N C K D E L M A B I J

Binomial Queues Remove_max 2-node po2 heap H G O F N C K D E L M A B I J unit po2 heap 4-node po2 heap 8-node power-of-2 heap 15-node left-heap-ordered complete binary tree

Summary Use of Priority Queues – dynamic Simple implementations – fast insert and fast remove_max Heaps – fast for everything except join HeapSort – guaranteed O(N lg N) time Binomial Queues and power-of-2 heaps – fast O(lg N) for insert, find_max, remove_max, and join; relationship to binary numbers

Next Exam 1 (Wed) – Ch 1-9 (NOT 10) Radix Sort (Ch 10) Binary Search Trees (Ch 12).

Data Structures & Algorithms Binomial Queues