Priority Queues. Fibonacci Heap

Transcription

1 ibonacci Heap hans to Sartaj Sahni for the original version of the slides Operation mae-heap insert find-min delete-min union decrease-ey delete Priority Queues Lined List Binary Binomial Heaps ibonacci * Relaxed is-empty Dijstra/Prim mae-heap V insert V delete-min E decrease-ey O( V ) O( E log V ) O( E + V log V ) ibonacci heaps Similar to binomial heaps, consists of a collection of trees, each arranged in a heap-order (each node is smaller than each of its children) Unlie binomial heaps, can have many trees of the same cardinality, and a tree does not have to have exactly i nodes. General Structure Very similar to Bionomail heaps Main structure: A collection of trees, each in a heap-order. All root are stored in a doubly connected list, called the roots-list. Every node points to one of its childrens. All the children are stored in a doubly connected list, called the sibling-list. A pointer min(h) always points to the min element. roots list sibling list Main idea laziness is welcomed. ry to postpone doing the hard wor, until no other solution wors. 7 ode Structure Very similar to Bionomail heaps Each node v stores its degree, a points to its parent, a points to a child, data, Pointers to left and right sibling used for circular doubly lined list of siblings, called the sibling list. ode Structure Each node v stores its also stores a flag ChildCut a flag (not existing in binomial heaps) rue if v has lost a child since v became a child of its current parent. We say that v is mared. Set to false by remove min, which is the only operation that maes one node a child of another. Undefined for a root node (not used) More in next slide

2 ibonacci Heap Representation roots list A Potential unction 7 sibling list Some nodes would be mared (to be explained later) We use the potential functions for the heap H Φ(H) = t(h) + m(h) Where t(h) is the number of trees in H And m(h) is the number of mared nodes in H. Insert(x) Create a new tree consisting of a single node v whose ey is x, Add v to the roots list. Actual time w i needed for the operation is umber of trees increased by. Changes in potential function Φ(H )-Φ(H)= Φ(H) = t(h ) + m(h )- t(h) - m(h)= So the amortized wor a i = w i + Φ(H) = DecreaseKey(theode, theamount) theode Decreased by 7 If theode is not a root and new ey < parent ey, remove subtree rooted at theode from its sibling list. Insert theode into roots-list. Perform cascading_cut from parent(theode ) (described later) Example without ChildCut DecreaseKey(theode, theamount) 7 Cascading Cut When theode is cut out of its sibling list in a decrease ey operation, follow path from parent of theode upward toward the root. Encountered nodes (other than root) with ChildCut = true are cut from their sibling lists and inserted into roots-list. Stop at first node with ChildCut = false. or this node, set ChildCut = true. (since it just lost exactly one child) In other words, if a node lost two children since it became a child, it must move itself from the the parent to the roots-list.

3 7 7 theode Decrease ey by. 7 7 ote a node that moves to the root lists loose its mar (becomes unmared). 7 Actual time complexity of the cascading_cut of a path of length is Θ(+) ( can be Θ(h) in the worst case, where h is the height ) Assume we specify the time of an elementary operations, so that this time w i is +. ote that the number of trees increases by +, and the number of mared nodes decreases by either or + Amortized time ote that the number of mared nodes decreases by or +, and the number of trees increased by +. Let H to be H denote the heap before and after the Decrease_min operation, then t(h ) = t(h)++ and m(h )=m(h)- he change in the potential function is (denoting H to be H after the Decrease_min ) is Φ(H ) - Φ(H) = ( t(h ) + m(h ) ( t(h) + m(h) ) = And a i =w i + Φ(H ) - Φ(H) = +(-+)= -+

4 Deletion of a node v Perform DecreaseKey(v) to value - Extract_min Perform ExtractMin(H) seen next. Remember - there is a pointer (min[h] ) pointing to the min. Set theode= min[h] Moved all children of theode to the the roots-list. his is done by merging their sibling list with the root list change their parent pointer to ULL If any of them was mared, set it to be unmared. Remove theode from its sibling list. ree theode. Perform Consolidate( H ) - merging trees. /* his is a good time to reduce the number of trees */ Extract_min example (.) Extract_min example () 7 7 ext comes the consolidation ime analysis for this part of the Extract_min operation Let deg(v) is the number of children of v. Lemma: (CLRS.3): he number of nodes in a tree the tree rooted at v is deg(v). deg(v) Conclusion : deg(v) =O(log n), for every node v. he actual time w i needed for disconnecting v from its children and adding them to the roots list O(log n)=deg(v) he amortized time for this operation is at most t(h )-t(h) + (m(h )-m(h)) = O(d(v)) = O(log n) (the number of has increased by deg(v), the number of mared nodes had decreased by deg(v) +) So the amortized time a i for this part is O(log n) Union of two trees. (eed for Consolidation) (Similar (but not identical) operation was seen in the binomial heaps) Degree of a tree is defined as the degree of the root of the tree. Given two trees with the same degree of their roots, connect the root of one as a child of the other root. here is always a way to do so while maintaining the heap order: he root with larger ey becomes the child of the smaller root Point of potential confusion: or Binomial heaps, trees have the same size iff they have the same degree. ot true here

5 Extract_min cont: Consolidation. Each extract_min is followed by the consolidation operation: his operation repeatedly joins trees with same degree, using the treeunion operation: Repeatedly pic two trees with the same degree, and merge them: but trees are not sorted by degree, (as oppose to Binomial heaps) and there are many of them how can this be done efficiently???? (on board) inish when no two trees with the same degree exist. Recover the new minimum while doing so. Actual ime w i proportional to the number of trees (since every operation reduces the number of tress by one, and taes a constant time). ime analysis for consolidation he consolidation taes actual time t(h) time. In H, (after the consolidation) there is at most one tree for each possible degree of its root. ollowed from conclusion, t(h ) =O(log n). he number of mared nodes is not changed. Φ(H ) - Φ(H) = ( t(h ) + m(h ) ( t(h) + m(h) ) = t(h )-t(h) = O(log n)- t(h) he amortized wor is therefore t(h)+ ( O(log n) -t(h) ) = O(log n) ime analysis for Delete Deletion consists of first DecreaseKey (amortized time O() ) and then ExtractMin (amortized time O(log n) ) otal amortized wor: O(log n) oward proving lemma CLRS.3 Lemma:.3: he number of nodes in a tree rooted at v is deg(v). deg(v) Let =, = and + = + + Lemma.: Proof by induction. + = + Lemma.: Let x be a root, and let y,, y denote its children, in the order they joined x. hen deg[y i ] i-. (,3 ). Proof: When y i joined x, its degree was exactly i- Since y i jointed, its degree might have decrease by. i ibonacci numbers Let =, = and + = + + hen -. - Here = ( + ) / Easily chec by induction Proving lemma CLRS.3 Let s denote the minimum number of nodes at a tree of degree. Lemma (.3) : s - Proof: Let y,, y denote its children of a node x, in the order they joined x. hen (assuming by induction the claim holds for all i<) s + si + i = + i = + Showed all we wanted to ibonacci heaps