Hollow Heaps. Aarhus University, Denmark. Tel Aviv University, Israel. Princeton University, USA. July 7, 2015
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1 Hollow Heaps Thomas Dueholm Hansen 1 Haim Kaplan 2 Robert E. Tarjan 3 Uri Zwick 2 1 Department of Computer Science, Aarhus University, Denmark. 2 School of Computer Science, Tel Aviv University, Israel. 3 Department of Computer Science, Princeton University, USA. July 7, 2015 Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 1/21
2 Heap operations make-heap(): Make an empty heap. insert(e, k, h): Insert item e with key k in the heap h. find-min(h): Return the item with minimum key in h. delete-min(h): Remove the item with minimum key from h. decrease-key(e, k, h): Decrease the key of item e in h to k. delete(e, h): Remove the item e from h. meld(h 1, h 2 ): Return a heap containing all items in h 1 and h 2. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 2/21
3 Previous results Let n be the number of items in the heap. Binary heaps: insert, decrease-key, and delete-min each take O(log n) time in the worst case. Fibonacci heaps, Fredman and Tarjan (1987): Optimal amortized time bounds. insert : O(1) decrease-key : O(1) delete-min : O(log n) meld : O(1) Brodal (1996); Brodal, Lagogiannis, and Tarjan (2012): Optimal worst-case bounds. Numerous other heap variants have been proposed. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 3/21
4 Applications Dijkstra s algorithm (1959) for solving single source shortest paths problems with non-negative weights. Algorithms for finding undirected and directed minimum spanning trees (Gabow, Galil, Spencer, and Tarjan; 1986). Etc. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 4/21
5 Goal Goal: Find a simple and practical data structure with optimal worst-case guarantees. The design space is rich, and better algorithms may yet be found. Fibonacci heaps get enough exposure to motivate further study. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 5/21
6 Goal Goal: Find a simple and practical data structure with optimal worst-case guarantees. The design space is rich, and better algorithms may yet be found. Fibonacci heaps get enough exposure to motivate further study. Pairing heaps (Fredman, Sedgewick, Sleator, and Tarjan; 1986) outperform Fibonacci heaps in simplicity and performance. Fredman (1999): decrease-key requires Ω(log log n) amortized time. Pettie (2005): O(2 2 log log n ) upper bound for decrease-key. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 5/21
7 Heap-order The key of a node is at least as large as the key of its parent Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 6/21
8 Linking trees while preserving heap-order A B = 8 B A Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 7/21
9 Characteristics of Fibonacci heaps A Fibonacci heap is a set S of heap-ordered trees. It maintains a pointer to the node with minimum key. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 8/21
10 Characteristics of Fibonacci heaps A Fibonacci heap is a set S of heap-ordered trees. It maintains a pointer to the node with minimum key. Each node has a rank that matches its degree, and nodes are possibly marked. decrease-key may trigger cascading cuts: Mark the first unmarked ancestor, and cut all ancestors in between. 4 pointers per node: Parent, child, left sibling, and right sibling. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 8/21
11 Characteristics of Fibonacci heaps A Fibonacci heap is a set S of heap-ordered trees. It maintains a pointer to the node with minimum key. Each node has a rank that matches its degree, and nodes are possibly marked. decrease-key may trigger cascading cuts: Mark the first unmarked ancestor, and cut all ancestors in between. 4 pointers per node: Parent, child, left sibling, and right sibling. The amortized time per delete-min is bounded by the maximum rank. By induction, the number of nodes in a tree of rank k is at least F k+1 = 1 + F 1 + F F k 1 = O(1.618 k ). The maximum rank is at most log ϕ n = O(log n). Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 8/21
12 Results We introduce Hollow heaps, a simple alternative to Fibonacci heaps. The distinguishing characteristics of hollow heaps are: Each heap is a single heap-ordered directed acyclic graph (DAG) that represents all comparisons explicitly. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 9/21
13 Results We introduce Hollow heaps, a simple alternative to Fibonacci heaps. The distinguishing characteristics of hollow heaps are: Each heap is a single heap-ordered directed acyclic graph (DAG) that represents all comparisons explicitly. There are no parent pointers, and decrease-key takes O(1) time worst-case and does no cascading cuts. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 9/21
14 Results We introduce Hollow heaps, a simple alternative to Fibonacci heaps. The distinguishing characteristics of hollow heaps are: Each heap is a single heap-ordered directed acyclic graph (DAG) that represents all comparisons explicitly. There are no parent pointers, and decrease-key takes O(1) time worst-case and does no cascading cuts. Children are stored in singly linked lists. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 9/21
15 Results We introduce Hollow heaps, a simple alternative to Fibonacci heaps. The distinguishing characteristics of hollow heaps are: Each heap is a single heap-ordered directed acyclic graph (DAG) that represents all comparisons explicitly. There are no parent pointers, and decrease-key takes O(1) time worst-case and does no cascading cuts. Children are stored in singly linked lists. The heap contains hollow nodes, and delete-min takes O(log N) amortized time, where N is the total number of nodes. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 9/21
16 Results We introduce Hollow heaps, a simple alternative to Fibonacci heaps. The distinguishing characteristics of hollow heaps are: Each heap is a single heap-ordered directed acyclic graph (DAG) that represents all comparisons explicitly. There are no parent pointers, and decrease-key takes O(1) time worst-case and does no cascading cuts. Children are stored in singly linked lists. The heap contains hollow nodes, and delete-min takes O(log N) amortized time, where N is the total number of nodes. The structure is exogenous rather than endogenous: nodes hold items rather than being items. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 9/21
17 Using hollow nodes: deletion in linked lists Doubly linked list: Singly linked list: Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 10/21
18 Using hollow nodes: deletion in linked lists Doubly linked list: Singly linked list: Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 10/21
19 Using hollow nodes: deletion in linked lists Doubly linked list: Singly linked list: Deletions in a singly linked list create hollow nodes. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 10/21
20 Using hollow nodes: deletion in linked lists Doubly linked list: 4 2 Singly linked list: 4 2 Deletions in a singly linked list create hollow nodes. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 10/21
21 Using hollow nodes: deletion in linked lists Doubly linked list: 4 2 Singly linked list: 4 2 Deletions in a singly linked list create hollow nodes. They can only be removed from nodes pointing to them. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 10/21
22 Naive hollow heaps The heap is referenced by its root. meld(h 1, h 2 ): Link h 1 and h 2. insert(e, k, h): Create a node x with key k and item e. Link x and h. find-min(h): Return the item of h. decrease-key(e, k, h): Let x be the node containing e. Move e to a new node y with key k, making x hollow. Link y and h. delete-min(h): Delete h and all its immediate hollow descendants. Link the exposed full nodes. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 11/21
23 Naive hollow heaps The heap is referenced by its root. meld(h 1, h 2 ): Link h 1 and h 2. insert(e, k, h): Create a node x with key k and item e. Link x and h. find-min(h): Return the item of h. decrease-key(e, k, h): Let x be the node containing e. Move e to a new node y with key k, making x hollow. Link y and h. delete-min(h): Delete h and all its immediate hollow descendants. Link the exposed full nodes. Note: This version is inefficient. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 11/21
24 Making hollow heaps efficient Assign an integer rank to every node. Initially nodes have rank 0. Ranked link: Link two nodes of the same rank and increase the rank of the winner by 1. Unranked link: Link two nodes without changing their ranks. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 12/21
25 Making hollow heaps efficient Assign an integer rank to every node. Initially nodes have rank 0. Ranked link: Link two nodes of the same rank and increase the rank of the winner by 1. Unranked link: Link two nodes without changing their ranks. We allow hollow nodes to have up to two parents. Heap order: The key of a node is at least as large as the keys of both its parents. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 12/21
26 Hollow heaps meld(h 1, h 2 ): Link h 1 and h 2. insert(e, k, h): Create a node x with key k, item e, and rank x.rank = 0. Link x and h. find-min(h): Return the item of h. decrease-key(e, k, h): Let x be the node containing e. Move e to a new node y with key k and rank y.rank = max{0, x.rank 2}. Make x hollow and make y the second parent of x. Link y and h. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 13/21
27 Hollow heaps meld(h 1, h 2 ): Link h 1 and h 2. insert(e, k, h): Create a node x with key k, item e, and rank x.rank = 0. Link x and h. find-min(h): Return the item of h. decrease-key(e, k, h): Let x be the node containing e. Move e to a new node y with key k and rank y.rank = max{0, x.rank 2}. Make x hollow and make y the second parent of x. Link y and h. delete-min(h): Delete h and all its immediate hollow descendants that only have one parent. While possible, link the exposed full nodes with ranked links. Link the remaining nodes with unranked links. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 13/21
28 Example The heap after successive insertions of items with keys 14, 11, 5, 9, 0, 8, 10, 3, 6, 12, 13, Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
29 Example The heap after successive insertions of items with keys 14, 11, 5, 9, 0, 8, 10, 3, 6, 12, 13, Next: delete-min 14 Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
30 Example After delete-min Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
31 Example After delete-min Next: Decrease 5 to Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
32 Example After decreasing 5 to Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
33 Example After decreasing 5 to Next: Decrease 3 to 2 and 8 to 7. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
34 Example After decreasing 3 to 2 and 8 to Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
35 Example After decreasing 3 to 2 and 8 to Next: delete-min Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
36 Example After delete-min Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
37 Example After delete-min Next: delete-min Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
38 Example After delete-min Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 14/21
39 Results Let N be the number of nodes. We prove these amortized time bounds for hollow heaps: insert : O(1) decrease-key : O(1) delete-min : O(log N) meld : O(1) N is at most the number of items n plus the number of performed decrease-key operations. In many applications N = O(poly(n)), and otherwise the bound can be improved by occasionally rebuilding the heap. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 15/21
40 Eager hollow heaps Eager implementation: No second parents. Nodes are moved immediately by decrease-key. decrease-key(e, k, h): Let x be the node containing e. Move e to a new node y with key k and rank y.rank = max{0, x.rank 2}, making x hollow. Move all but the two highest ranked children of x to y. Link y and h. delete-min(h): Delete h and all its immediate hollow descendants. While possible, link the exposed full nodes with ranked links. Link the remaining nodes with unranked links. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 16/21
41 Bounding the maximum rank A full node of rank k has k ranked children with ranks 0,..., k 1. A hollow node of rank k has 2 ranked children of ranks k 1 and k 2 (or fewer if k 1). Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 17/21
42 Bounding the maximum rank A full node of rank k has k ranked children with ranks 0,..., k 1. A hollow node of rank k has 2 ranked children of ranks k 1 and k 2 (or fewer if k 1). By induction, the number of nodes in a tree of rank k is at least F k+3 1 = 1 + (F k+2 1) + (F k+1 1). k k 1 k 2 The maximum rank is at most log ϕ N = O(log N). Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 17/21
43 Analysis Potential argument: All unranked children and children of hollow nodes have 1 unit of potential each. The potential pays for linking and for deleting hollow nodes. insert: Create 1 new unranked child (+1). decrease-key: Create 1 new unranked child and 2 new children of a hollow node (+3). delete-min: Create at most O(log N) new unranked children. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 18/21
44 Analysis Potential argument: All unranked children and children of hollow nodes have 1 unit of potential each. The potential pays for linking and for deleting hollow nodes. insert: Create 1 new unranked child (+1). decrease-key: Create 1 new unranked child and 2 new children of a hollow node (+3). delete-min: Create at most O(log N) new unranked children. The analysis of lazy hollow heaps is essentially the same. In this case nodes are assigned virtual children corresponding to actual children in the eager version. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 18/21
45 Lazy hollow heap, implementation 3 pointers per node: Child, right sibling, and second parent (can be replaced by a single bit). 2 additional pointers to and from items: Nodes must hold items rather than be items; the structure is exogenous rather than endogenous. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 19/21
46 Lazy hollow heap, implementation 3 pointers per node: Child, right sibling, and second parent (can be replaced by a single bit). 2 additional pointers to and from items: Nodes must hold items rather than be items; the structure is exogenous rather than endogenous. e Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 19/21
47 Lazy hollow heap, implementation 3 pointers per node: Child, right sibling, and second parent (can be replaced by a single bit). 2 additional pointers to and from items: Nodes must hold items rather than be items; the structure is exogenous rather than endogenous. e Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 19/21
48 Lazy hollow heap, implementation 3 pointers per node: Child, right sibling, and second parent (can be replaced by a single bit). 2 additional pointers to and from items: Nodes must hold items rather than be items; the structure is exogenous rather than endogenous. e y Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 19/21
49 Lazy hollow heap, implementation Invariant: A hollow node is the first child of its second parent. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 20/21
50 Lazy hollow heap, implementation Invariant: A hollow node is the first child of its second parent. When traversing a list of children, we reach the end when either: There is no next sibling. The second parent pointer points back to the parent whose children we are traversing. Second parent pointers also show whether nodes have 1 or 2 parents. Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 20/21
51 Future work An experimental study of hollow heaps. Non-cascading Fibonacci heap without hollow nodes? Pairing heaps (Fredman, Sedgewick, Sleator, and Tarjan; 1986) outperform Fibonacci heaps in practice, although decrease-key requires Ω(log log n) time. What can be said about hollow pairing heaps? Array-based heaps outperform pointer-based heaps in practice, but the worst-case bounds are non-optimal. Can we get the best of both worlds? Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 21/21
52 Future work An experimental study of hollow heaps. Non-cascading Fibonacci heap without hollow nodes? Pairing heaps (Fredman, Sedgewick, Sleator, and Tarjan; 1986) outperform Fibonacci heaps in practice, although decrease-key requires Ω(log log n) time. What can be said about hollow pairing heaps? Array-based heaps outperform pointer-based heaps in practice, but the worst-case bounds are non-optimal. Can we get the best of both worlds? Thank you for listening! Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 21/21
53 Initial experiments Larkin, Sen, and Tarjan (2014) made an experimental study of various heaps. We used their framework to make some initial tests of hollow heaps with randomly generated benchmarks. 2,000,000 insertions and deletions, and 5,000,000 decrease-key operations. Fibonacci heap: ms Pairing heap: ms Lazy hollow heap: ms Hansen, Kaplan, Tarjan, and Zwick Hollow Heaps 22/21
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