1 / 114

Fundamental Algorithms Chapter 2: Advanced Heaps

Fundamental Algorithms Chapter 2: Advanced Heaps. Christian Scheideler WS 2012. Contents. A heap implements a priority queue . We will consider the following heaps : Binomial heap Fibonacci heap Radix heap. Priority Queue. 8. 5. 15. 12. 7. 3. Priority Queue. insert(10).

logan-irwin
Download Presentation

Fundamental Algorithms Chapter 2: Advanced Heaps

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fundamental AlgorithmsChapter 2: Advanced Heaps Christian Scheideler WS 2012 Chapter 2

  2. Contents A heapimplements a priorityqueue. We will considerthefollowingheaps: • Binomial heap • Fibonacci heap • Radix heap Chapter 2

  3. Priority Queue 8 5 15 12 7 3 Chapter 2

  4. Priority Queue insert(10) 8 5 15 12 7 3 10 Chapter 2

  5. Priority Queue min() outputs 3 (minimal element) 8 5 15 12 7 3 10 Chapter 2

  6. Priority Queue deleteMin() 8 5 15 12 7 3 10 Chapter 2

  7. Priority Queue decreaseKey(12,3) 8 5 15 12 9 7 10 Chapter 2

  8. Priority Queue delete(8) 8 5 15 9 7 10 Chapter 2

  9. Priority Queue merge(Q,Q’) 8 10 & 7 5 15 9 9 8 5 10 7 Chapter 2

  10. Priority Queue M:set ofelements in priorityqueue Every element eidentified by key(e). Operations: • M.build({e1,…,en}): M:={e1,…,en} • M.insert(e: Element): M:=M∪{e} • M.min: outputs e∈M with minimal key(e) • M.deleteMin: likeM.min, but additionallyM:=M∖{e}, for that ewith minimal key(e) Chapter 2

  11. Extended Priority Queue Additional operations: • M.delete(e: Element): M:=M∖{e} • M.decreaseKey(e:Element, ): key(e):=key(e)- • M.merge(M´): M:=M∪M´ Note: in deleteanddecreaseKeywehavedirectaccesstothecorrespondingelementandtherefore do not havetosearchfor it. Chapter 2

  12. Priority Queue • Priority Queue based on unsorted list: • build({e1,…,en}): time O(n) • insert(e): O(1) • min, deleteMin: O(n) • Priority Queue based on sorted array: • build({e1,…,en}): time O(n log n) (needed for sorting) • insert(e): O(n) (rearrange elements in array) • min, deleteMin: O(1) Better structure needed than list or array! Chapter 2

  13. Binary Heap Idee:use binary tree instead of list Preserve two invariants: • Form invariant:completebinary tree up to lowest level • Heap invariant: e1 key(e1)≤min{key(e2),key(e3)} e2 e3 Chapter 2

  14. Binary Heap Example: 4 Heap invariant 8 5 11 9 12 18 15 17 Form invariant Chapter 2

  15. Binary Heap Representation of binary tree via array: e1 e2 e3 e4 e5 e6 e7 e8 e9 e1 e2 e3 e3 e4 e5 e6 e7 e8 e9 Chapter 2

  16. Binary Heap Representation of binary tree via array: • H: Array [1..N]of Element (N>n) • Children of e in H[i]: in H[2i], H[2i+1] • Form invariant:H[1],…,H[n] occupied • Heap invariant: key(H[i])≤min{key(H[2i]),key(H[2i+1])} e1 e2 e3 e3 e4 e5 e6 e7 e8 e9 Chapter 2

  17. Binary Heap Representation of binary tree via array: insert(e): • Form invariant:n:=n+1; H[n]:=e • Heap invariant:switch ewith parent until key(H[⌊k/2⌋])≤key(e)for e in H[k]ore in H[1] e1 e2 e3 e3 e4 e5 e6 e7 e8 e9 e10 Chapter 2

  18. Insert Operation Procedure insert(e: Element)n:=n+1; H[n]:=esiftUp(n) Procedure siftUp(i: Integer)while i>1 and key(H[⌊i/2⌋])>key(H[i]) doH[i] ↔H[⌊i/2⌋]i:=⌊i/2⌋ Runtime:O(log n) Chapter 2

  19. Insert Operation - Correctness 3 3 5 8 5 8 10 9 12 15 10 9 12 15 11 18 11 18 4 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  20. Insert Operation - Correctness 3 3 5 8 5 8 10 9 12 15 10 4 12 15 11 18 4 11 18 9 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  21. Insert Operation - Correctness 3 3 5 8 4 8 10 4 12 15 10 5 12 15 11 18 9 11 18 9 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  22. Insert Operation - Correctness 3 3 4 8 4 8 10 5 12 15 10 5 12 15 11 18 9 11 18 9 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  23. Binary Heap deleteMin: • Form invariant:H[1]:=H[n]; n:=n-1 • Heap invariant: start with e in H[1].Switch e with the child with minimum key until H[k]≤min{H[2k],H[2k+1]}for the current position kof eor eis in a leaf e1 e2 e3 e3 e4 e5 e6 e7 e8 e9 Chapter 2

  24. Binary Heap Runtime: O(log n) Procedure deleteMin(): Elemente:=H[1]; H[1]:=H[n]; n:=n-1siftDown(1)return e Procedure siftDown(i: Integer)while 2i<=n do if 2i+1>n then m:=2i // m: pos. of the minimum child else if key(H[2i])<key(H[2i+1]) then m:=2i else m:=2i+1 if key(H[i])<=key(H[m]) then return // heap inv. holdsH[i] ↔H[m]; i:=m Chapter 2

  25. deleteMin Operation - Correctness 3 18 5 8 5 8 10 9 12 15 10 9 12 15 11 18 11 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  26. deleteMin Operation - Correctness 18 5 5 8 18 8 10 9 12 15 10 9 12 15 11 11 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  27. deleteMin Operation - Correctness 5 5 18 8 9 8 10 9 12 15 10 18 12 15 11 11 Invariant: H[k]is minimal w.r.t. subtree of H[k] : nodes that may violate invariant Chapter 2

  28. Binary Heap Build({e1,…,en}): • Naive implementation: via ninsert(e) operations. Runtime O(n log n) • Better implementation:Set H[i]:=eifor all i.Call siftDown(i)for i=⌊n/2⌋ down to 1.Runtime (with k=⌈log n⌉):O(1≤l<k 2l(k-l)) = O(2kj≥1j/2j) = O(n) Chapter 2

  29. Binary Heap Set H[i]:=eifor all i.Call siftDown(i)for i=⌊n/2⌋ down to 1. invariant violated Invariant: ∀j>i:H[j] min w.r.t. subtree of H[j] Chapter 2

  30. Binary Heap Runtime: • Build({e1,…,en}): O(n) • Insert(e): O(log n) • Min: O(1) • deleteMin: O(log n) Chapter 2

  31. Extended Priority Queue Additional Operations: • M.delete(e: Element): M:=M∖{e} • M.decreaseKey(e:Element, ): key(e):=key(e)- • M.merge(M´): M:=M∪M´ • delete and decreaseKey can be implemented with runtime O(log n)in binary heap (if position of eis known) • merge is expensive ((n)time)! Chapter 2

  32. Binomial Heap Binomial heap is based on binomial trees Binomial tree has to satisfy: • Form invariant(r: rank): • Heap invariant(key(Parent)≤key(Children)) r=0 r=1 r →r+1 r r Chapter 2

  33. Binomial Heap Examples of correct Binomial trees: r=1 r=2 r=3 r=0 4 4 4 4 10 10 10 7 6 6 11 8 8 20 24 Chapter 2

  34. Binomial Heap Properties of Binomial trees: • 2rnodes • maximum degreer(at root) • root deleted: Binomial tree decomposes into Binomial trees of rank 0tor-1 r=0 r=1 r →r+1 r r number of neighbors Chapter 2

  35. Binomial Heap Example for decomposition into Binomial trees of rank 0tor-1 rank 3 4 ranks 2 1 0 7 6 10 11 8 20 24 Chapter 2

  36. Binomial Heap Binomial Heap: • linked list of Binomial trees • for each rank at most 1 Binomial tree • pointer to root with minimal key 2 4 5 7 9 numbers: ranks Chapter 2

  37. Binomial Heap Example of a correct Binomial heap: min-pointer 9 3 4 15 10 7 6 11 8 20 Binomial tree ofrank r=1 24 Chapter 2

  38. Binomial Heap Merge of Binomial heaps H1and H2: like binary addition 2 H1 5 7 ranks 10100100 2 3 H2 5 + 101100 4 6 11010000 7 Chapter 2

  39. Example of Merge Operation 2 5 6 7 2 3 3 5 4 H1 numbers denotethe ranks Make sure that the heap invariant is preservedby the merging! H2 outcome Chapter 2

  40. Binomial Heap Runtime of merge operation: O(log n) because • thelargest rank in a Binomialheapwithnelementsatmostlog n, and • atmostoneBinomialtreeisallowedforeach rank value Bi:Binomial tree of rank i • insert(e): mergeexistingheapwithB0 (containingonlyelemente) • min: use min-pointer, time O(1) • deleteMin: let the min-pointer pointtotherootofBi.Deleting theroot in Biresults in BinomialtreesB0,…,Bi-1.These have to be merged back into Binomial heap. Thus, theinsertanddeleteMinoperationscanbereducedtothemergeoperation, whichimplies a runtimeofO(log n). Chapter 2

  41. Exampleof Insert Operation Insert(8): 9 3 4 8 & 15 10 6 11 Chapter 2

  42. Exampleof Insert Operation Outcomeof Insert(8): 3 15 4 8 10 9 6 11 Chapter 2

  43. Binomial Heap • decreaseKey(e,): perform siftUp operation in Binomial tree starting with e, update min-pointer. Time O(log n) • delete(e): (min-pointer does not point to e) set key(e):= -and perform siftUpoperation starting with e until eis in a root; then continue like in deleteMinwhenremovinge (but withoutupdatingthe min-pointer!).Time O(log n) Chapter 2

  44. ExampleofdecreaseKey decreaseKey(24,19): 9 3 4 15 10 7 6 11 8 20 24 Chapter 2

  45. ExampleofdecreaseKey OutcomeofdecreaseKey(24,19): 9 3 4 15 10 5 6 11 8 7 20 Chapter 2

  46. Fibonacci Heap • Based on Binomial trees, but it allows lazy merge and lazy delete. • Lazy merge: no merging of Binomial trees of the same rank during merge, only concatenation of the two lists • Lazy delete: creates incomplete Binomial trees Chapter 2

  47. Fibonacci Heap Tree in a Binomial heap: 4 7 6 10 8 20 11 24 Chapter 2

  48. Fibonacci Heap Tree in a Fibonacci heap: 4 Every parent only knows first and last child of list 7 6 10 Every childknows itsparent 8 20 11 List of siblings 24 Chapter 2

  49. Fibonacci Heap Every node remembers: • Element stored in the node • Parent node • Left and right sibling in sibling list • First and last child in its list of children • Its rank (number of children) Thus, every node stores ≤5 pointers. Chapter 2

  50. 2 2 5 5 7 7 2 2 3 3 5 5 Fibonacci Heap Lazy merge of results in min & min Chapter 2

More Related