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Data Structures Haim Kaplan & Uri Zwick December 2013 Binary Heaps 1.

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1 Data Structures Haim Kaplan & Uri Zwick December 2013 Binary Heaps 1

2 Required ADT Maintain items with keys subject to Insert(x,Q) min(Q) Deletemin(Q) Decrease-key(x,Q,Δ) Delete(x,Q) 2

3 Required ADT Decrease-key(x,Q,Δ): Can simulate by Delete(x,Q), insert(x-Δ,Q) We can use Red-Black trees to implement all operations in O(log n) time We want to implement Decrease-key in O(1) (amortized) time 3

4 Motivation Dijkstra’s algorithm for single source shortest path Prim’s algorithm for minimum spanning trees 4

5 Motivation Want to find the shortest route from New York to San Francisco Model the road-map with a graph 5

6 A Graph G=(V,E) V is a set of vertices E is a set of edges (pairs of vertices) 6

7 Model driving distances by weights on the edges 2 19 9 1 5 13 17 10 14 8 21 V is a set of vertices E is a set of edges (pairs of vertices) 7

8 Source and destination V is a set of vertices E is a set of edges (pairs of vertices) 2 19 9 1 5 13 17 10 14 8 21 8

9 Dijkstra’s algorithm Assume all weights are non-negative Finds the shortest path from some fixed vertex s to every other vertex 9

10 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Example 10

11 Maintain an upper bound d(v) on the shortest path to v 0 ∞ ∞ ∞ ∞ s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Example 11

12 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Maintain an upper bound d(v) on the shortest path to v 0 ∞ ∞ ∞ ∞ A node is either scanned ( in S) or labeled ( in Q) Initially S =  and Q = V 12

13 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Pick a vertex v in Q with minimum d(v) and add it to S 0 ∞ ∞ ∞ ∞ Initially S =  and Q = V 13

14 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Pick a vertex v in Q with minimum d(v) and add it to S 0 ∞ ∞ ∞ ∞ Initially S =  and Q = V v w 15 For every edge (v,w) where w in Q relax(v,w) 14

15 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 Relax(v,w) If d(v) + w(v,w) < d(w) then d(w) ← d(v) + w(v,w) π(w) ← v 0 ∞ ∞ ∞ ∞ v w For every edge (v,w) where w in Q relax(v,w) 15

16 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 ∞ ∞ ∞ ∞ S = {s} Relax(s,s 4 ) 16

17 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 ∞ ∞ ∞ Relax(s,s 4 ) S = {s} 17

18 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 ∞ ∞ ∞ Relax(s,s 4 ) Relax(s,s 3 ) S = {s} 18

19 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 ∞ ∞ 10 Relax(s,s 4 ) Relax(s,s 3 ) S = {s} 19

20 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 ∞ ∞ 10 Relax(s,s 4 ) Relax(s,s 3 ) Relax(s,s 1 ) S = {s} 20

21 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 10 Relax(s,s 4 ) Relax(s,s 3 ) Relax(s,s 1 ) S = {s} 21

22 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 10 S = {s} Pick a vertex v in Q with minimum d(v) and add it to S 22

23 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 10 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S 23

24 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 10 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 1,s 3 ) 24

25 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 9 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 1,s 3 ) 25

26 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 ∞ 9 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 1,s 3 ) Relax(s 1,s 2 ) 26

27 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 9 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 1,s 3 ) Relax(s 1,s 2 ) 27

28 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 9 S = {s,s 1 } Pick a vertex v in Q with minimum d(v) and add it to S 28

29 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 9 S = {s,s 1,s 2 } Pick a vertex v in Q with minimum d(v) and add it to S 29

30 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 9 S = {s,s 1,s 2 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 2,s 3 ) 30

31 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2 } Pick a vertex v in Q with minimum d(v) and add it to S Relax(s 2,s 3 ) 31

32 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2 } Pick a vertex v in Q with minimum d(v) and add it to S 32

33 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2,s 3 } Pick a vertex v in Q with minimum d(v) and add it to S 33

34 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2,s 3,s 4 } Pick a vertex v in Q with minimum d(v) and add it to S 34

35 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2,s 3,s 4 } When Q =  then the d() values are the distances from s The π function gives the shortest path tree 35

36 s s1s1 5 10 3 1 4 2 s3s3 s2s2 s4s4 15 8 0 5 6 8 S = {s,s 1,s 2,s 3,s 4 } When Q =  then the d() values are the distances from s The π function gives the shortest path tree 36

37 Implementation of Dijkstra’s algorithm We need to find efficiently the vertex with minimum d() in Q We need to update d() values of vertices in Q 37

38 Required ADT Maintain items with keys subject to Insert(x,Q) min(Q) Deletemin(Q) Decrease-key(x,Q,Δ) 38

39 Required ADT Insert(x,Q) min(Q) Deletemin(Q) Decrease-key(x,Q,Δ): Can simulate by Delete(x,Q), insert(x-Δ,Q) 39

40 Insert(x,Q) min(Q) Deletemin(Q) Decrease-key(x,Q,Δ): Can simulate by Delete(x,Q), insert(x-Δ,Q) How many times we do these operations ? n = |V| n n m = |E| Total running time: O(m log n) → O(m + n log n) 40

41 Heap Heap is  An almost complete binary tree  The items at the children of v are greater than the one at v Heap-ordered tree 23 7 17 3365 19 24 2629 7940 41

42 Representing a heap with an array  Get the elements from top to bottom, from left to right Array Representation 23 7 17 3365 19 24 2629 723172429261965334079 40 Q 42

43 Array Representation 23 7 17 3365 19 24 2629 723172429261965334079 40 Left(i): 2i+1 Right(i): 2i+2 Parent(i):  (i-1)/2  Q 43 012345678910

44 Heap Representation I info 29 key 11 Q array len 9 n Store associated information directly in the array Should only be used only for succinct information john

45 Heap Representation II info key 11 Q array len 9 n Store pointers to associated information 29

46 4 4 info key pos x Heap Representation III 11 Q array len 9 n Allow decrease-key operations

47 Find the minimum 23 7 17 3365 19 24 2629 723172429261965334079 40 Return Q[0] Q 47

48 Delete the minimum 23 7 17 3365 19 24 2629 723172429261965334079 40 Q 48

49 Delete the minimum 2317 3365 19 24 2629 23172429261965334079 40 Q 49

50 Delete the minimum 23 79 17 3365 19 24 2629 79231724292619653340 Q 50

51 Delete the minimum 23 17 79 3365 19 24 2629 17237924292619653340 Q 51

52 Delete the minimum 23 17 19 3365 79 24 2629 17231924292679653340 Q 52

53 Delete the minimum 23 17 19 3365 79 24 2629 17231924292679653340 Q Q[0] ← Q[size(Q)-1] Heapify-down(Q,0) size(Q) ← size(Q)-1 53

54 Heapify-down(Q,i) l ← left(i) r ← right(i) smallest ← i if l < size(Q) and Q[l] < Q[smallest] then smallest ← l if r < size(Q) and Q[r] < Q[smallest] then smallest ← r if smallest > i then Q[i] ↔ Q[smallest] Heapify-down(Q, smallest) 54

55 Inserting an item 23 17 19 3365 79 24 2629 17231924292679653340 Q Insert(15,Q) 55

56 Inserting an item 23 17 19 3365 79 24 2629 1723192429267965334015 40 Q Insert(15,Q) Q[size(Q)] ← 15 15 Heapify-up(Q,size(Q)) size(Q) ← size(Q) + 1 56

57 Heapify-up 23 17 19 3365 79 24 2615 1723192415267965334029 40 Q 29 57

58 15 17 19 3365 79 24 2623 1715192423267965334029 40 Q 29 58 Heapify-up

59 17 15 19 3365 79 24 2623 1517192423267965334029 40 Q 29 59 Heapify-up

60 Q Heapify-up(Q, i) while i > 0 and Q[i] < Q[parent(i)] do Q[i] ↔ Q[parent(i)] i ← parent(i) 17 15 19 3365 79 24 2623 4029 1517192423267965334029 60

61 Other operations Decrease-key(x,Q,Δ) Delete(x,Q) Can implement them easily using heapify 61

62 Heapsort (Williams, Floyd, 1964) Put the elements in an array Turn the array into a heap Do a delete-min and put the deleted element at the last position of the array Reverse the array, or use a max-heap 62

63 65 79 26 3323 29 24 1519 796526241915292333407 7 Q Turn an array into a heap 63 0 1 2 3 456 7 8 910

64 65 79 26 3323 29 24 1519 796526241915292333407 7 Q Heapify-down(Q,4) 64 4 Turn an array into a heap

65 65 79 26 3323 29 24 157 796526247152923334019 40 Q Heapify-down(Q,4) 65 Turn an array into a heap

66 65 79 26 3323 29 24 157 796526247152923334019 40 Q Heapify-down(Q,3) 66 3 Turn an array into a heap

67 65 79 26 3324 29 23 157 796526237152924334019 40 Q Heapify-down(Q,3) 67 Turn an array into a heap

68 65 79 26 3324 29 23 157 796526237152924334019 40 Q Heapify-down(Q,2) 68 2 Turn an array into a heap

69 65 79 15 3324 29 23 267 796515237262924334019 40 Q Heapify-down(Q,2) 69 Turn an array into a heap

70 65 79 15 3324 29 23 267 796515237262924334019 40 Q Heapify-down(Q,1) 70 1 Turn an array into a heap

71 7 79 15 3324 29 23 2665 797152365262924334019 40 Q Heapify-down(Q,1) 71 Turn an array into a heap

72 7 79 15 3324 29 23 2619 797152319262924334065 40 Q Heapify-down(Q,1) 72 Turn an array into a heap

73 7 79 15 3324 29 23 2619 797152319262924334065 40 Q Heapify-down(Q,0) 73 0 Turn an array into a heap

74 79 7 15 3324 29 23 2619 779152319262924334065 40 Q Heapify-down(Q,0) 74 Turn an array into a heap

75 19 7 15 3324 29 23 2679 719152379262924334065 40 Q Heapify-down(Q,0) 75 Turn an array into a heap

76 19 7 15 3324 29 23 2640 719152340262924337965 79 Q Heapify-down(Q,0) 76 Turn an array into a heap

77 23 7 15 3379 29 24 2619 6540 How much time does it take to build the heap this way ? Total time = 77 At most n/2 nodes heapified at height 1 At most n/4 nodes heapified at height 2 At most n/8 nodes heapified at height 3

78 23 7 15 3379 29 24 2619 6540 How much time does it take to build the heap this way ? Total time = 78 At most n/2 nodes heapified at height 1 At most n/4 nodes heapified at height 2 At most n/8 nodes heapified at height 3

79 Summary We can build the heap in linear time We still have to deletemin the elements one by one in order to sort that will take O(nlog(n)) 79

80 An example Given an array of length n, find the k smallest numbers. 80

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