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(J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 1 / 18
Heaps
Heaps
An array A[1..n] is called a heap if for all i ≥ 1,
A[i ] ≥ A[2i ] if 2i ≤ n
and
A[i ] ≥ A[2i + 1] if 2i + 1 ≤ n
16
14 10
8 7 9 3
2 4 1
1
2 3
4 5 6 7
8 9 10
16 14 10 8 7 9 3 2 4 1
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 2 / 18
Heaps
Root of the tree: A[1]
Consider a node with index i :
Parent of i has index bi/2c: parent(i) = bi/2cLeft child of i has index 2i : left(i) = 2i
Right child of i has index 2i + 1: right(i) = 2i + 1
A[1..n] is a heap if for all 1 < i ≤ n, A[parent(i)] ≥ A[i ].
16
14 10
8 7 9 3
2 4 1
1
2 3
4 5 6 7
8 9 10
16 14 10 8 7 9 3 2 4 1
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 3 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h
(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1
< 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)c
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
What is the height of a heap?
h
level 0 (height h): 1 nodelevel 1 (height h− 1): 2 nodeslevel 2 (height h− 2): 4 nodes
level h− 1 (height 1): 2h−1 nodes
level h (height 0): between 1 and 2h nodes
1 + 2 + 22 + ...+ 2h−1 + 1 ≤n ≤ 1 + 2 + 22 + ...+ 2h−1 + 2h(2h − 1
)+ 1 ≤n ≤ 2h+1 − 1 < 2h+1
2h ≤n < 2h+1
h ≤ log2(n) < h + 1
h =blog2(n)cJ.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 4 / 18
Heaps
From now on: we consider an array A[1..N] with an integer n where1 ≤ n ≤ N.
A[1..n] is a heap which contains n integers and A[(n + 1)..N] containsgarbage.
heap garbage
1 n (n + 1) N
1. Finding a maximum elementMaximum(A): returns A[1]
This takes O(1) time.
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 5 / 18
Heaps
2. Increase the value of A[i ] to x .
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14 10
8 7 9 3
2 4 1
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
2. Increase the value of A[i ] to x .
16
14 10
8 7 9 3
2 4 1
1
2 3
4 5 6 7
8 9 10
Increase key(A, 9, 15)
Increase this key to 15
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
2. Increase the value of A[i ] to x .
16
14 10
8 7 9 3
2 15 1
1
2 3
4 5 6 7
8 9 10
Increase key(A, 9, 15)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
2. Increase the value of A[i ] to x .
16
14 10
8
7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Increase key(A, 9, 15)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
2. Increase the value of A[i ] to x .
16
14
10
8
7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Increase key(A, 9, 15)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
2. Increase the value of A[i ] to x .
16
14
10
8
7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Increase key(A, 9, 15)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 6 / 18
Heaps
Algorithm 1 Increase key(A, i , x)
Require: x ≥ A[i ]1: A[i ] = x2: // The following while-loop is sometimes called percolate.3: while i > 1 and A[parent(i)] < A[i ] do4: swap A[i ] and A[parent(i)]5: i = parent(i)6: end while
This takes O(h) = O(log(n)) time.
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 7 / 18
Heaps
3. Insert a new value in A.
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7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Insert(A, 13)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 8 / 18
Heaps
3. Insert a new value in A.
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7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Insert(A, 13)
−∞11
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 8 / 18
Heaps
3. Insert a new value in A.
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10
8
7 9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Insert(A, 13)
1311
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 8 / 18
Heaps
3. Insert a new value in A.
16
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10
8 7
9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Insert(A, 13)
13
11
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 8 / 18
Heaps
3. Insert a new value in A.
16
14
10
8 7
9 3
2
15
1
1
2 3
4 5 6 7
8 9 10
Insert(A, 13)
13
11
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 8 / 18
Heaps
Algorithm 2 Insert(A, x)
Require:
1 ≤ n < N
and A[1..n] is a heap.
1: n = n + 12: A[n] = −∞3: Increase key(A, n, x)
This takes O(h) = O(log(n)) time.
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 9 / 18
Heaps
4. Heapify(A, i) (sometimes called sift-down): this operation assumesthat
1 ≤ i ≤ n,
the subtree rooted at left(i) is a heap
and the subtree rooted at right(i) is a heap.
At termination, the subtree rooted at i is a heap.
i
Heap Heap
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 10 / 18
Heaps
16
4 10
14 7 9 3
2 8 1
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 11 / 18
Heaps
16
4 10
14 7 9 3
2 8 1
1
2 3
4 5 6 7
8 9 10
Heapify(A, 2)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 11 / 18
Heaps
16
4
1014
7 9 3
2 8 1
1
2 3
4 5 6 7
8 9 10
Heapify(A, 2)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 11 / 18
Heaps
16
4
1014
7 9 3
2
8
1
1
2 3
4 5 6 7
8 9 10
Heapify(A, 2)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 11 / 18
Heaps
Algorithm 3 Heapify(A, i)Require:
1 ≤ i ≤ n,
the subtree rooted at left(i) is a heap
and the subtree rooted at right(i) is a heap.
1: ` = left(i)2: r = right(i)3: if ` ≤ n and A[`] > A[i ] then4: max = `5: else6: max = i7: end if8: if r ≤ n and A[r ] > A[max] then9: max = r10: end if11: if max 6= i then12: swap A[i ] and A[max]13: Heapify(A,max)14: end if
This takes O(h) = O(log(n)) time.J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 12 / 18
Heaps
5. Build heap(A)
4
1 3
2 16 9 10
14 8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2 16 9 10
14 8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2 16 9 10
14 8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2 16 9 10
14 8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2 16 9 10
14 8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2
16 9 1014
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2
16 9 1014
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16 9
10
14
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16 9
10
14
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1 3
2
16
9
10
14
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16
9
10
14
8
7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16
9
10
14
8
7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16
9
10
14
8
7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4
1
3
2
16
9
1014
8
7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4 1
3
2
16
9
1014
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
5. Build heap(A)
4 1
3
2
16
9
1014
8 7
1
2 3
4 5 6 7
8 9 10
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 13 / 18
Heaps
Algorithm 4 Build heap(A)
1: for i = bn/2c to 1 do2: Heapify(A, i)3: end for
How many steps to build a heap?
There is 20 node at height h.There are 21 nodes at height h − 1.There are 22 nodes at height h − 2.
...There are 2h−1 nodes at height 1.
h · 20 + (h − 1) · 21 + (h − 2) · 22 + ...+ 1 · 2h−1
=h−1∑i=0
(h − i) · 2i
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 14 / 18
Heaps
Algorithm 4 Build heap(A)
1: for i = bn/2c to 1 do2: Heapify(A, i)3: end for
How many steps to build a heap?
There is 20 node at height h.There are 21 nodes at height h − 1.There are 22 nodes at height h − 2.
...There are 2h−1 nodes at height 1.
h · 20 + (h − 1) · 21 + (h − 2) · 22 + ...+ 1 · 2h−1
=h−1∑i=0
(h − i) · 2i
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 14 / 18
Heaps
Algorithm 4 Build heap(A)
1: for i = bn/2c to 1 do2: Heapify(A, i)3: end for
How many steps to build a heap?
There is 20 node at height h.There are 21 nodes at height h − 1.There are 22 nodes at height h − 2.
...There are 2h−1 nodes at height 1.
h · 20 + (h − 1) · 21 + (h − 2) · 22 + ...+ 1 · 2h−1
=h−1∑i=0
(h − i) · 2i
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 14 / 18
Heaps
h−1∑i=0
(h − i) · 2i
=h−1∑i=0
h · 2i −h−1∑i=0
i · 2i
= hh−1∑i=0
2i −h−1∑i=0
i · 2i
= h(2h − 1
)−((h − 2) · 2h + 2
)= 2 · 2h − h − 2
= O(n)
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 15 / 18
Heaps
6. Extract max A
Algorithm 5 Extract max(A)
1: max = A[1]2: A[1] = A[n]3: n = n − 14: Heapify(A, 1)5: return max
This takes O(h) = O(log(n)) time.
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 16 / 18
Heaps
We have discussed max-heaps.
There is their symmetric counterpart: min-heaps,
where A[parent(i)] ≤ A[i ] (1 < i ≤ n).
J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 17 / 18
Heaps
)J.-L. De Carufel (U. of O.) Design & Analysis of Algorithms Fall 2017 18 / 18
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