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Advanced Data Structures
CHAPTER 2
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Binary search trees
B-trees
Heaps and priority queues
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Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
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Binary Search Trees
Binary Search Trees: Nodes
left
parent right
key
internal node
left
key
parent right
root node
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key
parent right
leaf node
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Property of Binary Search Trees
For any node n in a binary search tree T,
• if p is a node in the left subtree of n, then
p.key ≤ n.key;
• if q is a node in the right subtree of n, then
n.key ≤ q.key.
T
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Example of Binary Search Trees
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Inorder Traversal of Binary Search Trees
inorder(T: node)
if T is not null,
then inorder(T.left)
print(T.key)
inorder(T.right)
return.
}
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Search in Binary Search Trees
search(n: node, k: key)
if n is null or n.key = k,
then return(n)
if k < n.key,
then search(n.left, k)
else search(n.right, k)
return.
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Insertion in Binary Search Trees
insert(T: node, z: node)
insert(T: node, z: node)
if T is null
y = null
The tree T is empty.
then T.root = z
x=T
return
while x is not null
if key[z] < key[T]
y=x
then if T.left is null
if z.key < x.key
then x = x.left Found the place, and then T.left = z
z.p = T
else x = x.right insert as the right
left child.
child.
else insert(x.left, z)
z.parent= y
else if T.right is null
if y = NIL
then x.right = z
then T.root= z
z.p = T
else if z.key < y.key
else insert(x.right, z)
then y.left= z
return
else y.right = z
Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
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Deletion in Binary Search Trees
• See textbook
Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
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B-trees
B-trees
• Index structures for large amount of data.
• All data cannot be resided in the main
memory.
• Data is stored in a disk.
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Disk Operations
• In a disk, data is
organized in disk
pages.
• To read data in a
disk,
• The disk rotates, and
the R/W head moves
to the page
containing the data.
• Then, the whole disk
page is read.
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
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B-trees: Nodes
Internal node
number
of keys
key1
key2
key3
keyn
false
leaf
keyn
true
leaf
Pointers to other nodes
leaf node
number
of keys
key1
key2
key3
Pointers to data pages
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Properties of B-trees
For any node n in a B-tree T
n p
…
key1
c
q
keyi
key1
keyi+1
…
keyi
…
false
keyi+1
…
false
• n.keyi ≤ c.key1 ≤ c.key2 ≤ … ≤ c.keyq ≤
n.keyi+1
• If n is a leaf node, the depth of n is h, where
h is the tree’s height.
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Properties of B-trees
• The minimum degree, t, of the B-tree is the lower bound
on the number of keys a node can contain. (t ≥ 2)
• Every node other than the root must have at least t − 1
keys.
• Every internal node other than the root thus has at least t
children.
• If the tree is nonempty, the root must have at least one
key.
• Every node can contain at most 2t − 1 keys. Therefore, an
internal node can have at most 2t children.
• We say that a node is full if it contains exactly 2t − 1
keys.
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Search in B-trees
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Search in B-trees
B-TREE-SEARCH(x, k)
i =1
► find a proper child
while i ≤ x.n and k > x.keyi do i =i + 1
if i ≤ x.n and k = x.keyi
then return (x, i )
if x.leaf
then return NIL
► recursively search in the proper subtree
else DISK-READ (x.ci)
return B-TREE-SEARCH (x.ci, k)
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Creating B-trees
B-TREE-CREATE(T )
x = ALLOCATE-NODE()
x.leaf = TRUE
x.n = 0
DISK-WRITE(x)
T.root = x
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Insertion in B-trees
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Insertion in B-trees
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Insertion in B-trees
B-TREE-INSERT(T, k)
r = T.root
if r.n = 2t − 1
► The root node r is full, then create a new root node
then s = ALLOCATE-NODE()
T.root = s;
s.leaf = FALSE
s.n = 0; s.c1 = r
► Split old root into two, and put under the new root
B-TREE-SPLIT-CHILD(s, 1, r)
B-TREE-INSERT-NONFULL(s, k)
else B-TREE-INSERT-NONFULL(r, k)
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Splitting Nodes in B-trees
B-TREE-SPLIT-CHILD(x, i, y)
►Create a new node
z = ALLOCATE-NODE()
►Move up other 1/2 of old node
for j = x.n + 1 downto i + 1
do x.cj+1 = x.cj
z.leaf = y.leaf
x.ci+1 = z
z.n = t − 1
for j = x.n downto i
►Move 1/2 of old node to new
do x.keyj+1 = x.keyj
for j = 1 to t − 1
x.keyi = y.keyt
do z.keyj = y.keyj+t
x.n = x.n + 1
if not y.leaf
DISK-WRITE(y)
then for j = 1 to t
DISK-WRITE(z)
do z.cj = y.cj+t
y.n = t − 1
DISK-WRITE(x)
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Binary Heaps
Binary Heaps
• Binary heap is an array
• can be viewed as a nearly
complete binary tree
• each node of the tree
corresponds to an element
of the array that stores the
value in the node.
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left
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right
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• The tree is completely
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filled on all levels except
possibly the lowest, which
is filled from the left up to
a point.
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Representation of Biinary Heaps
• An array A that represents a heap is an
object with two attributes:
• length[A], which is the number of elements in
the array
• heap-size[A], the number of elements in the
heap stored within array A.
heap-size
1 2 3 …
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Properties of Binary Heaps
• If a heap contains n elements, its height is
lg2 n.
• In a max-heaps
For every non-root node i, A[PARENT(i)] ≥ A[i]
• In a min-heaps
For every non-root node i, A[PARENT(i)] ≤ A[i]
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Examples
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Max heap
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Heapify
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Heapify
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Heapify
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Heapify
MAX-HEAPIFY(A, i )
l = LEFT(i )
r = RIGHT(i )
► Find node containing the largest value among i and its valid children.
if l ≤ heap-size[A] and A[l] > A[i ]
then largest = l
else largest = i
if r ≤ heap-size[A] and A[r] > A[largest]
then largest = r
► Swap the largest node and node i if the node i is not the largest
if largest  i
then exchange A[i ] and A[largest]
MAX-HEAPIFY(A, largest)
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Time Complexity for Heapify
• worst-case
• The left subtree is one-level taller than the right
subtree, and both are complete binary trees
• The left subtree contains (2n-1)/3 nodes and the right
subtree contains (n-2)/3.
Let T(n) be time spent to heapify an n-node heap.
T(n) = T((2n-1)/3) + k
T(n)  Ο(lg n)
T(n)  Ο(h)
• since the height h of a heap is in Ο(lg n)
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Building Max-heaps: BUILD-MAX-HEAP(A)
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Building Max-heaps: BUILD-MAX-HEAP(A)
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Building Max-heaps: BUILD-MAX-HEAP(A)
heap-size[A] = length[A]
► Start from the rightmost node in the level above leaves
for i = length[A]/2 downto 1
do
MAX-HEAPIFY(A, i )
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Time to Build Max-heaps
Let n be the heap size,
T(n) be the time spent in BUILD-MAX-HEAP for heap of size n,
t(h) be the time spent in MAX-HEAPIFY for the heap of height h,
k(h, n) be the number of subtrees of height h in the heap of size n.
lg n
lg n
h=0
h=0
T(n)   t(h) k(h, n) =
lg n
 c h n/2h+1 = c n  (h/2h)
2
h=0
lg n
T(n) = O(n  (h/2h))
h=0

Since  (h/2h)) = 2, T(n) = O(n).
h=0
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Heapsort
HEAPSORT(A)
BUILD-MAX-HEAP(A)
for i = length[A] downto 2
► Swap the max node with last in heap, and
► heapify heap, excluding the last (old max)
do exchange A[1] and A[i ]
heap-size[A] = heap-size[A] − 1
MAX-HEAPIFY(A, 1)
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Time for Heapsort
• The call to BUILD-MAX-HEAP takes O(n)
• Each of the n − 1 calls to MAX-HEAPIFY
takes O(lg n)
• HEAPSORT takes O(n lg n).
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Priority Queues
Priority Queues
• A priority queue is a max-heap with
operations for queues.
• Insert
• Extract-max
• Increase-key
HEAP-MAXIMUM(A)
return A[1]
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Extract-Max In Priority Queues
HEAP-EXTRACT-MAX(A)
if heap-size[A] < 1
then error “heap underflow”
► Take the maximum element from the root
max = A[1]
► Move the last element in the heap to the root
A[1] = A[heap-size[A]]
heap-size[A] = heap-size[A] − 1
► Heapify
MAX-HEAPIFY(A, 1)
return max
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Increase Key
HEAP-INCREASE-KEY(A, i, key)
if key < A[i ]
then error “new key < current key”
A[i ] = key
► Swap the increased element with its parent up toward
► the root until it is not greater than its parent
while i > 1 and A[PARENT(i)] < A[i ]
do exchange A[i] and A[PARENT(i)]
i = PARENT(i)
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Insert
MAX-HEAP-INSERT(A, key)
► Insert the minimum element
heap-size[A] = heap-size[A] + 1
A[heap-size[A]]=−∞
► Increase the minimum element
HEAP-INCREASE-KEY(A, heap-size[A], key)
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Binomial Heaps
Why Binomial Heaps
• Union two binary heaps takes (n).
• Can it be reduced?
• Use binomial heaps
• A binomial heap is a binomial tree that has
the property of heaps.
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Binomial Trees
• The binomial tree Bk is an ordered tree
which consists of two binomial trees Bk−1
that are linked together.
• the root of one is the leftmost child of the root
of the other.
• The binomial tree B0 consists of a single
node.
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Chapter 2: Advanced Data Structures
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Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein,
Introduction to Algorithms, MIT Press, 2001.
Examples
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Examples
level
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
Jaruloj Chongstitvatana
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Properties of Binomial Trees
For the binomial tree Bk,
• the number of nodes = 2k
• the height of the tree = k
• the number of nodes at depth i = kCi
• the degree of root = k (maximum degree)
• if the children of the root are numbered from
left to right by k − 1, k − 2, . . . , 0, child i is
the root of a subtree Bi.
See the proof in the textbook.
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Chapter 2: Advanced Data Structures
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Binomial Heaps
• A binomial heap is a set of binomial trees
with heap properties.
head[H]
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B0
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1
8
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18
11 17
B2
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14 29
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B3
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Chapter 2: Advanced Data Structures
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Nodes in Binomial Heaps
parent
key
degree
child
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sibling
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Examples of Nodes in Binomial Heaps
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Chapter 2: Advanced Data Structures
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Find Minimum in Binomial Heaps
y = NIL
x = H.head
min = ∞
while x  NIL
do if x.key < min
then min = x.key
y=x
x = x.sibling
return y
head
BINOMIAL-HEAP-MINIMUM(H)
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Maximum number of trees = lg n + 1,
the running time of BINOMIAL-HEAP-MINIMUM = O(lg n)
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Link Binomial Heaps
• Make node y the new
head of the linked list of
node z’s children in O(1)
time.
BINOMIAL-LINK(y, z)
y.p = z
y.sibling = z.child
z.child = y
z.degree = z.degree + 1
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B2
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Chapter 2: Advanced Data Structures
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Union Binomial Heaps
• Link the roots of all trees in both heaps, in a
non-decreasing order of the degree.
• Go through each tree in the heap, and
combine two trees of equal degree in the
heaps, according to one of the following four
cases.
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Chapter 2: Advanced Data Structures
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Union Binomial Heaps: case 1
•
x.degree < next-x.degree
•
move up to the next tree.
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
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Union Binomial Heaps: case 2
•
x.degree = next-x.degree = next-x.sibling.degree
•
move up to the next tree (and they will be linked in
next step)
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
Jaruloj Chongstitvatana
Chapter 2: Advanced Data Structures
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•
•
•
x.degree = next-x.degree
Link x and next-x.
Use the node with smaller key as the root.
From: Cormen, T., C. Leiserson,
R. Rivest, and C. Stein, Introduction
to Algorithms, MIT Press, 2001.
Union Binomial Heaps: case 3
x.key  next-x.key
x.key<next-x.key
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Chapter 2: Advanced Data Structures
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Binomial-Heap-Union
BINOMIAL-HEAP-UNION(H1, H2)
H = MAKE-BINOMIAL-HEAP()
H.head =
BINOMIAL-HEAP-MERGE(H1, H2)
Free the objects H1 and H2 but not
the lists they point to
► Cases 1 and 2
then prev-x = x
x = next-x
else if x.key ≤ next-x.key
► Case 3
then x.sibling = next-x.sibling
BINOMIAL-LINK(next-x, x)
► Case 4
else if prev-x = NIL
then H.head = next-x
else prev-x.sibling = next-x
if H.head = NIL
then return H
prev-x = NIL
BINOMIAL-LINK(x,next-x)
x = H.head
x = next-x
next-x = x.sibling
next-x = sibling[x]
while next-x  NIL
do if (x.degree  next-x.degree) or return H
(next-x.sibling  NIL and
next-x.sibling.degree=x.degree)
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Binomial Heaps: Insert a Node
BINOMIAL-HEAP-INSERT(H, x)
H’ = MAKE-BINOMIAL-HEAP()
x.p = NIL
x.child = NIL
x.sibling = NIL
x.degree = 0
H’.head = x
H = BINOMIAL-HEAP-UNION(H, H’ )
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Chapter 2: Advanced Data Structures
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Fibonacci Heaps
Why Fibonacci Heaps
• Every insertion in a binomial heap requires
the structure adjustment.
• Take long time (O(lg n)) for that.
• Do we need to adjust it that often?
• No, if you use Fibonacci heaps instead!
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Chapter 2: Advanced Data Structures
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Fibonacci Heaps
• A Fibonacci heap is a collection of minheap-ordered trees.
• The trees in a Fibonacci heap are not
constrained to be binomial trees.
• Trees within Fibonacci heaps are rooted but
unordered. (unlike trees within binomial
heaps, which are ordered)
• Circular, doubly linked lists are used in
Fibonacci heaps.
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Chapter 2: Advanced Data Structures
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Fibonacci Heaps:Example
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
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Properties of unordered binomial tree Uk
• Number of nodes = 2k
• Height of the tree = k
• Number of nodes at depth i = kCi
• The degree of root = k, which is greater than
that of any other node
• The children of the root are roots of subtrees
U0,U1, . . . ,Uk−1 in some order.
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Nodes in Fibonacci Heaps
parent
key
degree
mark
left
right
child
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Links in Fibonacci Heaps
From: Cormen, T., C. Leiserson, R. Rivest, and C. Stein, Introduction to Algorithms, MIT Press, 2001.
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Chapter 2: Advanced Data Structures
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Fibonacci Heaps: Insert a node
• Link the node in the root list of H
• Update min[H] if necessary
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Consolidate Heaps
• Find two roots x and y in the root list with
the same degree, where key[x] ≤ key[y].
• Link y to x, using FIB-HEAP-LINK, i.e.
remove y from the root list, and make y a
child of x.
• The field degree[x] is incremented, and the
mark on y, if any, is cleared.
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Chapter 2: Advanced Data Structures
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Time Complexity
Binary Heap
(worst-case)
Binomial Heap
(worst-case)
(1)
(1)
Fibonacci Heap
(amortized)
(1)
(lg n)
Ο(lg n)
(1)
(1)
Ο(lg n)
(1)
(lg n)
(lg n)
Ο(lg n)
(n)
Ο(lg n)
(1)
Decrease-key
(lg n)
(lg n)
(1)
delete
(lg n)
(lg n)
Ο(lg n)
Procedure
Make-heap
Insert
Minimum
Extract-min
Union
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Skip Lists
Skip Lists
• Alternative to binary search trees
• Probabilistic data structure
• Easy to implement
• Good performance in average-case
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Chapter 2: Advanced Data Structures
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Structure of Skip Lists
• Max. level H of the skip list
• An array H of header pointers
• A set of nodes such that a node N contains
H
• a key k and a set of values v,
• the height of h node N, where h<=H, and
• an array of h pointers.
2
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Skip List v.s. Binary search tree
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Search in a Skip List
Search(list, searchkey)
cn = list.head
►Start at the list head
►Start skip from top level to bottom
for i=list.level downto 1
do ► Forward until key of node exceeds search key
while ((cn.fwd[i]).key < searchkey)
do cn = cn.fwd[i]
cn = cn.fwd[1]
if cn.key = searchkey
then return(cn)
else return(null)
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Chapter 2: Advanced Data Structures
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Search in a Skip List
H
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Search for 11
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Search in a Skip List
H
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Search for 12
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Insert in a Skip List
update[3]
cn
Insert 5
update[1]
update[2]
cn
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Insert in a Skip List
cn = list.head
update[i].fwd[i] = node
►Search for a place to insert ► Insert the node from level 2 up
for i=list.maxlevel downto 1 ► Toss a coin
do while ((cn.fwd[i]).key <
while (rand() >.5)
node.key)
do addLevel(node)
do cn = cn.fwd[i]
i++
update[i]=cn
► Update link in level i
► The key is already in the list if (i <= list.level)
if cn.fwd[1].key = node.key
then node.fwd[i] = update[i].fwd[i]
then return(false)
update[i].fwd[i] = node
► Insert the node in level 1
else addLevel(list.head)
i=1
(list.head).fwd[i] = node
node.fwd[i] = update[i].fwd[i]
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Chapter 2: Advanced Data Structures
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Delete a node
• Only remove the node, and connect the lost
links.
• The list can become uneven.
• But it is more efficient than deletion in
binary search trees.
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Chapter 2: Advanced Data Structures
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