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Advanced Algorithm Analysis
CS-717Lecture No. 12
Muhammad Shahid
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A RAM is an idealized machine with aninfinitely large random-access memory
Instructions are executed one-by-one
Each instruction involves performing some
basic operation on two values in themachines memory
Random Access Machine (RAM)
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Basic operations include: assigning a value to a variable
computing any basic arithmetic operation (+,- ,,/)
on integer values of any size performing any comparison
accessing an element of an array
We assume that each basic operation takes
the same constant time to execute
Random Access Machine (RAM)
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Let a point p in 2-dimensional space be givenby its integer coordinates, = (. , . )
A point p is said to be dominated by point q if. . and . .
Given a set of n points, = {, . . ., } in2-space a point is said to be maximal if it is
not dominated by any other point in
2-dimension maxima
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Given a set of points = {, . . . , } in2-space, output the set of maximal points of
, i.e., those pointssuch that is not
dominated by any other point of
2-dimension maxima
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Let = {, . . ., } be the initial set ofpoints
For each point , test it against all otherpoints
If is not dominated by any other point, thenoutput it
Brute-Force Algorithm
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2-dimension maxima
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Maximal(int n, Point p[1n])
1. fori = 1 to n
2. maximal = true
3. forj = 1 to n4. if(i != j) and (p[i].x
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Maximal(int n, Point p[1n])1. fori = 1 to n
2. maximal = true
3. forj = 1 to n4. if(i != j) and (p[i].x
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Thus we might express the worst-caserunning time as a pair of nested summations,
one for the i-loop and the other for thej-loop:
Analysis of the Brute-Force Maxima
Advanced Algorithm Analysis (CS-717)13
1 1
1
2
2
( ) (2 4)
( ) (4 2)
( ) (4 2)
( ) 4 2
Worst-case Running Time= (n )
n n
i j
n
i
T n
T n n
T n n n
T n n n
AsymptoticGrowth Rate
of the function
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We will sweep a vertical line across the planefrom left to right
As we sweep this line, we will build astructure holding the maximal points lying tothe left of the sweep line
When the sweep line reaches the rightmostpoint of P , then we will have constructed the
complete set of maxima This approach of solving geometric problems
by sweeping a line across the plane is called"Plane Sweep"
Plane-Sweep Algorithm
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Sort the points in increasing order of their x-coordinates
For simplicity, let us assume that no twopoints have the same y-coordinate
Then we will advance the sweep-line frompoint to point in n discrete steps
As we encounter each new point, we will
update the current list of maximal points When the sweep line reaches the rightmost
point of P, we will have the complete set ofmaximal points
Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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Plane-Sweep Algorithm
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PLANE-SWEEP-MAXIMA(n, P[1..n])1. Sort P in increasing order by x;
2. stack stk;
3. fori 1 to n
4. do
5. while (!stk.Empty() && stk.Top().y P[i].y)
6. do stk.Pop();
7. stk.Push(P[i]);
8. output the contents of stack stk;
Plane-Sweep Algorithm
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We pop an element off the stack each timewe go through the inner while-loop
The for-loop iterates n times and the inner
while-loop also iterates n time for a total of(n)
Combined with the sorting, the runtime of
entire plane-sweep algorithm is ()
Analysis of Plane-Sweep Algorithm
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notation: asymptotic "equality": f(n)= (g(n)) implies: f(n) "=" g(n)
O notation(Big-O): asymptotic "less than": f(n)=O(g(n)) implies: f(n) " "g(n)
notation: asymptotic "greater than": f(n)= (g(n)) implies: f(n) "" g(n)
Asymptotic Notation
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-notation
Asymptotic notations (cont.)
Advanced Algorithm Analysis (CS-717)31
(g(n)) is the set of functions
with the same order of growth
as g(n)
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Example
Advanced Algorithm Analysis (CS-717)32
3 2 3f(n) = 10n + 5n + 17 (n )
3 3
3 3
1
2
0
10 ( ) (10 5 17)
10 ( ) 32
c 10
321
f n n
f n n
cn
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Example
Advanced Algorithm Analysis (CS-717)33
3 3f(n)=10n log ( )n n n
3 3
1 2 0
10 ( ) 11
c 10 11 1
f n n
c n
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O-notation:
Asymptotic notations
Advanced Algorithm Analysis (CS-717)34
( )
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- notation
Asymptotic notations (cont.)
Advanced Algorithm Analysis (CS-717)35
(g(n)) is the set of functions
with larger or same order of
growth as g(n)
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M S t
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Merge Sort
Advanced Algorithm Analysis (CS-717)37
8 6 3 5 2 9 4 1
8 6 3 5 2 9 4 1
2 9 4 1
8 6 3 5 2 9 4 1
3 58 6
M S t
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Merge Sort
Advanced Algorithm Analysis (CS-717)38
1 2 3 4 5 6 8 9
3 5 6 8
1 2 4 9
2 9 1 4
8 6 3 5 2 9 4 1
3 56 8
Al ith M S t
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Merge-sort(array A, int p, int r)1. if(p
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Algorithm Merge-Sort
Advanced Algorithm Analysis (CS-717)40
1. Merge(array A, int p, int q,int r)
2. int B[pr];3. int i =k =p;
4. intj =q+1
5. while(i
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Recurrence Relation
Advanced Algorithm Analysis (CS-717)41
1 1( )
( / 2 ( / 2 ( )
if nT n
T n T n n otherwise
This kind of equation is called RecurrenceEquation orRecurrence
R R l ti
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Recurrence Relation
Advanced Algorithm Analysis (CS-717)42
( ) ( / 2 ( / 2 ( )
(1) 1
(2) (1) (1) 2 1 1 2 4
(3) (2) (1) 3 4 1 3 8
(4) (2) (2) 4 4 4 4 12
(5) (3) (2) 5 8 4 5 17
(6) (3) (3) 6 8 8 6 22
(7) (4) (3) 7 12 8 7 27
(8) (4) (4) 8
T n T n T n n
T
T T T
T T T
T T T
T T T
T T T
T T T
T T T
12 12 8 32
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The Iteration Method
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If n is a power of 2 then let n = 2k
or k = log n
The Iteration Method
Advanced Algorithm Analysis (CS-717)47
k
k
k
k
logn
logn
logn
nT(n) = 2 T( ) ( )
2
nT(n) = 2 T( )2
nT(n) = 2 T( ) (logn)
2
nT(n) = 2 T( ) (logn)n
T(n) = nT(1) (logn)
T(n) = n (logn)
n n n n
kn
n
n
n
n
Recurrence Tree
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Recurrence Tree
Advanced Algorithm Analysis (CS-717)48
n
n/8
n/2 n/2
n/4 n/4
n/8 n/8 n/8 n/8
n/4 n/4
n/8 n/8 n/8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Time to Merge
2(n/2) = n
= n
+
+
4(n/4) = n
+
8(n/8) = n
+n(n/n) = n
Recurrence Tree
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Recurrence Tree
Advanced Algorithm Analysis (CS-717)49
n
n/8
n/2 n/2
n/4 n/4
n/8 n/8 n/8 n/8
n/4 n/4
n/8 n/8 n/8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Time to Merge
2(n/2) = n
= n
+
+
4(n/4) = n
+
8(n/8) = n
+n(n/n) = n
log(n)+1
levels
n(log(n)+1)
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SORTING
Advanced Algorithm Analysis (CS-717)50
Elementary Sorting Techniques
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Bubble Sort
Elementary Sorting Techniques
Advanced Algorithm Analysis (CS-717)51
Scan the array
Whenever two consecutive items are found
that are out of order, swap them Repeat until all consecutive items are in order
Elementary Sorting Techniques
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Insertion Sort
Elementary Sorting Techniques
Advanced Algorithm Analysis (CS-717)52
Assume that A[1...i 1] have already been
sorted
Insert A[i] into its proper position in this subarray
Create this position by shifting all larger
elements to the right
Elementary Sorting Techniques
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Selection Sort
Elementary Sorting Techniques
Advanced Algorithm Analysis (CS-717)53
Assume that A[1...i 1] contain the i 1
smallest elements in sorted order
Find the smallest element in A[i...n] Swap it with A[i]
Heap
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A heap is a left-complete binary tree thatconforms to the heap order
The heap order property: in a (min) heap, forevery node X, the key in the parent is smallerthan or equal to the key in X
In other words, the parent node has keysmaller than or equal to both of its childrennodes
Heap
Advanced Algorithm Analysis (CS-717)54
Heap
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Similarly, in a max heap, the parent has a keylarger than or equal both of its children
Thus the smallest key is in the root in a minheap; in the max heap, the largest is in the
root
Heap
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Heap
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Heap
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1014
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Basic Functions
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Access to nodes involves simple arithmeticoperations:
parent(i): returns /2 , the parent of . Theroot is at position 1 of the array
left(i): returns 2, index of left child of node
right(i): returns 2 + 1, the right child
Basic Functions
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Heapsort Algorithm
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We build a max heap out of the given array ofnumbers A[1..n]
We repeatedly extract the maximum item fromthe heap
Once the max item is removed, we are left with ahole at the root
To fix this, we will replace it with the last leaf intree
But now the heap order will very likely bedestroyed
We will apply a heapify procedure to the root torestore the heap
Heapsort Algorithm
Advanced Algorithm Analysis (CS-717)58
Heap Sort
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HEAPSORT( array A, int n)1. BUILD-HEAP(A, n)
2. m n
3. while (m 2)4. do SWAP(A[1],A[m])
5. m m 1
6. HEAPIFY(A, 1,m)
Heap Sort
Advanced Algorithm Analysis (CS-717)59
Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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sorted
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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heap violated
Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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heap violated
Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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heap violated
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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Heapsort Trace
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19 15 12 23 44 57 87
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sorted
Heapsort Trace
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Heapsort Trace
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15
12 15 19 23 44 57 87
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sorted
heap violated
Heapsort Trace
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Heapsort Trace
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15
12 15 19 23 44 57 87
1 2 3 4 5 6 7
sorted
Heapsort Trace
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eapso ace
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12
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1 2 3 4 5 6 7
sorted
Heapsort Trace
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p
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12 15 19 23 44 57 87
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sorted
Heapsort Trace
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p
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sorted
Analysis of Heapsort
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Heapify: The total time for heapify is O(log n)
BuildHeap:
BuildHeap takes (n) time
Heapsort:
Heapsort calls BuildHeap once. This takes (n)
Heapsort then extracts roughly n maximumelements from the heap
Each extract requires a constant amount ofwork (swap) and O(log n) heapify
Heapsort is thus O(n log n)
y p
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Quicksort
Advanced Algorithm Analysis (CS-717)82
Quicksort
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QUICKSORT( array A, int p, int r)1. if(r > p)
2. then
3. i a random index from [p...r]4. swap A[i] with A[p]
5. q PARTITION(A, p, r)
6. QUICKSORT(A, p, q 1)7. QUICKSORT(A, q + 1, r)
Advanced Algorithm Analysis (CS-717)83
Partition Algorithm
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PARTITION( array A, int p, int r)1. x A[p]
2. q p
3. fors p + 1 to r4. do if(A[s] < x)
5. then q q + 1
6. swap A[q] with A[s]7. swap A[p] with A[q]
8. return q
g
Advanced Algorithm Analysis (CS-717)84
Partition Algorithm
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g
Advanced Algorithm Analysis (CS-717)85
7 5 10 8 6 9 5 3
p
q
r
s
7 5 10 8 6 9 5 3p
q
r
s
7 5 10 8 6 9 5 3
p
q
r
s
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Partition Algorithm
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g
Advanced Algorithm Analysis (CS-717)87
7 5 6 5 10 9 8 3
p
q
r
s
7 5 6 5 3 9 8 10p
q
r
s
3 5 6 5 7 9 8 10
p
q
r
s
Analysis of Quicksort
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The running time of quicksort dependsheavily on the selection of the pivot
If the rank of the pivot is very large or very
small then the partition (BST) will be
unbalanced
Since the pivot is chosen randomly in our
algorithm, the expected running time is
( log )
The average case, quicksort runs in
( log ) time
y
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Counting Sort
Advanced Algorithm Analysis (CS-717)89
Counting Sort
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CountingSort(A, B, n, k)1. for i 0 to k
2. do C[i] 0
3. forj 1 to n
4. do C[A[j]] C[A[j]] +15. for i 1 to k
6. do C[i] C[i] + C[i-1]
7. forj
n downto 18. B[ C[A[j]] ] A[j]
9. C[ A[j] ] C[ A[j] ] -1
Advanced Algorithm Analysis (CS-717)90
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Counting Sort
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Advanced Algorithm Analysis (CS-717)92
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
0 0 0 0 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)93
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
1 0 0 0 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)94
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
1 0 1 0 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
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Advanced Algorithm Analysis (CS-717)95
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 0 1 0 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)96
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 1 1 0 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)97
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 1 1 1 0 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)98
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 1 1 1 1 0 1
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)99
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 1 1 1 1 0 2
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)100
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 2 1 1 1 0 2
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
Counting Sort
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Advanced Algorithm Analysis (CS-717)101
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 2 1 2 1 0 2
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
k
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Counting Sort
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Advanced Algorithm Analysis (CS-717)103
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 2 2 2 1 0 2
1 2 3 4 5 6 7
C[1k]
1 2 3 4 5 6 7 8 9 10 11
2 4 6 8 9 9 11
1 2 3 4 5 6 7
C
fori = 2 to 7
do C[i] = C[i] + C[i-1]
Counting Sort
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Advanced Algorithm Analysis (CS-717)104
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 4 6 8 9 9 11
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 4 5 8 9 9 11
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3
Counting Sort
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Advanced Algorithm Analysis (CS-717)105
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 4 5 8 9 9 11
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 4 5 7 9 9 11
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 4
Counting Sort
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Advanced Algorithm Analysis (CS-717)106
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 4 5 7 9 9 11
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 3 5 7 9 9 11
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42
Counting Sort
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Advanced Algorithm Analysis (CS-717)107
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 3 5 7 9 9 11
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 3 5 7 9 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 7
Counting Sort
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Advanced Algorithm Analysis (CS-717)108
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 3 5 7 9 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 3 5 7 8 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 75
Counting Sort
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Advanced Algorithm Analysis (CS-717)109
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 3 5 7 8 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
2 3 5 6 8 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 754
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Counting Sort
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Advanced Algorithm Analysis (CS-717)111
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
2 2 5 6 8 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
1 2 5 6 8 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 75421
Counting Sort
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Advanced Algorithm Analysis (CS-717)112
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
1 2 5 6 8 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
1 2 4 6 8 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 7542 31
Counting Sort
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Advanced Algorithm Analysis (CS-717)113
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
1 2 4 6 8 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
0 2 4 6 8 9 10
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 7542 31 1
Counting Sort
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Advanced Algorithm Analysis (CS-717)114
7 1 3 1 2 4 5 7 2 4 3AInput
A[1n]
0 2 4 6 8 9 10
1 2 3 4 5 6 7
1 2 3 4 5 6 7 8 9 10 11
0 2 4 6 8 9 9
1 2 3 4 5 6 7
C
A
Output
B[1n]
1 2 3 4 5 6 7 8 9 10 11
C
3 42 7542 31 71
Analysis of Counting Sort
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CountingSort(A, B, n, k)
1. for i 0 to k
2. do C[i] 0
3. forj 1 to n
4. do C[A[j]] C[A[j]] +15. for i 1 to k
6. do C[i] C[i] + C[i-1]
7. forj n downto 1
8. B[ C[A[j]] ] A[j]
9. C[ A[j] ] C[ A[j] ] -1
Advanced Algorithm Analysis (CS-717)115
(k)
(n)
(k)
(n)
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Bucket Sort
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Bucket Sort
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Advanced Algorithm Analysis (CS-717)118
.78
.17
.39
.26
.72
.94
.21
.23
.68
/
.72
.94
/
//
.12
0
1
2
3
4
6
7
8
5
9
.12 .17 /
.21 .23 .26 /
.39 /
.68 /
/.78.72
.94 /
Step 2:concatenate
the lists
Bucket Sort
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Advanced Algorithm Analysis (CS-717)119
.78
.17
.39
.26
.72
.94
.21
.23
.68
.12
.12 .17 /
.21 .23 .26 /
.39 /
.68 /
/.78.72
.94 /
Step 2:concatenate
the lists
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Radix Sort
Advanced Algorithm Analysis (CS-717)120
Radix SortSTEP 1 STEP 2 STEP 3
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Advanced Algorithm Analysis (CS-717)121
576
494
194
296
278
176
954
49[4]
19[4]
95[4]
57[6]
29[6]
17[6]
27[8]
9[5]4
5[7]6
1[7]6
2[7]8
4[9]4
1[9]4
2[9]6
[1]76
[1]94
[2]78
[2]96
[4]94
[5]76
[9]54
176
194
278
296
494
576
954
STEP 1 STEP 2 STEP 3
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