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Why Sorting?
 “When in doubt, sort” – one of the
principles of algorithm design. Sorting
used as a subroutine in many of the
algorithms:
 Searching in databases: we can do binary
search on sorted data
 A large number of computer graphics and
computational geometry problems
 Closest pair, element uniqueness

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Why Sorting? (2)

 A large number of sorting algorithms are
developed representing different algorithm
design techniques.
 A lower bound for sorting W(n log n) is
used to prove lower bounds of other
problems

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Sorting Algorithms so far

 Insertion sort, selection sort
 Worst-case running time Q(n2); in-place
 Heap sort
 Worst-case running time Q(n log n).

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Divide and Conquer

 Divide-and-conquer method for algorithm
design:
 Divide: if the input size is too large to deal with in a
straightforward manner, divide the problem into two or
more disjoint subproblems
 Conquer: use divide and conquer recursively to solve
the subproblems
 Combine: take the solutions to the subproblems and
“merge” these solutions into a solution for the original
problem

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Merge Sort Algorithm
 Divide: If S has at least two elements (nothing
needs to be done if S has zero or one elements),
remove all the elements from S and put them
into two sequences, S1 and S2 , each containing
about half of the elements of S. (i.e. S1 contains
the first n/2elements and S2 contains the
remaining n/2elements).
 Conquer: Sort sequences S1 and S2 using
Merge Sort.
 Combine: Put back the elements into S by
merging the sorted sequences S1 and S2 into
one sorted sequence
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Merge Sort: Algorithm
Merge-Sort(A, p, r)
if p < r then
q(p+r)/2
Merge-Sort(A, p, q)
Merge-Sort(A, q+1, r)
Merge(A, p, q, r)

Merge(A, p, q, r)
Take the smallest of the two topmost elements of
sequences A[p..q] and A[q+1..r] and put into the
resulting sequence. Repeat this, until both sequences
are empty. Copy the resulting sequence into A[p..r].

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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MergeSort (Example)

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Merging Two Sequences (cont.)

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Merge Sort Revisited
 To sort n numbers
 if n=1 done!
 recursively sort 2 lists of
numbers n/2 and n/2
elements
 merge 2 sorted lists in Q(n)
time
 Strategy
 break problem into similar
(smaller) subproblems
 recursively solve
subproblems
 combine solutions to answer
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Recurrences
 Running times of algorithms with Recursive
calls can be described using recurrences
 A recurrence is an equation or inequality that
describes a function in terms of its value on
smaller inputs
 solving_trivial_problem if n  1
T (n)  
num_pieces T (n / subproblem_size_factor)  dividing  combining if n  1

 Example: Merge Sort

 Q(1) if n  1
T (n)  
2T (n / 2)  Q(n) if n  1 35

Solving Recurrences
 Repeated substitution method
 Expanding the recurrence by substitution and noticing
patterns
 Substitution method
 guessing the solutions
 verifying the solution by the mathematical induction

 Recursion-trees
 Master method
 templates for different classes of recurrences

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Repeated Substitution Method
 Let’s find the running time of merge sort (let’s
assume that n=2b, for some b).
 1 if n  1
T (n)  
2T (n / 2)  n if n  1

T (n)  2T  n / 2   n substitute
 2  2T  n / 4   n / 2   n expand
 22 T (n / 4)  2n substitute
 22 (2T (n / 8)  n / 4)  2n expand
 23 T (n / 8)  3n observe the pattern
T (n)  2i T (n / 2i )  in
 2lg n T (n / n)  n lg n  n  n lg n
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Repeated Substitution Method
 The procedure is straightforward:
 Substitute
 Expand
 Substitute
 Expand
…
 Observe a pattern and write how your expression
looks after the i-th substitution
 Find out what the value of i (e.g., lgn) should be to get
the base case of the recurrence (say T(1))
 Insert the value of T(1) and the expression of i into
your expression
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Java Implementation of Merge-Sort

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Java Implementation of MergeSort (cont.)
public class ListMergeSort implements SortObject {

public void sort(Sequence S, Comparator c) {
int n = S.size();
if (n < 2) return; //sequence with 0/1 element is sorted.
// divide
Sequence S1 = (Sequence)S.newContainer();
// put the first half of S into S1
for (int i=1; i <= (n+1)/2; i++) {
S1.insertLast(S.remove(S.first()));
}
Sequence S2 = (Sequence)S.newContainer();
// put the second half of S into S2
for (int i=1; i <= n/2; i++) {
S2.insertLast(S.remove(S.first()));
}
sort(S1,c); // recur
sort(S2,c);
merge(S1,S2,c,S); // conquer
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}

Java Implementation of MergeSort (cont.)

public void merge(Sequence S1, Sequence S2, Comparator c, Sequence S) {
while(!S1.isEmpty() && !S2.isEmpty()) {
if(c.isLessThanOrEqualTo(S1.first().element(),
S2.first().element())) {
// S1’s 1st elt <= S2’s 1st elt
S.insertLast(S1.remove(S1.first()));
}else { // S2’s 1st elt is the smaller one
}
}if(S1.isEmpty()) {
while(!S2.isEmpty()) {
}
}if(S2.isEmpty()) {
while(!S1.isEmpty()) {
}}}
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