Quicksort

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(No Transcript)
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Decision Problems
Optimization problems minimum, maximum,
smallest, largest
Traveling salesman, Clique, Vertex-Cover,
Independent Set, Knapsack
Satisfaction (SAT) problems
CNF, Constraint-SAT, Hamiltonian Circuit,
Circuit-SAT

• Only optimization problems use the minimum,
  maximum, smallest, largest value
• To formulate the decision problem
• As input parameter for Phase II of the non
  deterministic algorithms

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Computing an
Reduce and Conquer
Derek Drakes compute(a, n) if (n0)
then return 1 m ? floor(n/2) s ?
compute(a,m) if ( n is even) then return
s s if ( n is odd) then return a s
s
Divide and Conquer compute(a, n) if (n0)
then return 1 m ? floor(n/2) return
compute(a,m) compute(a, n m)
Brute Force compute(a, n) ans ? 1
for i ? 1 to n do ans ? ans a return
ans
Complexity O(n)
T(n) 2T(n/2) 1 T(1) 1 Complexity O(n)
T(n) T(n/2) 3 T(1) 1 Complexity O(log2n)
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Solving Recurrence Relations
T(n) 4T(n/2) n T(1) 1
T(n) 4T(n/2) n2 T(1) 1
4k kn2 with n 2k
4k 2k-1n 2n n with n 2k
O(n2 log2n)
O(n2 )
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Merge Sort is Stable

• We split A into two parts B (first half) and C
  (first half)
• Sort B and C separately
• When merging B and C, we compare elements in B
  against elements in C, if tides occur, we insert
  element of B back into A first and then elements
  in B

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Quicksort
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Quicksort

• An element of the array is chosen. We call it the
  pivot element.
• The array is rearranged such that
• - all the elements smaller than the
  pivot are moved before it
• - all the elements larger than the
  pivot are moved after it
• Then Quicksort is called recursively for these
  two parts.

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Quicksort - Algorithm

• Quicksort (AL..R)
• if L lt R then
• pivot Partition( AL..R)
• Quicksort (A1..L-1)
• Quicksort (AL1...R)

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Partition Algorithm

• Partition (AL..R)
• p ? AL i ? L j ? R 1
• while (i lt j) do
• repeat i ? i 1 until Ai p
• repeat j ? j - 1 until Aj p
• if (i lt j) then swap(Ai, Aj)
• swap(Aj,AL)
• return j

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Example
A 3 8 7 1 5 2 6 4
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Complexity Analysis

• Best Case every partition splits half of the
  array
• Worst Case one array is empty one has all
  elements
• Average case

O(n log2n)
O(n2 )
O(n log2n)
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Binary Search
BinarySearch(A1..n, el) Input A1..n sorted
in ascending order Output The position i such
that Ai el or -1 if el is not
in A1..n