Recurrences - Analysis of Algorithms Merge Sort and...

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Unformatted text preview: Analysis of Algorithms Merge Sort and Analysis of Recurrences c 2011 Aybar C. Acar & Ci˘dem G¨nd¨z Demir ¸g uu Compiled on: February 21, 2011 Analysis of Algorithms Divide and Conquer “Nothing is particularly hard if you divide it into small jobs.” – Henry Ford Many useful algorithms follow what is called the ‘‘divide-and-conquer” approach: Divide the problem into smaller subproblems. Conquer each subproblem by either: solving it, if simple enough or recursively dividing into smaller sub-subproblems Combine the results of the subproblems to get the result of the original problem. Analysis of Algorithms Sorting with Divide-and-Conquer Given an array A[0 . . . n − 1] of items, one way to approach the problem is: Divide: Bisect the array into two subarrays of size n/2 Conquer: Sort the subarrays directly or recursively bisect until they are trivial to sort. Combine: Merge the individual sorted results until you get back the sorted A This approach is called Merge Sort. Usually, the array is recursively divided into subarrays of size 1 Really trivial However, variants exist that only divide down to 5-7 items before conquering Use a different algorithm like insertion sort to finish the job. Called “hybrid merge sort” The “merge” step is crucial. Analysis of Algorithms The Merge Operation An array A, with two adjacent subarrays A[p , q ) and A[q , r ) that are sorted within themselves (called sorted runs): p A: ... ... q 1 4 5 7 r 2 3 6 8 ... ... How do you get a combined sorted run A[p , r )? Split A[p , r ) at q and copy into two temporary arrays, L and R: p 0 1 L: 1 4 1 4 5 2 3 4 5 7 7 2 r 3 6 8 ... ... 0 R: 1 2 3 4 2 3 6 8 8 ... ... 8 A: q The last items in L and R are sentinels, values that delimit the data. Assume no value can be larger than a sentinel. Analysis of Algorithms The Merge Operation (cont.) Merge and L and R back into A[p , r ) as a single sorted run: 2 3 1 4 5 7 4 0 1 2 3 4 2 3 6 8 8 8 8 8 8 8 8 8 8 8 1 8 0 p ... ... 1 q 2 4 3 5 4 7 5 2 r 6 3 7 6 8 ... ... Question What is the invariant here? At the end of k-th iteration, A[p , p + k ) contains the smallest k elements of L and R. Analysis of Algorithms The Merge Algorithm merge(A,p,q,r): nL ← q − p nR ← r − q create L[0 . . . nL ] and R [0 . . . nR ] L[nL ] ← ∞ R [nR ] ← ∞ for i ← 0 to nL − 1 do L [i ] ← A [p + i ] end for for j ← 0 to nR − 1 do R [j ] ← A [q + j ] end for i ←0j ←0 for k ← p to r − 1 do {Place sentinels} {Copy A[p , q ) to L} {Copy A[q , r ) to R} {Get the smaller top value of L or R} if L[i ] ≤ R [j ] then A [k ] ← L [i ] i ←i +1 else A [k ] ← R [j ] j ←j +1 end if end for Analysis of Algorithms Time Complexity of Merge Question What is the time T(n) for merge (A, p , q , r ) if n = (r − p )? T (n) = c1 + c2 n Question What is the Θ bound of T (n)? T (n) = Θ(n) Analysis of Algorithms The Merge Sort Algorithm mergesort(A,p,r): if p < r − 1 then q ← (p + r )/2 mergesort (A, p , q ) mergesort (A, q , r ) merge (A, p , q , r ) end if {Can’t split a single value!} {Divide} {Conquer} {Conquer} {Combine} For the whole array: mergesort (A, 0, n) Analysis of Algorithms Merge Sort: Random Input Analysis of Algorithms Some Observations The number of times mergesort recurses is only a function of how many times you can bisect A So, it is dependent only on n The running time of merge is Θ(n) regardless of the numbers themselves. Merge sort’s running time should depend only on n and not the configuration of the array A Compare with insertion sort... Analysis of Algorithms Merge Sort: Reverse Sorted Input Analysis of Algorithms The Time Requirement of Merge Sort mergesort(A,p,r): if p < r − 1 then {Θ(1)} {Θ(1)} {T ( n/2 )} {T ( n/2 )} {Θ(n)} q ← (p + r )/2 mergesort (A, p , q ) mergesort (A, q , r ) merge (A, p , q , r ) end if Can be shown with what is known as a “recurrence”: T (n ) = Θ(1) : n ≤ 1(Base case ) T ( n/2 ) + T ( n/2 ) + Θ(n) : n > 1(Recursion) This can be simplified with a few harmless assumptions: T (n) = 2T (n/2) + Θ(n) Analysis of Algorithms Finding Bounds on Recurrences: Substitution Method T (n) = 2T (n/2) + n How do we find the bound? The straightforward way is to “guess” the bound and verify. Called the “Substitution Method” Uses mathematical induction Used to find either the upper or lower bound. Analysis of Algorithms Substitution Method: Induction T (n) = 2T (n/2) + n Induction Step: Assume T (n) = O (n log n). Then, T (n0 ) ≤ cn0 log n0 for some c , n0 : n log (n/2) )+n 2 ≤ cn log (n/2) + n T (n) ≤ 2(c = cn log n − cn log 2 + n = cn log n − cn + n ≤ cn log n Analysis of Algorithms Basis of Induction T (n) ≤ cn log n Basis: Prove that this is true for n = 2 and n = 3. Assume T (1) = 1. From the recurrence, T (n) = 2T (n/2) + n: T (2) = 4 T (3) = 5 Then, T (2) ≤ c 2 log 2 T (3) ≤ c 3 log 3 So we can select n0 = 4 and c = 3. Important Notice that T (n) < cn log n does not hold for T (1): (T (1) = 1) > (c 1 log 1 = 0) But this is not a problem, since we only need to prove T (n) ≤ cn log n for n above some n0 . We only need T (2) and T (3) to compute T (n ≥ 4) and they hold. So we just choose n0 = 4. Analysis of Algorithms Caution You must obey the exact form of the inductive hypothesis lest you end up with false confirmations. Assume you guess T (n) = O (n): ≤ 2(c (n/2)) + n ≤ cn + n = T (n ) O (n) ⇐ WRONG!! To see this, assume that T(1) = 1, and using the original recurrence we know that T(2) = 4, T(4) = 12, T(8) = 32, T(16) = 80 &c. Hence, T (2) = 4 ≤ 2c only if c≥2 T (4) = 12 ≤ 4c only if c≥3 T (8) = 32 ≤ 8c only if c≥4 T (16) = 80 ≤ 16c . . . only if c≥5 . . . It is easy to see that there is no constant value of c and n0 such that the inequality will hold for any n ≥ n0 Analysis of Algorithms Substitution Method Substitution confirms that mergesort is indeed O (n log n). Substitution method cannot “solve” the recursion. It can only confirm your guess is correct. So how do we guess? Experience There are usually only a few possibilities Merge sort was obviously Ω(n) and O (n2 ). The reasonable guess was n log n, since it is between n and n2 . Other methods such as iteration method also called repeated substitution Analysis of Algorithms Iteration Method The iteration method expands the recurrence until it reaches the base case. By repeatedly substituting the recurrence into itself. T (n ) = 2T (n/2) + n T (n ) = 2(2T (n/4) + n/2) + n T (n ) = 4T (n/4) + 2n/2 + n T (n ) = 4(2T (n/8) + n/4) + n + n T (n ) = 8T (n/8) + n + n + n . . . = . . . T (n ) = 2k T (n/2k ) + kn When n = 2k , k = log n. So, T (n) = 2T (n/n) + (log n)n = nT (1) + n log n = n + n log n = Θ(n log n) Analysis of Algorithms Recursion trees Recursion trees are incremental graphical representations of the recursion of an algorithm. They can be seen as a graphical version of the iteration method. Useful in finding the form of the bound. Assume you have the recurrence: T (n) = 3T (n/4) + cn2 How is the first level of recursion going to work? cn2 T (n/4) T (n/4) T (n/4) Analysis of Algorithms Recursion Tree: Development cn2 c T ￿n￿ 16 ￿ n ￿2 T ￿n￿ 16 c 4 T ￿n￿ 16 T ￿n￿ 16 ￿ n ￿2 T c 4 ￿n￿ 16 T ￿n￿ 16 T Analysis of Algorithms ￿n￿ 16 ￿ n ￿2 T 4 ￿n￿ 16 T ￿n￿ 16 Recursion Tree: Final Tree cn2 c log4 n c ￿ n ￿2 16 c ￿ n ￿2 ￿ n ￿2 16 T (1) T (1) T (1) T (n) = cn2 + c 4 c ￿ n ￿2 16 c ￿ n ￿2 16 T (1) T (1) T (1) 32 3 cn + 16 16 cn2 ￿ n ￿2 c c 4 ￿ n ￿2 16 c ...................... ￿ n ￿2 16 c ￿ n ￿2 16 ￿ n ￿2 c ￿ n ￿2 16 T (1) T (1) T (1) 2 cn2 +. . .+ 3 16 32 cn 16 4 ￿ n ￿2 16 T (1) T (1) T (1) log4 n−1 Analysis of Algorithms c ￿ 3 16 ￿2 cn2 Θ(nlog4 3 ) cn2 +Θ(nlog4 3 ) Simplification of T(N) 32 cn + 16 T (n) = cn2 + + log4 n−1 3 16 log4 n−1 T (n ) = T (n ) = i =0 3 16 2 cn2 + . . . cn2 + Θ(nlog4 3 ) i cn2 + Θ(nlog4 3 ) i =0 3 log4 n 16 cn2 3 −1 16 ∞ i < = 3 16 3 16 + Θ(nlog4 3 ) cn2 + Θ(nlog4 3 ) 1 cn2 + Θ(nlog4 3 ) 3 1 − 16 = O (n2 ) Analysis of Algorithms ...
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Recurrences - Analysis of Algorithms Merge Sort and...

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