[ 11.2, 11.3, 11.4] Analysis of Algorithms. Complexity of Algorithms. 400 lecture note # Overview
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1 400 lecture note #0 [.2,.3,.4] Analysis of Algorithms Complexity of Algorithms 0. Overview The complexity of an algorithm refers to the amount of time and/or space it requires to execute. The analysis of an algorithm is the process by which we find an estimate for its complexity. The time needed to execute an algorithm is a function of the size of the input (n). Some typical n's are: o A numeric value (e.g. n in n!) o Number of elements in a list/array -- size of the list/array Various cases for complexity. Best-case -- minimum time required to process the input of size n 2. Worst-case -- maximum time 3. Average-case -- average time Example: Searching for a key in an array Problem: finding an element in an array of size n Input: key, array A = [x, x2,, xn], n Output: the index of key in A, (0 if key is not in the list) Program find_key( key, A, n ). index := 2. while index <= n 3. begin 4. if key = A[index] // equality test 5. then return index // key found, exit the program 6. index := index + // increment the index by 7. end 8. return 0 end find_key How fast can the key be found? o o o In the best case, key is at A[] and we only go through the loop once. In the worst case, key is at A[n] (or not even in the list) and we must go through the loop n times. On average, assuming that key can be anywhere in the list with equal probability, we would expect to go through the loop about n/2 times. So, the running time of depends most often on n. As n gets bigger, the running time grows with various rate (asymptotic growth). Example: times it takes to find the key in the algorithm above Best case Worst case Average case n times needed n times needed n times needed
2 Number of times a line is executed When we analyze an algorithm, we count how many times a particular (important) line is executed. Example:. for i := to n do 2. for j := to n do 3. x := x + // count this line The number of times the line 3 is executed when the input size was n: T(n) = n + n n = n * n = n 2 <-- n of them --> More examples:. for i := to 2n do 2. x := x + // count this line T(n) =. for i := to 2n do 2. for j := to n do 3. x := x + // count this line T(n) =. for i := to n do 2. for j := to i do 3. x := x + // count this line T(n) = 2. Logarithmic Growth Example :. i := n 2. while i >= do 3. begin 4. x := x + // count this line 5. i := i / 2 // i becomes half 6. end
3 iteration value of i (at the top of loop) number of times line 4 is executed n 2 n/2 3 n/2 2 k n/2 k- = k+ n/2 k = 0 0 (loop not executed) We are interested in what k is (because that's the number of times the line 4 is executed). In other words, T(n) = = (k * ) <-- k of them---> To derive k, we look at the relation at the last iteration (kth): So, T(n) = log2n + (where log2n is alternatively written as lgn). Example 2:. i := n 2. while i >= do 3. begin 4. for j := to n do 5. x := x + // count this line 5. i := i / 2 // i becomes half 6. end iteration 2 3 k k+ value of i (at the top of loop) number of times line 4 is executed So, T(n) = = (k * ) <---- k of them ---->
4 We know from example that k = lgn +. Therefore, we get T(n) = (k * ) = 3. Various Asymptotic Growth Functions Common growth functions (in an ascending order) T(n) Name constant lgn logarithmic n linear nlgn n log n n 2 quadratic n 3 cubic n m (general) polynomial (m is a fixed, non-negative integer; ; e.g. n 4, n 5 ) m n exponential (m >= 2; ; e.g. 2 n, 3 n ) n! factorial As the input size n becomes large, some growth functions grow very fast, while others grow slowly. n T(n) = lgn T(n) = n T(n) = n
5 4. Big-Oh (O), Omega (Ω) and Theta (Θ) Notations T(n) is often expressed as a function that has several terms. Each term also may have constant coefficients. For example, T(n) = 60n 2 + 5n + But we are mostly interested in an approximation -- order of the function (or order of growth). So we only look at the dominant term (60n 2 in the above) and throw away the coefficient (60). Then we get, for the above, T(n) = Θ(n 2 ) NOTE: Any function of the form n k is called a polynomial of degree k. There are three notations for expressing asymptotic growth:. Big-Oh -- e.g. O(n 2 ) -- for upper bound 2. Omega -- e.g. Ω(n 2 ) -- for lower bound 3. Theta -- e.g. Θ(n 2 ) -- for tight bound () Big-Oh NOTE: Throughout the rest of this document, n is a positive integer (i.e., n >= ). A function f(n) is of order at most g(n), noted f(n) = O(g(n)), if there exists c, n0 > 0 such that for all n >= n0, we have that f(n) <= c*g(n). Examples: a. 60n 2 + 5n + = O(n 2 )... NOTE: f(n) = 60n 2 + 5n +, and g(n) = n 2
6 b. c. n + nlgn = Ο(nlgn) d. n + nlgn = Ο(n 2 ) (2) Omega A function f(n) is of order at least g(n), noted f(n) = Ω(g(n)), if there exists c, n0 > 0 such that for all n >= n0, we have that f(n) >= c*g(n). Examples: a. 60n 2 + 5n + = Ω(n 2 ) b. 60n 2 + 5n + = Ω(n) c.
7 (3) Theta A function f(n) is of exact order g(n), noted f(n) = Θ(g(n)), if f(n) = O(g(n)) and f(n) = Ω(g(n)). Examples: a. 60n 2 + 5n + = Θ(n 2 ) From () a. and (2) a. above. b. 3n 2 + 2n lgn = Θ(n 2 ) 5. Example Algorithms. Linear search in an unsorted (or any) sequence Problem: finding an element in an array of size n Input: key, array A = [x, x2,, xn], n Output: the index of key in A, (0 if key is not in the list) Program find_key( key, A, n ). index := 2. while index <= n 3. begin 4. if key = A[index] then // count this comparison 5. return index 6. index := index + 7. end 8. return 0 end find_key Complexity Worst-case T(n) = = Θ( ) Best-case T(n) = = Θ( ) Average-case T(n) = = Θ( ) 2. Binary Search -- for sorted sequence (in ascending order) Example: Search for 6
8 The algorithm: Problem: find an element in a sorted list (whose index is 0 through n-) Input: key, L a list and n Output: The index of key in L, - otherwise procedure BinarySearch(key, L, n). low := 0 2. high := n- 3. while (low <= high) do 4. begin 5. mid = floor((low + high)/2) 6. if (key = L[mid]) then // count this comparison 7. return mid 8. else if (key < L[mid]) then 9. high := mid - 0. else. low := mid + 2. end 3. return - // key not found (fall through) end BinarySearch Complexity. Worst -case The worse case occurs when key is not present is the list. How many runs through the while loop will we do? Initially, we have n element to consider. At the end of the first loop, we are down to n/2. After the second loop, we have n/4. Next, n/8, etc. In general, at the end of the kth loop, we have n/2 k elements left to consider. How far can we keep dividing (what is k?)? Again, according to the algorithm we exit the loop when low > high. This means, in particular, that when we are down to element and process it, we stop the loop. So,
9 2. Best-case So we get T(n) = lgn, thus Θ(lgn). The best case happens when key is present at the middle of the array, namely L[n/2]. In this case, the loop executes only once, regardless of the size of array (i.e., n). So, we get T(n) =, thus Θ(). 3. Average-case
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