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Complexity of Algorithms Complexity of Algorithms Luong The Nhan, Tran Giang Son Algorithm Efficiency Big O notation Problems and common complexities P and NP Problems 2 1 Chapter 2 Complexity of Algo[.]
Complexity of Algorithms Luong The Nhan, Tran Giang Son Chapter Complexity of Algorithms Data Structures and Algorithms Algorithm Efficiency Big-O notation Problems and common complexities Luong The Nhan, Tran Giang Son Faculty of Computer Science and Engineering University of Technology, VNU-HCM P and NP Problems 2.1 Outcomes Complexity of Algorithms Luong The Nhan, Tran Giang Son • L.O.1.1 - Define concept “computational complexity” and its sepcial cases, best, average, and worst • L.O.1.2 - Analyze algorithms and use Big-O notation to characterize the computational complexity of algorithms composed by using the following control structures: sequence, branching, and iteration (not recursion) • L.O.1.3 - List, give examples, and compare complexity classes, for examples, constant, linear, etc • L.O.1.4 - Be aware of the trade-off between space and Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems time in solutions • L.O.1.5 - Describe strategies in algorithm design and problem solving 2.2 Contents Complexity of Algorithms Luong The Nhan, Tran Giang Son Algorithm Efficiency Big-O notation Algorithm Efficiency Big-O notation Problems and common complexities Problems and common complexities P and NP Problems P and NP Problems 2.3 Complexity of Algorithms Luong The Nhan, Tran Giang Son Algorithm Efficiency Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems 2.4 Algorithm Efficiency Complexity of Algorithms Luong The Nhan, Tran Giang Son • A problem often has many algorithms • Comparing two different algorithms ⇒ Computational complexity: measure of the difficulty degree (time and/or space) of an algorithm • • Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems How fast an algorithm is? How much memory does it cost? 2.5 Algorithm Efficiency Complexity of Algorithms Luong The Nhan, Tran Giang Son General format efficiency = f(n) n is the size of a problem (the key number that determines the size of input data) Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems 2.6 Complexity of Algorithms Linear Loops Luong The Nhan, Tran Giang Son for ( i = 0; i < 1000; i ++) application code The number of times the bodyof the loop is replicated is 1000 f(n) = n Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems for ( i = 0; i < 1000; i += 2) application code The number of times the bodyof the loop is replicated is 500 f(n) = n/2 2.7 Complexity of Algorithms Linear Loops time f (n) = n Luong The Nhan, Tran Giang Son Algorithm Efficiency Big-O notation f (n) = n/2 Problems and common complexities P and NP Problems n 2.8 Complexity of Algorithms Logarithmic Loops Luong The Nhan, Tran Giang Son Multiply loops i = while ( i = 1) application code i = i / P and NP Problems The number of times the body of the loop is replicated is f(n) = log2 n 2.9 Complexity of Algorithms Logarithmic Loops time Luong The Nhan, Tran Giang Son Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems f (n) = log2 n n 2.10 Complexity of Algorithms Standard Measures of Efficiency Luong The Nhan, Tran Giang Son Efficiency logarithmic linear linear log quadratic polynomial exponential factorial Big-O O(log2 n) O(n) O(n log2 n) O(n2 ) O(nk ) O(2n ) O(n!) Iterations 14 10 000 140 000 100002 10000k 2( 10000) 10000! Est Time microseconds 0.1 seconds seconds 15-20 hours intractable intractable Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems Assume instruction speed of microsecond and 10 instructions in loop n = 10000 2.19 Standard Measures of Efficiency n2 n log n time Complexity of Algorithms Luong The Nhan, Tran Giang Son n Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems log2 n n 2.20 ... 2.3 Complexity of Algorithms Luong The Nhan, Tran Giang Son Algorithm Efficiency Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems 2.4 Algorithm Efficiency Complexity. .. of an algorithm • • Algorithm Efficiency Big-O notation Problems and common complexities P and NP Problems How fast an algorithm is? How much memory does it cost? 2.5 Algorithm Efficiency Complexity. .. Asymptotic Complexity Complexity of Algorithms Luong The Nhan, Tran Giang Son • • • Algorithm efficiency is considered with only big problem sizes We are not concerned with an exact measurement of an algorithm? ??s