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BASICS 2

Office hour 改成 三 9-10. BASICS 2. Prof. Michael Tsai 2012/3/6. 作業一的程式題不容易寫呀 ! 要趕快開始 !. Definition [Omega ]: { : there exist positive constants and such that for all } “f of n is omega of g of n ” 可以想成是 Lower Bound. Asymptotic notation – Omega. since for all 1 .

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BASICS 2

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  1. Office hour 改成三9-10 BASICS 2 Prof. Michael Tsai 2012/3/6 作業一的程式題不容易寫呀!要趕快開始!

  2. Definition [Omega]: {: there exist positive constants and such that for all } “f of n is omega of g of n” 可以想成是Lower Bound Asymptotic notation – Omega

  3. since for all 1. since for all 1. since for all . since for all 1. since for all 1. Examples for all

  4. Examples • (1) • (1) for all

  5. Omega is a lower bound. Should be as large a function as possible. Theorem: If and , then . (證明: 請自己證證看!) Discussion

  6. Definition [Theta]: {: there exist positive constants such that for all } “f of n is theta of g of n” 三明治夾心, 同時具有和 Asymptotic Notation – Theta Theta Platform GMC Terrain

  7. 用圖表示 • Big Oh • 紅色 • Omega • 藍色 • Theta • 紅色藍色都要

  8. since for all and for all . Example for all

  9. More precise than both the “big oh” and omega notations • It is true if and only g(n) is both an upper and lower bound on f(n). • Coefficient = 1 • (no good) • (ok!) • Theorem: If and , then . • (證明: 請自己證證看!) Discussion

  10. 那麼, 怎麼知道程式的時間複雜度? • 看每個statement執行幾次, 畫表來計算 • 有時候複雜度不只跟input大小不一定有關係 • (像是binary search, 跟裡面實際數字是什麼有關) • 這時候可以取 • Worst case:給一組input可以使此algorithm執行最慢; 最慢的執行時間是多少? • Best case: 給一組input可以使此algorithm執行最快; 最快的執行時間是多少? • Average case: 考慮所有case的執行時間, 平均是多少? • 實際測量程式執行時間最實際! • (但是常常不可行,此時asymptotic analysis是最好的方法)

  11. Example 1

  12. intbinsearch(int list[], intsearchnum, int left, int right) { int middle; while(left<=right) { middle=(left+right)/2; switch(COMPARE(list[middle], searchnum)) { case -1: left=middle+1; break; case 0: return middle; case 1: right=middle-1; } } return -1; } Example 2

  13. Worst case: Best case: searchnum=13; 0 1 2 3 4 5 6 7 8 9 10 11 1 3 4 4 7 11 13 13 13 18 19 6 right left middle middle=(left+right)/2; left=middle+1; Half the searching window every iteration.

  14. 兜基??

  15. n很大的時候右邊比左邊好 但是n會很大嗎? 有時候不會 如果n永遠小於 ?? 結論:要看n大小 (實際上寫程式的時候適用) 哪一個好…

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