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The previous examples illustrate that processes can differ considerably in the rates at which they consume computational resources. One convenient way to describe this difference is to use the notion of order of growth to obtain a gross measure of the resources required by a process as the inputs become larger.

Let n be a parameter that measures the size of the problem, and let R(n) be the amount of resources the process requires for a problem of size n. In our previous examples we took n to be the number for which a given function is to be computed, but there are other possibilities. For instance, if our goal is to compute an approximation to the square root of a number, we might take n to be the number of digits accuracy required. For matrix multiplication we might take n to be the number of rows in the matrices. In general there are a number of properties of the problem with respect to which it will be desirable to analyze a given process. Similarly, R(n) might measure the number of internal storage registers used, the number of elementary machine operations performed, and so on. In computers that do only a fixed number of operations at a time, the time required will be proportional to the number of elementary machine operations performed.

We say that R(n) has order of growth Θ(f(n)), written R(n) = Θ(f(n)) (pronounced “theta of f(n)”), if there are positive constants k1 and k2 independent of n such that

k sub 1 * f(n) <= R(n) <= k sub 2 * f(n)

for any sufficiently large value of n. (In other words, for large n, the value R(n) is sandwiched between k1f(n) and k2f(n).)

For instance, with the linear recursive process for computing factorial described in section 1.2.1 the number of steps grows proportionally to the input n. Thus, the steps required for this process grows as Θ(n). We also saw that the space required grows as Θ(n). For the iterative factorial, the number of steps is still Θ(n) but the space is Θ(1) — that is, constant.[36] The tree-recursive Fibonacci computation requires Θ(φn) steps and space Θ(n), where φ is the golden ratio described in section 1.2.2.

Orders of growth provide only a crude description of the behavior of a process. For example, a process requiring n2 steps and a process requiring 1000n2 steps and a process requiring 3n2 + 10n + 17 steps all have Θ(n2) order of growth. On the other hand, order of growth provides a useful indication of how we may expect the behavior of the process to change as we change the size of the problem. For a Θ(n) (linear) process, doubling the size will roughly double the amount of resources used. For an exponential process, each increment in problem size will multiply the resource utilization by a constant factor. In the remainder of section 1.2 we will examine two algorithms whose order of growth is logarithmic, so that doubling the problem size increases the resource requirement by a constant amount.

[36] These statements mask a great deal of oversimplification. For instance, if we count process steps as “machine operations” we are making the assumption that the number of machine operations needed to perform, say, a multiplication is independent of the size of the numbers to be multiplied, which is false if the numbers are sufficiently large. Similar remarks hold for the estimates of space. Like the design and description of a process, the analysis of a process can be carried out at various levels of abstraction. [back]

Exercises

Exercise 1.14

Draw the tree illustrating the process generated by the count-change procedure of section 1.2.2 in making change for 11 cents. What are the orders of growth of the space and number of steps used by this process as the amount to be changed increases?

Exercise 1.15

The sine of an angle (specified in radians) can be computed by making use of the approximation sin xx if x is sufficiently small, and the trigonometric identity

sin x = 3 sin (x/3) - 4 sin^3 (x/3)

to reduce the size of the argument of sin. (For purposes of this exercise an angle is considered “sufficiently small” if its magnitude is not greater than 0.1 radians.) These ideas are incorporated in the following procedures:

(define (cube x) (* x x x))
(define (p x) (- (* 3 x) (* 4 (cube x))))
(define (sine angle)
   (if (not (> (abs angle) 0.1))
       angle
       (p (sine (/ angle 3.0)))))
  1. How many times is the procedure p applied when (sine 12.15) is evaluated?
  2. What is the order of growth in space and number of steps (as a function of a) used by the process generated by the sine procedure when (sine a) is evaluated?

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xunjie ワイスロングファッションブランドを立ち上げました忘れられない好奇心をオープンしました旅、
スミス·ストリートのビジネスコンサルティング、
展覧会はプライベートジェットであろう、
monster beats ͨ؜ プレザント写真やその他の関連情報を提供するファッション業界だけでなく、
またオーストラリアに数百万人のファン数百人の注目を集めネットはいくつかの伝説を追加しました。
主張して​​いるラングウェンXiの女性のファッションさん知的な好意と維持皆高貴な特別な味。
gaga milano

1.2.3 Orders of Growth | SICP in Clojure

Incredible tons of great tips!
xunjie 市場全体は収縮傾向基づく調整を滑らかにするために価格を表示し続けた。
モードを変更するので、
2011年に中国のオンラインショッピング市場の取引は、

1.2.3 Orders of Growth | SICP in Clojure

Nicely put. Appreciate it!
xunjie 3 Koushouを崇拝……パール村に代わって、
今日の職場多数と忙しい生活の中で、
消費者に販売される衣類のすべての部分に、

1.2.3 Orders of Growth | SICP in Clojure

Regards. Very good information.
xunjie ブラジルメリッサゼリーの靴を選ぶ(メリッサ)の靴の女の子は、
衣類の輸入カリブの国の輸出加工区からの米国は13.4%までの平均減少した。
王女を構築するために子供の頃の夢のように、

1.2.3 Orders of Growth | SICP in Clojure

Incredible all kinds of awesome data!
xunjie 髪をおかっぱ - 38歳のシャーリーズ·セロン(シャーリーズ·セロン)いつでも異端児。
嗅覚の鋭い感覚と韓国Kelierでランジェリー市場に関与して、
強力なバッキングストアのセキュリティシステムは、

1.2.3 Orders of Growth | SICP in Clojure

Wonderful advice, Thanks.
xunjie フランスMansha 9元韓国人女性店は非常に簡単なことであるのは、
ドライブだけで30分から卸売ヒイラギの葉の衣類に位置集め、
低コストで高品質のサービス、

1.2.3 Orders of Growth | SICP in Clojure

You actually mentioned it really well!
xunjie 市Guantingbingzhuan企業786で提供される統計によると東莞市は、
異なる消費の製造プロセスに従って適切な料金処理を支払う必要があります。
メディアとの標準化事務総長チーXiaoxia靴インタビュー錦江履物フェア、

1.2.3 Orders of Growth | SICP in Clojure

Wonderful information. Cheers.
xunjie 上質紙の日記のために爪。
すべてが完成されていることを確認します。
一意のIDと相まって太陽漂白光感のあるソリッドカラーのファブリック、

1.2.3 Orders of Growth | SICP in Clojure

With thanks. Loads of advice!

xunjie アイデアはI-赤ちゃんと一緒にブランドの特性を作成するために組み合わせる5000年中国の文化と外国の先進的な保育園保育園です現代の母親のライフスタイル:安全、
ファッショナブルなドレスになっているので、
ビジネス哲学の無限の追求、

1.2.3 Orders of Growth | SICP in Clojure

Thank you, An abundance of material.

xunjie

1.2.3 Orders of Growth | SICP in Clojure

Thank you, Quite a lot of data!

xunjie

1.2.3 Orders of Growth | SICP in Clojure

You reported it superbly.
xunjie 町が10に達した5000元企業以上の年間販売収入を含むニット衣類の500以上の企業を持って、
福建省の造船産業のセグメント化された血管は、
昨年(7.3億ドル)8.8%の増加、 tory burch

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