Pythonによる検索アルゴリズム

Pythonの基本

2022.10.20 2022.09.10

異なるデータ構造に格納されたデータを検索することは、ほとんどすべてのアプリケーションで重要な部分です。

検索時に利用できるアルゴリズムは数多くあり、それぞれが異なる実装を持ち、仕事を成し遂げるために異なるデータ構造に依存しています。

特定のタスクに対して特定のアルゴリズムを選択できることは、開発者にとって重要なスキルであり、高速で信頼性が高く安定したアプリケーションと、単純なリクエストで崩れてしまうアプリケーションとの違いを意味します。

メンバーシップ演算子
線形探索
二項演算子
ジャンプ検索
フィボナッチ検索
指数探索
補間探索

メンバーシップオペレーター

アルゴリズムは、絶え間ない進化と、さまざまな領域の根本的な問題に対する最も効率的な解決策を見つける必要性の結果として、時間とともに発展し最適化されます。

コンピュータサイエンスの領域で最も一般的な問題の1つは、コレクションを検索して、与えられたオブジェクトがコレクションに存在するかどうかを判断することです。

ほとんどすべてのプログラミング言語は、基本的な検索アルゴリズムの独自の実装を持っています。通常は、与えられたアイテムのコレクションでアイテムが見つかったときに True または False という Boolean 値を返す関数として実装されています。

Pythonでは、オブジェクトを検索する最も簡単な方法は、メンバーシップ演算子を使うことです。これは、与えられたオブジェクトがコレクションのメンバーであるかどうかを判断することができるので、そのように名付けられました。

これらの演算子は、文字列、リスト、タプルなど、Pythonのあらゆる反復可能なデータ構造で使用することができます。

in – 与えられた要素が構造体の一部である場合、 True を返します。
not in – 与えられた要素が構造体の一部でない場合、 True を返します。

&gt;&gt;&gt; 'apple' in ['orange', 'apple', 'grape']
True
&gt;&gt;&gt; 't' in 'stackabuse'
True
&gt;&gt;&gt; 'q' in 'stackabuse'
False
&gt;&gt;&gt; 'q' not in 'stackabuse'
True

文字列の中に部分文字列があるかどうかを調べたり、2つの文字列、リスト、タプルのオブジェクトが交差しているかどうかを調べるだけなら、メンバーシップ演算子で十分です。

ほとんどの場合、アイテムが存在するかどうかの判断に加えて、シーケンス内のアイテムの位置が必要です。メンバーシップ演算子はこの要件を満たしません。

組み込みの演算子に依存せず、より高速に、あるいはより効率的に値を探索することができる探索アルゴリズムが数多く存在します。さらに、単に要素の存在を決定できるだけでなく、コレクション内の要素の位置など、より多くの情報を得ることができます。

線形探索

線形探索は最も単純な探索アルゴリズムの1つであり、最も理解しやすいものです。Pythonの in 演算子の実装を発展させたものと考えることができます。

このアルゴリズムは、配列に対して繰り返し処理を行い、ある項目が見つかったら、その項目が最初に現れるインデックスを返すというものです。

def LinearSearch(lys, element):
    for i in range (len(lys)):
        if lys[i] == element:
            return i
    return -1

ですから、この関数で計算すると

&gt;&gt;&gt; print(LinearSearch([1,2,3,4,5,2,1], 2))

コードを実行すると、次のように表示されます。

これは私たちが探している項目の最初の出現のインデックスです。Pythonのインデックスは0ベースであることを覚えておいてください。

線形探索の時間計算量は O(n) で、入力リスト lys の項目数に応じて実行時間が増加することを意味します。

線形探索は実際にはあまり使われません。なぜなら、内蔵のメソッドや既存の演算子を使っても同じ効率を得ることができ、他の探索アルゴリズムと比べても高速で効率的ではないからです。

線形探索は、他の多くの探索アルゴリズムとは異なり、探索を始める前にコレクションをソートする必要がないので、ソートされていないコレクションで最初に現れる項目を見つける必要がある場合に適している。

バイナリサーチ

バイナリサーチは、Divide and Conquer 法に従います。線形探索よりも高速ですが、アルゴリズムを実行する前に配列がソートされている必要があります。

ソートされた配列から値 val を探すとすると，このアルゴリズムは val と配列の真ん中の要素（ここでは mid と呼ぶ）の値を比較します．

この値を mid と呼ぶことにします． * mid が探している要素である場合（ベストケース），そのインデックスを返します．
* もしそうでなければ、 val が mid よりも小さいか大きいかによって、 val が mid のどちら側にあるか判断し、配列のもう一方の側は破棄します。
* その後、再帰的または反復的に同じ手順を踏んで、 mid に新しい値を選び、 val と比較し、アルゴリズムの各反復で可能なマッチの半分を破棄します。

バイナリサーチアルゴリズムは、再帰的または反復的に記述することができます。Python では一般に再帰的な方が遅いです。新しいスタックフレームの割り当てが必要だからです。

良い探索アルゴリズムは可能な限り高速で正確であるべきなので、バイナリサーチの反復的な実装を考えてみましょう。

def BinarySearch(lys, val):
    first = 0
    last = len(lys)-1
    index = -1
    while (first &lt;= last) and (index == -1):
        mid = (first+last)//2
        if lys[mid] == val:
            index = mid
        else:
            if val<lys[mid]: +1="" -1="" ```="" compute:="" else:="" first="mid" function="" if=" index="" last="mid" return=" the=" to=" use="" we=" &gt;&gt; BinarySearch([10,20,30,40,50], 20)

We get the result:

Which is the index of the value that we are searching for.

The action that the algorithm performs next in each iteration is one of several possibilities:

Returning the index of the current element
Searching through the left half of the array
Searching through the right half of the array

We can only pick one possibility per iteration, and our pool of possible matches gets divided by two in each iteration. This makes the time complexity of binary search O(log n).

One drawback of binary search is that if there are multiple occurrences of an element in the array, it does not return the index of the first element, but rather the index of the element closest to the middle:

&gt;&gt;&gt; print(BinarySearch([4,4,4,4], 4))

Running this piece of code will result in the index of the middle element:

For comparison performing a linear search on the same array would return:

Which is the index of the first element. However, we cannot categorically say that binary search does not work if an array contains the same element twice – it can work just like linear search and return the first occurrence of the element in some cases.

If we perform binary search on the array [1,2,3,4,4,5] for instance, and search for 4, we would get 3 as the result.

Binary search is quite commonly used in practice because it is efficient and fast when compared to linear search. However, it does have some shortcomings, such as its reliance on the // operator. There are many other divide and conquer search algorithms that are derived from binary search, let’s examine a few of those next.

Jump Search

Jump Search is similar to binary search in that it works on a sorted array, and uses a similar divide and conquer approach to search through it.

It can be classified as an improvement of the linear search algorithm since it depends on linear search to perform the actual comparison when searching for a value.

Given a sorted array, instead of searching through the array elements incrementally, we search in jumps. So in our input list lys, if we have a jump size of jump our algorithm will consider elements in the order lys[0], lys[0+jump], lys[0+2jump], lys[0+3jump] and so on.

With each jump, we store the previous value we looked at and its index. When we find a set of values where lys[i]<element<lys[i+jump], (lys,=”” +=”” -=”” 0=”” 1,=”” <=”val” ``=""lys[i+jump]=""lys[i]`=”” a=”” and=” as=”” def=” element=”” if=”” import=”” in=” jump=” int(math. sqrt(length))” jump)=” jumpsearch=” left=” left,=” left-most=” length=” linear=” lys[left]=” lys[right]=” math=” our=” perform=” right=” min(length” right-most=” search=” set:=” the=” val):=” we=”” while=” with=””= val:
ブレーク
left += jump;
if left = length or lys[left] val:
return -1
right = min(length – 1, right)
i = 左
while i <= right and lys[i] <= val:
if lys[i] == val:
return i
i += 1
return -1



Since this is a complex algorithm, let's consider the step-by-step computation of jump search with this input:

print(JumpSearch([1,2,3,4,5,6,7,8,9], 5))



* Jump search would first determine the jump size by computing `math.sqrt(len(lys))`. Since we have 9 elements, the jump size would be √9 = 3.
* Next, we compute the value of the `right` variable, which is the minimum of the length of the array minus 1, or the value of `left+jump`, which in our case would be 0+3= 3. Since 3 is smaller than 8 we use 3 as the value of `right`.
* Now we check whether our search element, 5, is between `lys[0]` and `lys[3]`. Since 5 is not between 1 and 4, we move on.
* Next, we do the calculations again and check whether our search element is between `lys[3]` and `lys[6]`, where 6 is 3+jump. Since 5 is between 4 and 7, we do a linear search on the elements between `lys[3]` and `lys[6]` and return the index of our element as:



The time complexity of jump search is O(√n), where √n is the jump size, and n is the length of the list, placing jump search between the linear search and binary search algorithms in terms of efficiency.


The single most important advantage of jump search when compared to binary search is that it does not rely on the division operator (`/`).


In most CPUs, using the division operator is costly when compared to other basic arithmetic operations (addition, subtraction, and multiplication), because the implementation of the division algorithm is iterative.


The cost by itself is very small, but when the number of elements to search through is very large, and the number of division operations that we need to perform increases, the cost can add up incrementally. Therefore jump search is better than binary search when there is a large number of elements in a system where even a small increase in speed matters.


To make jump search faster, we could use binary search or another internal jump search to search through the blocks, instead of relying on the much slower linear search.


### Fibonacci Search


Fibonacci search is another divide and conquer algorithm which bears similarities to both binary search and jump search. It gets its name because it uses Fibonacci numbers to calculate the block size or search range in each step.


Fibonacci numbers start with zero and follow the pattern 0, 1, 1, 2, 3, 5, 8, 13, 21... where each element is the addition of the two numbers that immediately precede it.


The algorithm works with three Fibonacci numbers at a time. Let's call the three numbers `fibM`, `fibM_minus_1`, and `fibM_minus_2` where `fibM_minus_1` and `fibM_minus_2` are the two numbers immediately before `fibM` in the sequence:

fibM = fibM_minus_1 + fibM_minus_2



We initialize the values to 0,1, and 1 or the first three numbers in the Fibonacci sequence to avoid getting an index error in the case where our search array `lys` contains a very small number of items.


Then we choose the smallest number of the Fibonacci sequence that is greater than or equal to the number of elements in our search array `lys`, as the value of `fibM`, and the two Fibonacci numbers immediately before it as the values of `fibM_minus_1` and `fibM_minus_2`. While the array has elements remaining and the value of `fibM` is greater than one, we:


* Compare `val` with the value of the block in the range up to `fibM_minus_2`, and return the index of the element if it matches.
* If the value is greater than the element we are currently looking at, we move the values of `fibM`, `fibM_minus_1` and `fibM_minus_2` two steps down in the Fibonacci sequence, and reset the index to the index of the element.
* If the value is less than the element we are currently looking at, we move the values of `fibM`, `fibM_minus_1` and `fibM_minus_2` one step down in the Fibonacci sequence.


Let's take a look at the Python implementation of this algorithm:

def FibonacciSearch(lys, val):
fibM_minus_2 = 0
fibM_minus_1 = 1
fibM = fibM_minus_1 + fibM_minus_2
while (fibM < len(lys)):
fibM_minus_2 = fibM_minus_1
fibM_minus_1 = fibM
fibM = fibM_minus_1 + fibM_minus_2
index = -1;
while (fibM 1):
i = min(index + fibM_minus_2, (len(lys)-1))
if (lys[i] < val):
fibM = fibM_minus_1
fibM_minus_1 = fibM_minus_2
fibM_minus_2 = fibM – fibM_minus_1
index = i
elif (lys[i] val):
fibM = fibM_minus_2
fibM_minus_1 = fibM_minus_1 – fibM_minus_2
fibM_minus_2 = fibM – fibM_minus_1
else :
return i
if(fibM_minus_1 and index < (len(lys)-1) and lys[index+1] == val):
return index+1;
return -1



If we use the FibonacciSearch function to compute:

print(FibonacciSearch([1,2,3,4,5,6,7,8,9,10,11], 6))



Let's take a look at the step-by-step process of this search:


* Determining the smallest Fibonacci number greater than or equal to the length of the list as `fibM`; in this case, the smallest Fibonacci number meeting our requirements is 13.
* The values would be assigned as:
    + fibM = 13
    + fibM_minus_1 = 8
    + fibM_minus_2 = 5
    + index = -1
* Next, we check the element `lys[4]` where 4 is the minimum of -1+5 . Since the value of `lys[4]` is 5, which is smaller than the value we are searching for, we move the Fibonacci numbers one step down in the sequence, making the values:
    + fibM = 8
    + fibM_minus_1 = 5
    + fibM_minus_2 = 3
    + index = 4
* Next, we check the element `lys[7]` where 7 is the minimum of 4+3. Since the value of `lys[7]` is 8, which is greater than the value we are searching for, we move the Fibonacci numbers two steps down in the sequence.
    + fibM = 3
    + fibM_minus_1 = 2
    + fibM_minus_2 = 1
    + index = 4
* Now we check the element `lys[5]` where 5 is the minimum of 4+1 . The value of `lys[5]` is 6, which is the value we are searching for!


The result, as expected is:



The time complexity for Fibonacci search is O(log n); the same as binary search. This means the algorithm is faster than both linear search and jump search in most cases.


Fibonacci search can be used when we have a very large number of elements to search through, and we want to reduce the inefficiency associated with using an algorithm which relies on the division operator.


An additional advantage of using Fibonacci search is that it can accommodate input arrays that are too large to be held in CPU cache or RAM, because it searches through elements in increasing step sizes, and not in a fixed size.


### Exponential Search


Exponential search is another search algorithm that can be implemented quite simply in Python, compared to jump search and Fibonacci search which are both a bit complex. It is also known by the names galloping search, doubling search and Struzik search.


Exponential search depends on binary search to perform the final comparison of values. The algorithm works by:


* Determining the range where the element we're looking for is likely to be
* Using binary search for the range to find the exact index of the item


The Python implementation of the exponential search algorithm is:

def ExponentialSearch(lys, val):
if lys[0] == val:
return 0
index = 1
while index < len(lys) and lys[index] <= val:
index = index * 2
return BinarySearch( arr[:min(index, len(lys))], val)



If we use the function to find the value of:

print(ExponentialSearch([1,2,3,4,5,6,7,8],3))を実行。



The algorithm works by:


* Checking whether the first element in the list matches the value we are searching for - since `lys[0]` is 1 and we are searching for 3, we set the index to 1 and move on.
* Going through all the elements in the list, and while the item at the index'th position is less than or equal to our value, exponentially increasing the value of `index` in multiples of two:
    + index = 1, `lys[1]` is 2, which is less than 3, so the index is multiplied by 2 and set to 2.
    + index = 2, `lys[2]` is 3, which is equal to 3, so the index is multiplied by 2 and set to 4.
    + index = 4, `lys[4]` is 5, which is greater than 3; the loop is broken at this point.
* It then performs a binary search by slicing the list; `arr[:4]`. In Python, this means that the sub list will contain all elements up to the 4th element, so we're actually calling:

BinarySearch([1,2,3,4],3)



which would return:



Which is the index of the element we are searching for in both the original list, and the sliced list that we pass on to the binary search algorithm.


Exponential search runs in O(log i) time, where i is the index of the item we are searching for. In its worst case, the time complexity is O(log n), when the last item is the item we are searching for (n being the length of the array).


Exponential search works better than binary search when the element we are searching for is closer to the beginning of the array. In practice, we use exponential search because it is one of the most efficient search algorithms for unbounded or infinite arrays.


### Interpolation Search


Interpolation search is another divide and conquer algorithm, similar to binary search. Unlike binary search, it does not always begin searching at the middle. Interpolation search calculates the probable position of the element we are searching for using the formula:

index = low + [(val-lys[low])*(high-low) / (lys[high]-lys[low]) ]である。



Where the variables are:


* lys - our input array
* val - the element we are searching for
* index - the probable index of the search element. This is computed to be a higher value when val is closer in value to the element at the end of the array (`lys[high]`), and lower when val is closer in value to the element at the start of the array (`lys[low]`)
* low - the starting index of the array
* high - the last index of the array


The algorithm searches by calculating the value of `index`:


* If a match is found (when `lys[index] == val`), the index is returned
* If the value of `val` is less than `lys[index]`, the value for the index is re-calculated using the formula for the left sub-array
* If the value of `val` is greater than `lys[index]`, the value for the index is re-calculated using the formula for the right sub-array


Let's go ahead and implement the Interpolation search using Python:

def InterpolationSearch(lys, val):
low = 0
high = (len(lys) – 1)
while low <= high and val = lys[low] and val <= lys[high]:
index = low + int((float(high – low) / ( lys[high] – lys[low])) * ( val – lys[low]))
if lys[index] == val:
indexを返す
if lys[index] < val:
low = index + 1;
else:
high = index – 1;
1を返す



If we use the function to compute:

print(InterpolationSearch([1,2,3,4,5,6,7,8], 6))



Our initial values would be:


* val = 6,
* low = 0,
* high = 7,
* lys[low] = 1,
* lys[high] = 8,
* index = 0 + [(6-1)*(7-0)/(8-1)] = 5


Since `lys[5]` is 6, which is the value we are searching for, we stop executing and return the result:



If we have a large number of elements, and our index cannot be computed in one iteration, we keep on re-calculating values for index after adjusting the values of high and low in our formula.


The time complexity of interpolation search is O(log log n) when values are uniformly distributed. If values are not uniformly distributed, the worst-case time complexity is O(n), the same as linear search.


Interpolation search works best on uniformly distributed, sorted arrays. Whereas binary search starts in the middle and always divides into two, interpolation search calculates the likely position of the element and checks the index, making it more likely to find the element in a smaller number of iterations.


### Why Use Python For Searching?


Python is highly readable and efficient when compared to older programming languages like Java, Fortran, C, C++ etc. One key advantage of using Python for implementing search algorithms is that you don't have to worry about casting or explicit typing.


In Python, most of the search algorithms we discussed will work just as well if we're searching for a String. Keep in mind that we do have to make changes to the code for algorithms which use the search element for numeric calculations, like the interpolation search algorithm.


Python is also a good place to start if you want to compare the performance of different search algorithms for your dataset; building a prototype in Python is easier and faster because you can do more with fewer lines of code.


To compare the performance of our implemented search algorithms against a dataset, we can use the time library in Python:

インポート時間

start = time.time()

ここで関数を呼び出す

end = time.time()
print(start-end)

“`

結論

コレクション内の要素を検索する方法はたくさんあります。この記事では、いくつかの検索アルゴリズムとそのPythonでの実装を説明しました。

どのアルゴリズムを使うかは、検索するデータに基づいています。入力配列は、私たちの実装ではすべて lys と呼んでいます。

もし、ソートされていない配列を検索したい場合や、検索対象の変数が最初に現れる場所を探したい場合は、線形探索が最も適しています。
ソートされた配列を検索したい場合、多くの選択肢がありますが、最もシンプルで高速な方法はバイナリサーチです。
ソートされた配列を除算演算子なしで検索したい場合、ジャンプ検索かフィボナッチ検索を使用できます。
検索する要素が配列の先頭に近いことが分かっている場合は、指数探索を使用できます。
ソートされた配列も一様に分布している場合、最も高速で効率的な検索アルゴリズムは補間検索になります。

もし、ソートされた配列にどのアルゴリズムを使うか迷ったら、Pythonのtimeライブラリを使ってそれぞれを試してみて、あなたのデータセットで最もうまくいくものを選んでください。

</element<lys[i+jump],[mid]:>となります。