<jupyter_start><jupyter_text>Depth-First SearchThe depth-first search algorithm adopts a strategy that suggests, as its name implies, to go "deeper" in the graph whenever possible. The strategy explores all the edges connected to the most recently discovered vertex $v$. Once all the edges are explored, the search backtracks to explore the remaining edges coming out of the vertex from which $v$ was discovered. This process repeats until all the vertices reachable from the source are reached.  Procedure```func DFS(G, u) {    u.visited = true    for each v in G.adj[u]        if v.visited == false            DFS(G, v)}```As we can see from the above psudocode, the DFS itself doesn't do anything other than traversing the graph. In many situations, that is not sufficient to solve a particular problem. This requires us to do extra work in each state of the traversing, which in turn may require extending the state representation, as we will see in the examples below.   An Extremely Simple Example **Minimum Depth of Binary Tree** ([Leetcode 111](https://leetcode.com/problems/minimum-depth-of-binary-tree/description/))Given a binary tree, find its minimum depth.The minimum depth is the number of nodes along the shortest path from the root node down to the nearest leaf node.Note: A leaf is a node with no children.**Sample**Given binary tree `[3,9,20,null,null,15,7]````    3   / \  9  20    /  \   15   7```**Output** Minimum depth=2**Analysis**Each node (vertex) has a depth, which is equal to the depth of its parent + 1. A depth of the tree is equivalent to the depth of a leaf node, therefore the minimum depth of a tree can be found by finding the minimum depth amongst all the leaf nodes.One approach is to calculate the depths for all nodes, starting from the root. In this approach, we denote $DFS(u, d)$ as calculating all the nodes' depths in the subtree rooted with $u$, whose depth is $d$. By repeatedly traversing all the states in this manner, we will consequently have all the depth assigned. $DFS(u, d) \to DFS(left_u, d+1), DFS(right_u, d+1)$When a node is a leaf node, we compare its depth with the current optimal answer, if it's lesser, update the answer. A leaf node is defined as a node without any children.<jupyter_code>optimal_answer = 1 << 20

def DFS(u, d):
    if u is None:
        return
    if u.left is None and u.right is None:
        optimal_answer = min(optimal_answer, d)
        return
    DFS(u.left, d + 1)
    DFS(u.right, d + 1)

DFS(root, 1)
if optimal_answer == 1 << 20:
    optimal_answer = 0<jupyter_output><empty_output><jupyter_text>Another approach can also achieve the same goal. Let's consider the state $f(u)$ as the minimum depth of the subtree rooted at $u$, by the problem statement and the definition of a binary tree, we can easily derive the state as $f(u)=min\big\{ f(left_u), f(right_u) \big\} + 1$, with the depth of a leaf node equals 1.<jupyter_code>def DFS(u):
    if u.left is None and u.right is None:
        return 1
    if u.left is not None:
        depth = DFS(u.left)
    if u.right is not None:
        depth = min(depth, DFS(u.right))
    depth += 1
    return depth

answer = DFS(root)<jupyter_output><empty_output><jupyter_text>There's more to thatThe depth-first search may have been discovered and composed as an algorithm running on a graph. However, with proper abstraction and mathematical modelling, the depth-first search algorithm can be utilized to solve nearly every problem, and the key to this is breaking the original problem into multiple states with topological orders, which can be abstracted as vertices in a graph. The depth-first search traverses on this graph until it reaches the final state. Let's see an example which, by the first glance, we would not likely to categorize it into a graph problem.  Number of Islands ([Leetcode 200](https://leetcode.com/problems/number-of-islands/description/))Given a 2d grid map of '1's (land) and '0's (water), count the number of islands. An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water.**Sample 1**```11110110101100000000```**Output:** 1**Sample 2**```11000110000010000011```**Output:** 3**Analysis**The problem asks to count the number of islands in a given 0-1 matrix where two cells belong to the same land if they are horizontally or vertically adjacent. One approach is to abstract each cell as a vertex in a graph, each has a value $\in \{0, 1\}$. For a "land" cell, there is an edge to its adjacent "land" cell for all 4 directions (up, down, left, right). After abstracting the whole matrix to a graph, we can repeatedly find a "land" vertex that has not been colorized yet, assign it and all its directly and indirectly connected vertices with a same color. After scanning through the whole graph, simply count the number of colors appearing in that graph and we get the number of islands. We can think of coloring a connected block of vertices as pouring water into one vertex and have the the water flood through all other reachable vertices. With this imagination in mind, we define the state as $f(u, c)$, where $u$ represents the current vertex we are pouring water into, and the color of the water is $c$. This methodology is called **flood fill**.```func flood_fill(u, c) {    color[u] = c    for v: u and v is connected and v is land        if (v has not been colored yet)            flood_fill(v, c)}func main_process(matrix) {    m = matrix.row_number    n = matrix.column_number    color = 0    for i = 1 to m        for j = 1 to n            if (matrix[i][j].color == null)                flood_fill(matrix[i][j], color)                color = color + 1}```<jupyter_code>class Solution:
    def valid_position(self, x, y):
        m = len(self.grid)
        n = len(self.grid[0]) if m else 0
        return 0 <= x < m and 0 <= y < n

    def flood_fill(self, x, y):
        if not self.valid_position(x, y) or self.grid[x][y] != '1' or self.visited[x][y]:
            return
        self.visited[x][y] = True
        steps = [(-1, 0), (0, 1), (1, 0), (0, -1)]
        for (xstep, ystep) in steps:
            x_, y_ = x + xstep, y + ystep
            self.flood_fill(x_, y_)

    def numIslands(self, grid):
        self.grid = grid
        self.visited = [[False for _ in range(len(line))] for line in grid]
        m = len(grid)
        n = len(grid[0]) if m else 0
        lands = 0
        for i in range(m):
            for j in range(n):
                if not self.visited[i][j] and grid[i][j] == '1':
                    self.flood_fill(i, j)
                    lands += 1
        return lands
s = Solution()
grid_raw = [
    '11000',
    '11000',
    '00100',
    '00011'
]
grid = []
for i, line in enumerate(grid_raw):
    grid.append([])
    for j, ch in enumerate(line):
        grid[i].append(ch)
answer = s.numIslands(grid)
print(answer)<jupyter_output>3<jupyter_text>Concatenated Word ([Leetcode 472](https://leetcode.com/problems/concatenated-words/description/))Given a list of words (without duplicates), please write a program that returns all concatenated words in the given list of words.A concatenated word is defined as a string that is comprised entirely of at least two shorter words in the given array.**Note**1. The number of elements of the given array will not exceed **10,000**2. The length sum of elements in the given array will not exceed **600,000**.3. All the input string will only include lower case letters.4. The returned elements order does not matter.**Sample Input**`["cat","cats","catsdogcats","dog","dogcatsdog","hippopotamuses","rat","ratcatdogcat"]`**Sample Output**`["catsdogcats","dogcatsdog","ratcatdogcat"]`**Analysis**Apparently we need to come up with an algorithm to check if each word is a concatenated word. We also know from the **Note** that there are at most 10K words, and by the Leetcode convention, the program must produce results in 1 second, that's approximately $10^7$ operations, which means an algorithm with the overall time complexity $O(n^2)$ and up is not accepted. Therefore, our algorithm for checking individual words should have the complexity upperbound of $O(logn)$.The process of validating a word can be broken up into 2 subroutines:1. check if there exists $i$ so that the prefix of word $word[0...i]$ is one of the words in the input list2. if (1) stands true, then all we need to do is to validate the rest of the word $word[i+1...n]$, meaning repeatedly practice the process on the remainder of the word.This composes a search graph, and the initial state is the original word. Having this initial state as the starting vertex in the search graph, we can derive the subsequent vertices, each of which has a state that holds a suffix of the original word, given the corresponding prefix is in the input list.**Our task then is converted to finding a path describing how the original word can be gradually disassembled using components in the given word list.**<jupyter_code>class Solution:
    def dfs(self, word, word_count):
        # if the current word's length = 0, it means that we have found all the components that 
        # concatenates to the original word, if it's more than one word, then we have successfully
        # found a right path that disassemble the original word
        if len(word) == 0:
            if word_count > 1:
                return True
            return False
        success = False
        # checking for all possible prefixes
        for prefix in self.words:
            if len(prefix) == 0:
                continue
            if not word.startswith(prefix):
                continue
            length = len(prefix)
            # if the remaining word can find a path, it means the current word can find a path, 
            # terminating the loop
            success = self.dfs(word[length:], word_count + 1)  
            if success:
                break
        return success

    def findAllConcatenatedWordsInADict(self, words):
        self.words = words
        answer = []
        for word in self.words:
            if self.dfs(word, 0):
                answer.append(word)
        return answer

s = Solution()
words = ["cat","cats","catsdogcats","dog","dogcatsdog","hippopotamuses","rat","ratcatdogcat"]
answer = s.findAllConcatenatedWordsInADict(words)
print(answer)<jupyter_output>['catsdogcats', 'dogcatsdog', 'ratcatdogcat']<jupyter_text>From the above code we can successfully solve the problem, but it has an important issue: the algorithm consumes a lot of time! Submitting it onto Leetcode will result in a TLE (Time Limit Exceeded). Let's take a look at an example:Input: `['cat', 'dog', 'catdog', 'dogcatdog']`After analysing "catdog" we know that this is a valid answer. However, when analysing "dogcatdog", after removing the prefix "dog", we need to re-analyse the remainder "catdog", which is unnecessary, since we know from previous assessment that "catdog" is valid, there's no need to revisit what we have already done. Therefore, a common technique in depth-first search can be adopted here to reduce the number of recursion calls: **Memoization (记忆化搜索)**. After each search result, we cache the state and its corresponding result to provide shortcuts for continuous search tasks.One thing to notice during adopting memoization is that we want to maximize its effect, meaning that we use cache as much as possible to reduce the total number of calculations. To achieve that, assuming a word $w$ is valid through concatenating its prefix $w_1$ and sufix $w_2$, we need to guarantee that $w_2$ is already cached. Naturally, the length of $w_2$ is less than $w$, therefore by reordering the word list we are processing from the shortest to longest, we can utilize memoization to the most.This is easily achievable:<jupyter_code>class MemoizationSolution:
    def dfs(self, word, word_count):
        # ****** check if the word is successfully verified earlier ******
        if word in self.cache:
            return True
        # ================================================================
        if len(word) == 0:
            if word_count > 1:
                return True
            return False
        success = False
        for prefix in self.words:
            if len(prefix) == 0:
                continue
            if not word.startswith(prefix):
                continue
            length = len(prefix)
            success = self.dfs(word[length:], word_count + 1)  
            if success:
                break
        return success

    def findAllConcatenatedWordsInADict(self, words):
        self.words = sorted(words, key=lambda x: len(x))  # sorting the word list by its elements' lengths
        self.cache = {}  # initialize the cache with empty set, using hashmap here to save time for "lookup"
        answer = []
        for word in self.words:
            if self.dfs(word, 0):
                answer.append(word)
                self.cache[word] = True  # after each success, cache the original word
        return answer

s1 = Solution()
s2 = MemoizationSolution()

words = ['a', 'b', 'c', 'ab', 'cab', 'ac', 'acb', 'aab', 'acbacb', 'abaab', 'cabacb', 'aacb', 'abacb', 'acbacbacb', 'abacba', 'acc', 'abaabacc', 'cab', 'accaacb', 'aabacc', 'cabcab', 'acbacbacbabaabacc', 'abaabaccacbacbacb', 'cabacbacb', 'abaabaacb', 'acbacc', 'abaabaacb', 'cabacbacbacbabaabacc', 'abacbacbacbacb', 'cacbacbacb', 'accaabacc', 'abacbaabacbacbacbacb', 'acbacbcab', 'acbacbcabcacbacbacb', 'abaabacccacbacbacb', 'aababaabacc', 'abaabaacbab', 'abaabaccacbacbacbabacbaabacbacbacbacb', 'bacbacc', 'cabcabacbacc', 'abaabaccacbacbacbabaab', 'bcabacbacbacbabaabacc', 'acabaabacccacbacbacb', 'acbacbaccaacb', 'abaabaacbabaabacc', 'abaabacbacbaccaacb', 'abacbcabacb', 'accc', 'acbacbcababaab', 'acbaccbacbacc']

%timeit answer1 = s1.findAllConcatenatedWordsInADict(words)
%timeit answer2 = s2.findAllConcatenatedWordsInADict(words)<jupyter_output>535 µs ± 17 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
299 µs ± 12.7 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)<jupyter_text>From the previous experiment we see that memoization helps speeding up the process to some extent. However, it's still not enough since the strategies we adopted so far can only be categorized as pruning, meaning that the overall number of vertices in the graph does not change, we merely just skipped some of the paths that we do not need to explore. To reduce the overall scale of the graph, we need to analyze the necessities of our blocks of codes. Originally, in one recursion call, we are matching each and every word in the given list with the beginning of the target word at the time. Therefore, with bad luck, we need to try out almost the entire list before we can draw a conclusion in one single recursion. Alternatively, we can converse this process to only try out a limited number of prefix words to match the word list since each word has an average length of 60. Moreover, if we construct the given word list with **hash table** data structure, the lookup time can be reduced to $O(1)$. Let's try that!<jupyter_code>class ConversedMemoizationSolution:
    def dfs(self, word, word_count):
        if word in self.cache:
            return True
        if len(word) == 0:
            if word_count > 1:
                return True
            return False

        success = False
        length = len(word)
        for i in range(1, length + 1):
            # test all the possible prefix words against the hash table we created
            prefix = word[:i]
            if not prefix in self.word_dict:
                continue
            success = self.dfs(word[i:], word_count + 1)
            if success:
                break
        return success


    def findAllConcatenatedWordsInADict(self, words):
        self.words = sorted(words, key=lambda x: len(x))
        self.cache = {}
        self.word_dict = {key: True for key in self.words}  # make a hash table out of the words
        answer = []
        for word in self.words:
            if self.dfs(word, 0):
                answer.append(word)
                self.cache[word] = True
        return answer

s1 = MemoizationSolution()
s2 = ConversedMemoizationSolution()

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words = origin[:4000]

%timeit answer1 = s1.findAllConcatenatedWordsInADict(words)
%timeit answer2 = s2.findAllConcatenatedWordsInADict(words)<jupyter_output>5.76 s ± 124 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
18 ms ± 569 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)<jupyter_text>The above trials tested the algorithm against 4000 words, and the results stated clearly that with the latest optimization, the running time is dramatically reduced. This means so far we have successfully solved the problem.However, a good computer engineer will ask the following question:> I know the problem is solved. But is this the best version of the code? And the answer is clearly **NO**. Let's think about the following issues:From the last optimization, we introduced a hash table, storing the input word list, using it to verify if a prefix is one of the given word. What if a prefix not only appears in the original word list, but it's also a valid concatenated word?Similarly, in the memoization optimization, we introduced another hash table caching the suffix that we calculated to be valid. What if a suffix does not appear in the cache, but it's one of the words in the input list?It's not that hard to derive the conclusion that the 2 hash tables are, de facto, analogous in their functionalities, thanks to the following reasons:1. a privously validated word can also serve as a valid prefix, and it's guaranteed to appear in the input list2. any word in the input list can be the prefix or suffix of another word during checking, even if itself is not a valid concatenated wordWith this understanding, we can optimize the code by combining the variables into 1:<jupyter_code>class Solution:
    def check(self, word):
        if word in self.dict:
            return True
        for i in range(len(word)):
            prefix = word[:i]
            if prefix in self.dict and self.check(word[i:]):
                return True
        return False


    def findAllConcatenatedWordsInADict(self, words):
        self.words = sorted(words, key=lambda w: len(w))
        self.words = list(filter(lambda w: len(w) > 0, self.words))
        if len(self.words) == 0:
            return []
        self.dict = {}
        answer = []
        for i, word in enumerate(self.words):
            if self.check(word):
                answer.append(word)
            self.dict[word] = True  # cache the word no matter if it's a concatenated word
        return answer

s1 = ConversedMemoizationSolution()
s2 = Solution()
    
answer1 = s1.findAllConcatenatedWordsInADict(words)
answer2 = s2.findAllConcatenatedWordsInADict(words)

assert answer1 == answer2
print("Answer is correct!")<jupyter_output>Answer is correct!
{'fraction_non_alphanumeric': 0.224930018112959, 'fraction_numerical': 0.002140622427136506, 'mean_word_length': 26.25173883778326, 'pattern_counts': {'":': 0, '<': 32, '<?xml version=': 0, '>': 29, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 1, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '7', 'dir': 'jupyter-structured-clean-dedup', 'id': '425526', 'license': 'MIT', 'n_tokens_mistral': 39908, 'n_tokens_neox': 37443, 'n_words': 2486, 'path': 'lectures/lecture2.ipynb', 'repo_name': 'AlgorithmClub/Algorithms', 'size': '121460'}