Day 19: add explanation

How the "search" method of the "Trie" class works

19/python/explanation.md

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+## Explanation of the `search` method of the `Trie` class
+
+Here's the `search` method of the `Trie` class:
+
+```python
+def search(self, string: str, debug: bool = False) -> bool:
+    n = len(string)
+    dp = [True] + [False] * n
+
+    for i in range(n):
+        if debug:
+            print(dp)
+
+        if not dp[i]:
+            continue
+
+        node = self.root
+        j = i
+        while j < n and string[j] in node.children:
+            node = node.children[string[j]]
+            j += 1
+            if node.is_end:
+                dp[j] = True
+                if j == n:
+                    return True
+    return dp[n]
+```
+
+Say that we have the following trie – where `*` denotes a a node that marks the end of a pattern:
+
+```
+b*
+ r*
+ w
+  u*
+g*
+ b*
+r*
+ b*
+w
+ r*
+```
+
+Here's the output when we try to search for the string `brwrr`:
+
+```
+Searching for string: brwrr
+Initial dp: [True, False, False, False, False, False]
+
+Trying from position 0 (brwrr)
+  Looking for patterns:
+    Found pattern b -> marking dp[1] = True
+    Found pattern br -> marking dp[2] = True
+  dp after position 0: [True, True, True, False, False, False]
+
+Trying from position 1 (rwrr)
+  Looking for patterns:
+    Found pattern r -> marking dp[2] = True
+  dp after position 1: [True, True, True, False, False, False]
+
+Trying from position 2 (wrr)
+  Looking for patterns:
+    Found pattern wr -> marking dp[4] = True
+  dp after position 2: [True, True, True, False, True, False]
+
+Trying from position 3 (rr)
+  Skipping pos 3 since dp[3] is False
+
+Trying from position 4 (r)
+  Looking for patterns:
+    Found pattern r -> marking dp[5] = True
+  dp after position 4: [True, True, True, False, True, True]
+
+Final dp: [True, True, True, False, True, True]
+```
+
+It's crucial to understand what the list `dp` represents: `dp[i] = True` means that we can build the substring `string[0:i]` (where `i` is **not** included, as per Python indexing/slicing rules) using available patterns.
+
+For example, considering the string `brwrr`:
+
+- `dp[0] = True`: can build empty string, which is always possible
+- `dp[1] = True`: can build "b" (since we have a pattern "b")
+- `dp[2] = True`: can build "br" (since we have a pattern "br")
+- `dp[3] = False`: cannot build "brw" because "w" only appears in pattern "wr", never alone
+- `dp[4] = True`: can build "brwr" (since we have a pattern "br" + "wr")
+- `dp[5] = True`: can build "brwrr" (since we have a pattern "br" + "wr" + "r")