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Word Ladder (Length of shortest chain to reach a target word)
• Difficulty Level : Medium
• Last Updated : 05 Apr, 2021

Given a dictionary, and two words ‘start’ and ‘target’ (both of same length). Find length of the smallest chain from ‘start’ to ‘target’ if it exists, such that adjacent words in the chain only differ by one character and each word in the chain is a valid word i.e., it exists in the dictionary. It may be assumed that the ‘target’ word exists in dictionary and length of all dictionary words is same.

Example:

```Input: Dictionary = {POON, PLEE, SAME, POIE, PLEA, PLIE, POIN}
start = TOON
target = PLEA
Output: 7
TOON - POON - POIN - POIE - PLIE - PLEE - PLEA

Input: Dictionary = {ABCD, EBAD, EBCD, XYZA}
Start = ABCV
End = EBAD
Output: 4
ABCV - ABCD - EBCD - EBAD```

Solution: The idea is to use BFS
Approach:

1. Start from the given start word.
2. Push the word in the queue
3. Run a loop until the queue is empty
4. Traverse all words that adjacent (differ by one character) to it and push the word in a queue (for BFS)
5. Keep doing so until we find the target word or we have traversed all words.

Below are the implementations of the above idea.

## C++

 `// C++ program to find length``// of the shortest chain``// transformation from source``// to target``#include ``using` `namespace` `std;` `// Returns length of shortest chain``// to reach 'target' from 'start'``// using minimum number of adjacent``// moves.  D is dictionary``int` `shortestChainLen(``string start, string target,``set& D)``{``  ` `    ``if``(start == target)``      ``return` `0;` `    ``// If the target string is not``    ``// present in the dictionary``    ``if` `(D.find(target) == D.end())``        ``return` `0;` `    ``// To store the current chain length``    ``// and the length of the words``    ``int` `level = 0, wordlength = start.size();` `    ``// Push the starting word into the queue``    ``queue Q;``    ``Q.push(start);` `    ``// While the queue is non-empty``    ``while` `(!Q.empty()) {` `        ``// Increment the chain length``        ``++level;` `        ``// Current size of the queue``        ``int` `sizeofQ = Q.size();` `        ``// Since the queue is being updated while``        ``// it is being traversed so only the``        ``// elements which were already present``        ``// in the queue before the start of this``        ``// loop will be traversed for now``        ``for` `(``int` `i = 0; i < sizeofQ; ++i) {` `            ``// Remove the first word from the queue``            ``string word = Q.front();``            ``Q.pop();` `            ``// For every character of the word``            ``for` `(``int` `pos = 0; pos < wordlength; ++pos) {` `                ``// Retain the original character``                ``// at the current position``                ``char` `orig_char = word[pos];` `                ``// Replace the current character with``                ``// every possible lowercase alphabet``                ``for` `(``char` `c = ``'a'``; c <= ``'z'``; ++c) {``                    ``word[pos] = c;` `                    ``// If the new word is equal``                    ``// to the target word``                    ``if` `(word == target)``                        ``return` `level + 1;` `                    ``// Remove the word from the set``                    ``// if it is found in it``                    ``if` `(D.find(word) == D.end())``                        ``continue``;``                    ``D.erase(word);` `                    ``// And push the newly generated word``                    ``// which will be a part of the chain``                    ``Q.push(word);``                ``}` `                ``// Restore the original character``                ``// at the current position``                ``word[pos] = orig_char;``            ``}``        ``}``    ``}` `    ``return` `0;``}` `// Driver program``int` `main()``{``    ``// make dictionary``    ``set D;``    ``D.insert(``"poon"``);``    ``D.insert(``"plee"``);``    ``D.insert(``"same"``);``    ``D.insert(``"poie"``);``    ``D.insert(``"plie"``);``    ``D.insert(``"poin"``);``    ``D.insert(``"plea"``);``    ``string start = ``"toon"``;``    ``string target = ``"plea"``;``    ``cout << ``"Length of shortest chain is: "``         ``<< shortestChainLen(start, target, D);``    ``return` `0;``}`

## Java

 `// Java program to find length``// of the shortest chain``// transformation from source``// to target``import` `java.util.*;` `class` `GFG``{` `// Returns length of shortest chain``// to reach 'target' from 'start'``// using minimum number of adjacent moves.``// D is dictionary``static` `int` `shortestChainLen(String start,``                            ``String target,``                            ``Set D)``{` `     ``if``(start == target)``      ``return` `0``;``    ``// If the target String is not``    ``// present in the dictionary``    ``if` `(!D.contains(target))``        ``return` `0``;` `    ``// To store the current chain length``    ``// and the length of the words``    ``int` `level = ``0``, wordlength = start.length();` `    ``// Push the starting word into the queue``    ``Queue Q = ``new` `LinkedList<>();``    ``Q.add(start);` `    ``// While the queue is non-empty``    ``while` `(!Q.isEmpty())``    ``{` `        ``// Increment the chain length``        ``++level;` `        ``// Current size of the queue``        ``int` `sizeofQ = Q.size();` `        ``// Since the queue is being updated while``        ``// it is being traversed so only the``        ``// elements which were already present``        ``// in the queue before the start of this``        ``// loop will be traversed for now``        ``for` `(``int` `i = ``0``; i < sizeofQ; ++i)``        ``{` `            ``// Remove the first word from the queue``            ``char` `[]word = Q.peek().toCharArray();``            ``Q.remove();` `            ``// For every character of the word``            ``for` `(``int` `pos = ``0``; pos < wordlength; ++pos)``            ``{` `                ``// Retain the original character``                ``// at the current position``                ``char` `orig_char = word[pos];` `                ``// Replace the current character with``                ``// every possible lowercase alphabet``                ``for` `(``char` `c = ``'a'``; c <= ``'z'``; ++c)``                ``{``                    ``word[pos] = c;` `                    ``// If the new word is equal``                    ``// to the target word``                    ``if` `(String.valueOf(word).equals(target))``                        ``return` `level + ``1``;` `                    ``// Remove the word from the set``                    ``// if it is found in it``                    ``if` `(!D.contains(String.valueOf(word)))``                        ``continue``;``                    ``D.remove(String.valueOf(word));` `                    ``// And push the newly generated word``                    ``// which will be a part of the chain``                    ``Q.add(String.valueOf(word));``                ``}` `                ``// Restore the original character``                ``// at the current position``                ``word[pos] = orig_char;``            ``}``        ``}``    ``}` `    ``return` `0``;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``// make dictionary``    ``Set D = ``new` `HashSet();``    ``D.add(``"poon"``);``    ``D.add(``"plee"``);``    ``D.add(``"same"``);``    ``D.add(``"poie"``);``    ``D.add(``"plie"``);``    ``D.add(``"poin"``);``    ``D.add(``"plea"``);``    ``String start = ``"toon"``;``    ``String target = ``"plea"``;``    ``System.out.print(``"Length of shortest chain is: "``        ``+ shortestChainLen(start, target, D));``}``}` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 program to find length of the``# shortest chain transformation from source``# to target``from` `collections ``import` `deque` `# Returns length of shortest chain``# to reach 'target' from 'start'``# using minimum number of adjacent``# moves. D is dictionary``def` `shortestChainLen(start, target, D):``    ` `    ``if` `start ``=``=` `target:``      ``return` `0``    ``# If the target is not``    ``# present in the dictionary``    ``if` `target ``not` `in` `D:``        ``return` `0` `    ``# To store the current chain length``    ``# and the length of the words``    ``level, wordlength ``=` `0``, ``len``(start)` `    ``# Push the starting word into the queue``    ``Q ``=`  `deque()``    ``Q.append(start)` `    ``# While the queue is non-empty``    ``while` `(``len``(Q) > ``0``):``        ` `        ``# Increment the chain length``        ``level ``+``=` `1` `        ``# Current size of the queue``        ``sizeofQ ``=` `len``(Q)` `        ``# Since the queue is being updated while``        ``# it is being traversed so only the``        ``# elements which were already present``        ``# in the queue before the start of this``        ``# loop will be traversed for now``        ``for` `i ``in` `range``(sizeofQ):` `            ``# Remove the first word from the queue``            ``word ``=` `[j ``for` `j ``in` `Q.popleft()]``            ``#Q.pop()` `            ``# For every character of the word``            ``for` `pos ``in` `range``(wordlength):``                ` `                ``# Retain the original character``                ``# at the current position``                ``orig_char ``=` `word[pos]` `                ``# Replace the current character with``                ``# every possible lowercase alphabet``                ``for` `c ``in` `range``(``ord``(``'a'``), ``ord``(``'z'``)``+``1``):``                    ``word[pos] ``=` `chr``(c)` `                    ``# If the new word is equal``                    ``# to the target word``                    ``if` `("".join(word) ``=``=` `target):``                        ``return` `level ``+` `1` `                    ``# Remove the word from the set``                    ``# if it is found in it``                    ``if` `("".join(word) ``not` `in` `D):``                        ``continue``                        ` `                    ``del` `D["".join(word)]` `                    ``# And push the newly generated word``                    ``# which will be a part of the chain``                    ``Q.append("".join(word))` `                ``# Restore the original character``                ``# at the current position``                ``word[pos] ``=` `orig_char` `    ``return` `0` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``# Make dictionary``    ``D ``=` `{}``    ``D[``"poon"``] ``=` `1``    ``D[``"plee"``] ``=` `1``    ``D[``"same"``] ``=` `1``    ``D[``"poie"``] ``=` `1``    ``D[``"plie"``] ``=` `1``    ``D[``"poin"``] ``=` `1``    ``D[``"plea"``] ``=` `1``    ``start ``=` `"toon"``    ``target ``=` `"plea"``    ` `    ``print``(``"Length of shortest chain is: "``,``    ``shortestChainLen(start, target, D))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program to find length of the shortest chain``// transformation from source to target``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{` `// Returns length of shortest chain``// to reach 'target' from 'start'``// using minimum number of adjacent moves.``// D is dictionary``static` `int` `shortestChainLen(String start,``                            ``String target,``                            ``HashSet D)``{` `     ``if``(start == target)``       ``return` `0;``    ``// If the target String is not``    ``// present in the dictionary``    ``if` `(!D.Contains(target))``        ``return` `0;` `    ``// To store the current chain length``    ``// and the length of the words``    ``int` `level = 0, wordlength = start.Length;` `    ``// Push the starting word into the queue``    ``List Q = ``new` `List();``    ``Q.Add(start);` `    ``// While the queue is non-empty``    ``while` `(Q.Count != 0)``    ``{` `        ``// Increment the chain length``        ``++level;` `        ``// Current size of the queue``        ``int` `sizeofQ = Q.Count;` `        ``// Since the queue is being updated while``        ``// it is being traversed so only the``        ``// elements which were already present``        ``// in the queue before the start of this``        ``// loop will be traversed for now``        ``for` `(``int` `i = 0; i < sizeofQ; ++i)``        ``{` `            ``// Remove the first word from the queue``            ``char` `[]word = Q.ToCharArray();``            ``Q.RemoveAt(0);` `            ``// For every character of the word``            ``for` `(``int` `pos = 0; pos < wordlength; ++pos)``            ``{` `                ``// Retain the original character``                ``// at the current position``                ``char` `orig_char = word[pos];` `                ``// Replace the current character with``                ``// every possible lowercase alphabet``                ``for` `(``char` `c = ``'a'``; c <= ``'z'``; ++c)``                ``{``                    ``word[pos] = c;` `                    ``// If the new word is equal``                    ``// to the target word``                    ``if` `(String.Join(``""``, word).Equals(target))``                        ``return` `level + 1;` `                    ``// Remove the word from the set``                    ``// if it is found in it``                    ``if` `(!D.Contains(String.Join(``""``, word)))``                        ``continue``;``                    ``D.Remove(String.Join(``""``, word));` `                    ``// And push the newly generated word``                    ``// which will be a part of the chain``                    ``Q.Add(String.Join(``""``, word));``                ``}` `                ``// Restore the original character``                ``// at the current position``                ``word[pos] = orig_char;``            ``}``        ``}``    ``}``    ``return` `0;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``// make dictionary``    ``HashSet D = ``new` `HashSet();``    ``D.Add(``"poon"``);``    ``D.Add(``"plee"``);``    ``D.Add(``"same"``);``    ``D.Add(``"poie"``);``    ``D.Add(``"plie"``);``    ``D.Add(``"poin"``);``    ``D.Add(``"plea"``);``    ``String start = ``"toon"``;``    ``String target = ``"plea"``;``    ``Console.Write(``"Length of shortest chain is: "``        ``+ shortestChainLen(start, target, D));``}``}` `// This code is contributed by PrinciRaj1992`
Output

`Length of shortest chain is: 7`

Complexity Analysis:

• Time complexity: O(n²m), where m is the number of entries originally in the dictionary and n is the size of the string.

• Auxiliary Space:O(m*n), where m are the strings are stored in queue.
So the space Complexity is O(m*n).

Alternate Implementation: (Maintaining the mapping of the intermediate words and the original word):

Below is an alternative implementation to the above approach.

Here, in this approach, we find out all the intermediate words of the start word and the words in the given list of dictionary and maintain a map of the intermediate word and a vector of the original word (map<string, vector<string>>). For instance, for the word “POON”, the intermediate words are “*OON” , “P*ON”, “PO*N”, “POO*”. Then, we perform BFS traversal starting with the start word and push a pair of start word and the distance (pair(word, distance)) to the queue until we reach the target word. Then, the distance is our answer.

## C++

 `// C++ program to find length``// of the shortest chain``// transformation from source``// to target``#include ``using` `namespace` `std;` `// Returns length of shortest chain``// to reach 'target' from 'start'``// using minimum number of adjacent``// moves.  D is dictionary``int` `shortestChainLen(``string start, string target,``set& D)``{``  ` `   ``if``(start == target)``      ``return` `0;``  ` `  ``// Map of intermediate words and``  ``// the list of original words``  ``map> umap;` `  ``// Find all the intermediate``  ``// words for the start word``  ``for``(``int` `i = 0; i < start.size(); i++)``  ``{``    ``string str = start.substr(0,i) + ``"*"` `+``                        ``start.substr(i+1);``    ``umap[str].push_back(start);``  ``}` `  ``// Find all the intermediate words for``  ``// the words in the given Set``  ``for``(``auto` `it = D.begin(); it != D.end(); it++)``  ``{``    ``string word = *it;``    ``for``(``int` `j = 0; j < word.size(); j++)``    ``{``      ``string str = word.substr(0,j) + ``"*"` `+``                          ``word.substr(j+1);``      ``umap[str].push_back(word);``    ``}``  ``}` `  ``// Perform BFS and push (word, distance)``  ``queue> q;` `  ``map visited;` `  ``q.push(make_pair(start,1));``  ``visited[start] = 1;` `  ``// Traverse until queue is empty``  ``while``(!q.empty())``  ``{``    ``pair p = q.front();``    ``q.pop();` `    ``string word = p.first;``    ``int` `dist = p.second;` `    ``// If target word is found``    ``if``(word == target)``    ``{``      ``return` `dist;``    ``}` `    ``// Finding intermediate words for``    ``// the word in front of queue``    ``for``(``int` `i = 0; i < word.size(); i++)``    ``{``      ``string str = word.substr(0,i) + ``"*"` `+``                           ``word.substr(i+1);` `      ``vector vect = umap[str];``      ``for``(``int` `j = 0; j < vect.size(); j++)``      ``{``        ``// If the word is not visited``        ``if``(visited[vect[j]] == 0)``        ``{``          ``visited[vect[j]] = 1;``          ``q.push(make_pair(vect[j], dist + 1));``        ``}``      ``}``    ``}` `  ``}` `    ``return` `0;``}` `// Driver code``int` `main()``{``    ``// Make dictionary``    ``set D;``    ``D.insert(``"poon"``);``    ``D.insert(``"plee"``);``    ``D.insert(``"same"``);``    ``D.insert(``"poie"``);``    ``D.insert(``"plie"``);``    ``D.insert(``"poin"``);``    ``D.insert(``"plea"``);``    ``string start = ``"toon"``;``    ``string target = ``"plea"``;``    ``cout << ``"Length of shortest chain is: "``         ``<< shortestChainLen(start, target, D);``    ``return` `0;``}`
Output
`Length of shortest chain is: 7`

Thanks to Gaurav Ahirwar and Rajnish Kumar Jha for the above solution.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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