205. Isomorphic Strings - Explanation

Description

You are given two strings s and t, determine if they are isomorphic.

Two strings s and t are isomorphic if the characters in s can be replaced to get t.

All occurrences of a character must be replaced with another character while preserving the order of characters. No two characters may map to the same character, but a character may map to itself.

Example 1:

Input: s = "egg", t = "add"

Output: true

Explanation: The strings s and t can be made identical by:

Mapping 'e' to 'a'.
Mapping 'g' to 'd'.

Example 2:

Input: s = "foo", t = "bar"

Output: false

Explanation: The strings s and t can not be made identical as 'o' needs to be mapped to both 'a' and 'r'.

Example 3:

Input: s = "paper", t = "title"

Output: true

Constraints:

0 <= s.length == t.length <= 50,000
s and t consist of any valid ascii character.

Topics

Hash Table String

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Prerequisites

Before attempting this problem, you should be comfortable with:

Hash Maps - Storing and looking up key-value pairs in O(1) time
String Iteration - Traversing characters in a string by index
Bijective Mapping - Understanding one-to-one correspondence between two sets

1. Hash Map (Two Pass)

Intuition

Two strings are isomorphic if there's a one-to-one mapping between their characters. We need to ensure that each character in s maps to exactly one character in t, and vice versa. A single pass checking only s -> t mapping isn't enough because two different characters in s could map to the same character in t. By running the check twice (once for (s, t) and once for (t, s)), we guarantee the mapping is bijective.

Algorithm

Create a helper function that checks if characters from one string map consistently to another:
- Use a hash map to store the character mappings.
- For each character, check if an existing mapping conflicts with the current pair.
Call the helper twice: once with (s, t) and once with (t, s).
Return true only if both checks pass.

class Solution:
    def helper(self, s: str, t: str) -> bool:
        mp = {}
        for i in range(len(s)):
            if (s[i] in mp) and (mp[s[i]] != t[i]):
                return False
            mp[s[i]] = t[i]
        return True

    def isIsomorphic(self, s: str, t: str) -> bool:
        return self.helper(s, t) and self.helper(t, s)

public class Solution {
    private boolean helper(String s, String t) {
        HashMap<Character, Character> map = new HashMap<>();
        for (int i = 0; i < s.length(); i++) {
            char sc = s.charAt(i);
            char tc = t.charAt(i);
            if (map.containsKey(sc) && map.get(sc) != tc) {
                return false;
            }
            map.put(sc, tc);
        }
        return true;
    }

    public boolean isIsomorphic(String s, String t) {
        return helper(s, t) && helper(t, s);
    }
}

class Solution {
private:
    bool helper(const string& s, const string& t) {
        unordered_map<char, char> map;
        for (int i = 0; i < s.length(); i++) {
            if (map.count(s[i]) && map[s[i]] != t[i]) {
                return false;
            }
            map[s[i]] = t[i];
        }
        return true;
    }

public:
    bool isIsomorphic(string s, string t) {
        return helper(s, t) && helper(t, s);
    }
};

class Solution {
    /**
     * @param {string} s
     * @param {string} t
     * @return {boolean}
     */
    helper(s, t) {
        const map = new Map();
        for (let i = 0; i < s.length; i++) {
            if (map.has(s[i]) && map.get(s[i]) !== t[i]) {
                return false;
            }
            map.set(s[i], t[i]);
        }
        return true;
    }

    /**
     * @param {string} s
     * @param {string} t
     * @return {boolean}
     */
    isIsomorphic(s, t) {
        return this.helper(s, t) && this.helper(t, s);
    }
}

public class Solution {
    private bool Helper(string s, string t) {
        Dictionary<char, char> mp = new Dictionary<char, char>();
        for (int i = 0; i < s.Length; i++) {
            if (mp.ContainsKey(s[i]) && mp[s[i]] != t[i]) {
                return false;
            }
            mp[s[i]] = t[i];
        }
        return true;
    }

    public bool IsIsomorphic(string s, string t) {
        return Helper(s, t) && Helper(t, s);
    }
}

func isIsomorphic(s string, t string) bool {
    helper := func(s, t string) bool {
        mp := make(map[byte]byte)
        for i := 0; i < len(s); i++ {
            if val, ok := mp[s[i]]; ok && val != t[i] {
                return false
            }
            mp[s[i]] = t[i]
        }
        return true
    }
    return helper(s, t) && helper(t, s)
}

class Solution {
    private fun helper(s: String, t: String): Boolean {
        val mp = mutableMapOf<Char, Char>()
        for (i in s.indices) {
            if (mp.containsKey(s[i]) && mp[s[i]] != t[i]) {
                return false
            }
            mp[s[i]] = t[i]
        }
        return true
    }

    fun isIsomorphic(s: String, t: String): Boolean {
        return helper(s, t) && helper(t, s)
    }
}

class Solution {
    private func helper(_ s: String, _ t: String) -> Bool {
        let sArr = Array(s)
        let tArr = Array(t)
        var mp = [Character: Character]()
        for i in 0..<sArr.count {
            if let val = mp[sArr[i]], val != tArr[i] {
                return false
            }
            mp[sArr[i]] = tArr[i]
        }
        return true
    }

    func isIsomorphic(_ s: String, _ t: String) -> Bool {
        return helper(s, t) && helper(t, s)
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(m)$

Where $n$ is the length of the input string and $m$ is the number of unique characters in the strings.

2. Hash Map (One Pass)

Intuition

We can verify both mapping directions simultaneously in a single pass. By maintaining two hash maps, one for s -> t and one for t -> s, we check at each position that neither mapping is violated. If a character in s was previously mapped to a different character in t, or if a character in t was previously mapped to a different character in s, the strings aren't isomorphic.

Algorithm

Create two hash maps: mapST for s -> t and mapTS for t -> s.
Iterate through both strings simultaneously:
- If s[i] already maps to something other than t[i], return false.
- If t[i] already maps to something other than s[i], return false.
- Otherwise, record both mappings.
If the loop completes without conflicts, return true.

class Solution:
    def isIsomorphic(self, s: str, t: str) -> bool:
        mapST, mapTS = {}, {}

        for i in range(len(s)):
            c1, c2 = s[i], t[i]
            if ((c1 in mapST and mapST[c1] != c2) or
                (c2 in mapTS and mapTS[c2] != c1)):
                return False
            mapST[c1] = c2
            mapTS[c2] = c1

        return True

public class Solution {
    public boolean isIsomorphic(String s, String t) {
        HashMap<Character, Character> mapST = new HashMap<>();
        HashMap<Character, Character> mapTS = new HashMap<>();

        for (int i = 0; i < s.length(); i++) {
            char c1 = s.charAt(i), c2 = t.charAt(i);

            if ((mapST.containsKey(c1) && mapST.get(c1) != c2) ||
                (mapTS.containsKey(c2) && mapTS.get(c2) != c1)) {
                return false;
            }

            mapST.put(c1, c2);
            mapTS.put(c2, c1);
        }

        return true;
    }
}

class Solution {
public:
    bool isIsomorphic(string s, string t) {
        unordered_map<char, char> mapST, mapTS;

        for (int i = 0; i < s.size(); i++) {
            char c1 = s[i], c2 = t[i];

            if ((mapST.count(c1) && mapST[c1] != c2) ||
                (mapTS.count(c2) && mapTS[c2] != c1)) {
                return false;
            }

            mapST[c1] = c2;
            mapTS[c2] = c1;
        }

        return true;
    }
};

class Solution {
    /**
     * @param {string} s
     * @param {string} t
     * @return {boolean}
     */
    isIsomorphic(s, t) {
        const mapTS = new Map();
        const mapST = new Map();

        for (let i = 0; i < s.length; i++) {
            const c1 = s[i],
                c2 = t[i];

            if (
                (mapST.has(c1) && mapST.get(c1) !== c2) ||
                (mapTS.has(c2) && mapTS.get(c2) !== c1)
            ) {
                return false;
            }

            mapST.set(c1, c2);
            mapTS.set(c2, c1);
        }

        return true;
    }
}

public class Solution {
    public bool IsIsomorphic(string s, string t) {
        Dictionary<char, char> mapST = new Dictionary<char, char>();
        Dictionary<char, char> mapTS = new Dictionary<char, char>();

        for (int i = 0; i < s.Length; i++) {
            char c1 = s[i], c2 = t[i];

            if ((mapST.ContainsKey(c1) && mapST[c1] != c2) ||
                (mapTS.ContainsKey(c2) && mapTS[c2] != c1)) {
                return false;
            }

            mapST[c1] = c2;
            mapTS[c2] = c1;
        }

        return true;
    }
}

func isIsomorphic(s string, t string) bool {
    mapST := make(map[byte]byte)
    mapTS := make(map[byte]byte)

    for i := 0; i < len(s); i++ {
        c1, c2 := s[i], t[i]

        if val, ok := mapST[c1]; ok && val != c2 {
            return false
        }
        if val, ok := mapTS[c2]; ok && val != c1 {
            return false
        }

        mapST[c1] = c2
        mapTS[c2] = c1
    }

    return true
}

class Solution {
    fun isIsomorphic(s: String, t: String): Boolean {
        val mapST = mutableMapOf<Char, Char>()
        val mapTS = mutableMapOf<Char, Char>()

        for (i in s.indices) {
            val c1 = s[i]
            val c2 = t[i]

            if (mapST.containsKey(c1) && mapST[c1] != c2) {
                return false
            }
            if (mapTS.containsKey(c2) && mapTS[c2] != c1) {
                return false
            }

            mapST[c1] = c2
            mapTS[c2] = c1
        }

        return true
    }
}

class Solution {
    func isIsomorphic(_ s: String, _ t: String) -> Bool {
        let sArr = Array(s)
        let tArr = Array(t)
        var mapST = [Character: Character]()
        var mapTS = [Character: Character]()

        for i in 0..<sArr.count {
            let c1 = sArr[i]
            let c2 = tArr[i]

            if let val = mapST[c1], val != c2 {
                return false
            }
            if let val = mapTS[c2], val != c1 {
                return false
            }

            mapST[c1] = c2
            mapTS[c2] = c1
        }

        return true
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(m)$

Where $n$ is the length of the input string and $m$ is the number of unique characters in the strings.

Common Pitfalls

Only Checking One Direction of Mapping

Checking only that each character in s maps to a unique character in t is insufficient. You must also verify the reverse: that each character in t maps to a unique character in s. For example, s = "ab" and t = "aa" would pass a one-way check but fails the bidirectional requirement since both 'a' and 'b' map to 'a'.

Assuming Same Character Sets

The two strings may contain completely different character sets. Do not assume that a character appearing in s must also appear in t, or vice versa. The mapping must be established dynamically as you iterate through both strings.