1898. Maximum Number of Removable Characters - Explanation

Prerequisites

Before attempting this problem, you should be comfortable with:

Binary Search - Efficiently searching a sorted or monotonic search space by halving the range
Subsequence Checking (Two Pointers) - Verifying if one string is a subsequence of another using two pointers
Hash Set - Tracking removed indices for O(1) lookup during subsequence validation

1. Brute Force

Intuition

We want to find the maximum number of characters we can remove from s (in the order given by removable) while keeping p as a subsequence of s.

The simplest approach is to remove characters one at a time. After each removal, check if p is still a subsequence. Once p is no longer a subsequence, stop and return the count of successful removals.

Algorithm

Maintain a set marked of indices that have been removed.
Iterate through removable:
- Add the current index to marked.
- Check if p is a subsequence of s (skipping marked indices).
- If p is still a subsequence, increment the result counter.
- If not, break out of the loop.
Return the count of successful removals.

class Solution:
    def maximumRemovals(self, s: str, p: str, removable: List[int]) -> int:
        n, m = len(s), len(p)
        marked = set()
        res = 0

        for removeIdx in removable:
            marked.add(removeIdx)

            sIdx = pIdx = 0
            while pIdx < m and sIdx < n:
                if sIdx not in marked and s[sIdx] == p[pIdx]:
                    pIdx += 1
                sIdx += 1
            if pIdx != m:
                break
            res += 1

        return res

public class Solution {
    public int maximumRemovals(String s, String p, int[] removable) {
        int n = s.length(), m = p.length();
        Set<Integer> marked = new HashSet<>();
        int res = 0;

        for (int removeIdx : removable) {
            marked.add(removeIdx);

            int sIdx = 0, pIdx = 0;
            while (pIdx < m && sIdx < n) {
                if (!marked.contains(sIdx) && s.charAt(sIdx) == p.charAt(pIdx)) {
                    pIdx++;
                }
                sIdx++;
            }

            if (pIdx != m) break;
            res++;
        }

        return res;
    }
}

class Solution {
public:
    int maximumRemovals(string s, string p, vector<int>& removable) {
        int n = s.size(), m = p.size();
        unordered_set<int> marked;
        int res = 0;

        for (int removeIdx : removable) {
            marked.insert(removeIdx);

            int sIdx = 0, pIdx = 0;
            while (pIdx < m && sIdx < n) {
                if (marked.find(sIdx) == marked.end() && s[sIdx] == p[pIdx]) {
                    pIdx++;
                }
                sIdx++;
            }

            if (pIdx != m) break;
            res++;
        }

        return res;
    }
};

class Solution {
    /**
     * @param {string} s
     * @param {string} p
     * @param {number[]} removable
     * @return {number}
     */
    maximumRemovals(s, p, removable) {
        let n = s.length,
            m = p.length;
        let marked = new Set();
        let res = 0;

        for (let removeIdx of removable) {
            marked.add(removeIdx);

            let sIdx = 0,
                pIdx = 0;
            while (pIdx < m && sIdx < n) {
                if (!marked.has(sIdx) && s[sIdx] === p[pIdx]) {
                    pIdx++;
                }
                sIdx++;
            }

            if (pIdx !== m) break;
            res++;
        }

        return res;
    }
}

func maximumRemovals(s string, p string, removable []int) int {
    n, m := len(s), len(p)
    marked := make(map[int]bool)
    res := 0

    for _, removeIdx := range removable {
        marked[removeIdx] = true

        sIdx, pIdx := 0, 0
        for pIdx < m && sIdx < n {
            if !marked[sIdx] && s[sIdx] == p[pIdx] {
                pIdx++
            }
            sIdx++
        }

        if pIdx != m {
            break
        }
        res++
    }

    return res
}

class Solution {
    fun maximumRemovals(s: String, p: String, removable: IntArray): Int {
        val n = s.length
        val m = p.length
        val marked = mutableSetOf<Int>()
        var res = 0

        for (removeIdx in removable) {
            marked.add(removeIdx)

            var sIdx = 0
            var pIdx = 0
            while (pIdx < m && sIdx < n) {
                if (sIdx !in marked && s[sIdx] == p[pIdx]) {
                    pIdx++
                }
                sIdx++
            }

            if (pIdx != m) break
            res++
        }

        return res
    }
}

class Solution {
    func maximumRemovals(_ s: String, _ p: String, _ removable: [Int]) -> Int {
        let sArr = Array(s)
        let pArr = Array(p)
        let n = sArr.count
        let m = pArr.count
        var marked = Set<Int>()
        var res = 0

        for removeIdx in removable {
            marked.insert(removeIdx)

            var sIdx = 0
            var pIdx = 0
            while pIdx < m && sIdx < n {
                if !marked.contains(sIdx) && sArr[sIdx] == pArr[pIdx] {
                    pIdx += 1
                }
                sIdx += 1
            }

            if pIdx != m { break }
            res += 1
        }

        return res
    }
}

Time & Space Complexity

Time complexity: $O(k * (n + m))$
Space complexity: $O(k)$

Where $n$ and $m$ are the lengths of the given strings $s$ and $p$ respectively. $k$ is the size of the array $removable$ .

2. Binary Search + Hash Set

Intuition

If removing the first k characters still allows p to be a subsequence, then removing fewer than k characters will also work. Conversely, if removing k characters breaks the subsequence property, removing more will certainly break it too.

This monotonic property makes binary search applicable. We search for the largest k such that after removing removable[0..k], p remains a subsequence.

Algorithm

Binary search over the number of removals k (range: 0 to length of removable - 1).
For each midpoint m:
- Create a set of removed indices from removable[0..m].
- Check if p is a subsequence of s (skipping removed indices).
- If yes, search the right half (try more removals).
- If no, search the left half (try fewer removals).
Track the maximum valid k found.
Return the result.

class Solution:
    def maximumRemovals(self, s: str, p: str, removable: List[int]) -> int:
        def isSubseq(s, subseq, removed):
            i1 = i2 = 0
            while i1 < len(s) and i2 < len(subseq):
                if i1 in removed or s[i1] != subseq[i2]:
                    i1 += 1
                    continue
                i1 += 1
                i2 += 1
            return i2 == len(subseq)

        res = 0
        l, r = 0, len(removable) - 1
        while l <= r:
            m = (l + r) // 2
            removed = set(removable[:m + 1])
            if isSubseq(s, p, removed):
                res = max(res, m + 1)
                l = m + 1
            else:
                r = m - 1

        return res

public class Solution {
    public int maximumRemovals(String s, String p, int[] removable) {
        int res = 0, l = 0, r = removable.length - 1;

        while (l <= r) {
            int m = (l + r) / 2;
            Set<Integer> removed = new HashSet<>();
            for (int i = 0; i <= m; i++) {
                removed.add(removable[i]);
            }

            if (isSubseq(s, p, removed)) {
                res = Math.max(res, m + 1);
                l = m + 1;
            } else {
                r = m - 1;
            }
        }

        return res;
    }

    private boolean isSubseq(String s, String subseq, Set<Integer> removed) {
        int i1 = 0, i2 = 0;
        while (i1 < s.length() && i2 < subseq.length()) {
            if (removed.contains(i1) || s.charAt(i1) != subseq.charAt(i2)) {
                i1++;
                continue;
            }
            i1++;
            i2++;
        }
        return i2 == subseq.length();
    }
}

class Solution {
public:
    int maximumRemovals(string s, string p, vector<int>& removable) {
        int res = 0, l = 0, r = removable.size() - 1;

        while (l <= r) {
            int m = (l + r) / 2;
            unordered_set<int> removed(removable.begin(), removable.begin() + m + 1);

            if (isSubseq(s, p, removed)) {
                res = max(res, m + 1);
                l = m + 1;
            } else {
                r = m - 1;
            }
        }

        return res;
    }

private:
    bool isSubseq(string& s, string& subseq, unordered_set<int>& removed) {
        int i1 = 0, i2 = 0;
        while (i1 < s.size() && i2 < subseq.size()) {
            if (removed.count(i1) || s[i1] != subseq[i2]) {
                i1++;
                continue;
            }
            i1++;
            i2++;
        }
        return i2 == subseq.size();
    }
};

class Solution {
    /**
     * @param {string} s
     * @param {string} p
     * @param {number[]} removable
     * @return {number}
     */
    maximumRemovals(s, p, removable) {
        let res = 0,
            l = 0,
            r = removable.length - 1;

        while (l <= r) {
            let m = Math.floor((l + r) / 2);
            let removed = new Set(removable.slice(0, m + 1));

            if (this.isSubseq(s, p, removed)) {
                res = Math.max(res, m + 1);
                l = m + 1;
            } else {
                r = m - 1;
            }
        }

        return res;
    }

    /**
     * @param {string} s
     * @param {string} subseq
     * @param {Set<number>} removed
     * @return {boolean}
     */
    isSubseq(s, subseq, removed) {
        let i1 = 0,
            i2 = 0;
        while (i1 < s.length && i2 < subseq.length) {
            if (removed.has(i1) || s[i1] !== subseq[i2]) {
                i1++;
                continue;
            }
            i1++;
            i2++;
        }
        return i2 === subseq.length;
    }
}

func maximumRemovals(s string, p string, removable []int) int {
    res, l, r := 0, 0, len(removable)-1

    isSubseq := func(s, subseq string, removed map[int]bool) bool {
        i1, i2 := 0, 0
        for i1 < len(s) && i2 < len(subseq) {
            if removed[i1] || s[i1] != subseq[i2] {
                i1++
                continue
            }
            i1++
            i2++
        }
        return i2 == len(subseq)
    }

    for l <= r {
        m := (l + r) / 2
        removed := make(map[int]bool)
        for i := 0; i <= m; i++ {
            removed[removable[i]] = true
        }

        if isSubseq(s, p, removed) {
            if m+1 > res {
                res = m + 1
            }
            l = m + 1
        } else {
            r = m - 1
        }
    }

    return res
}

class Solution {
    fun maximumRemovals(s: String, p: String, removable: IntArray): Int {
        var res = 0
        var l = 0
        var r = removable.size - 1

        while (l <= r) {
            val m = (l + r) / 2
            val removed = removable.sliceArray(0..m).toHashSet()

            if (isSubseq(s, p, removed)) {
                res = maxOf(res, m + 1)
                l = m + 1
            } else {
                r = m - 1
            }
        }

        return res
    }

    private fun isSubseq(s: String, subseq: String, removed: Set<Int>): Boolean {
        var i1 = 0
        var i2 = 0
        while (i1 < s.length && i2 < subseq.length) {
            if (i1 in removed || s[i1] != subseq[i2]) {
                i1++
                continue
            }
            i1++
            i2++
        }
        return i2 == subseq.length
    }
}

class Solution {
    func maximumRemovals(_ s: String, _ p: String, _ removable: [Int]) -> Int {
        let sArr = Array(s)
        let pArr = Array(p)
        var res = 0
        var l = 0
        var r = removable.count - 1

        func isSubseq(_ removed: Set<Int>) -> Bool {
            var i1 = 0
            var i2 = 0
            while i1 < sArr.count && i2 < pArr.count {
                if removed.contains(i1) || sArr[i1] != pArr[i2] {
                    i1 += 1
                    continue
                }
                i1 += 1
                i2 += 1
            }
            return i2 == pArr.count
        }

        while l <= r {
            let m = (l + r) / 2
            let removed = Set(removable[0...m])

            if isSubseq(removed) {
                res = max(res, m + 1)
                l = m + 1
            } else {
                r = m - 1
            }
        }

        return res
    }
}

Time & Space Complexity

Time complexity: $O((n + m) * \log k)$
Space complexity: $O(k)$

Where $n$ and $m$ are the lengths of the given strings $s$ and $p$ respectively. $k$ is the size of the array $removable$ .

3. Binary Search

Intuition

Instead of using a hash set to track removed indices (which adds overhead), we can directly modify the string by replacing removed characters with a placeholder character (like '#'). This avoids hash lookups during the subsequence check.

The binary search logic remains the same, but the subsequence check becomes simpler since we just compare characters, skipping any '#' naturally by checking for equality.

Algorithm

Binary search over the number of removals with l = 0 and r = length of removable.
For each midpoint mid:
- Create a copy of s as a character array.
- Mark positions removable[0..mid] as '#'.
- Check if p is a subsequence (characters equal to '#' will never match).
- If yes, move l = mid + 1.
- If no, move r = mid.
Return l as the maximum number of removals.

class Solution:
    def maximumRemovals(self, s: str, p: str, removable: List[int]) -> int:
        l, r = 0, len(removable)
        n, m = len(s), len(p)

        def isSubseq(tmpS):
            i1 = i2 = 0
            while i1 < n and i2 < m:
                if tmpS[i1] == p[i2]:
                    i2 += 1
                i1 += 1
            return i2 == m

        while l < r:
            mid = l + (r - l) // 2
            tmpS = list(s)

            for i in range(mid + 1):
                tmpS[removable[i]] = '#'

            if isSubseq(tmpS):
                l = mid + 1
            else:
                r = mid

        return l

public class Solution {
    public int maximumRemovals(String s, String p, int[] removable) {
        int l = 0, r = removable.length;
        int n = s.length(), m = p.length();

        while (l < r) {
            int mid = l + (r - l) / 2;
            char[] tmpS = s.toCharArray();

            for (int i = 0; i <= mid; i++) {
                tmpS[removable[i]] = '#';
            }

            if (isSubseq(tmpS, p)) {
                l = mid + 1;
            } else {
                r = mid;
            }
        }

        return l;
    }

    private boolean isSubseq(char[] s, String p) {
        int i1 = 0, i2 = 0, n = s.length, m = p.length();

        while (i1 < n && i2 < m) {
            if (s[i1] == p.charAt(i2)) {
                i2++;
            }
            i1++;
        }
        return i2 == m;
    }
}

class Solution {
public:
    int maximumRemovals(string s, string p, vector<int>& removable) {
        int l = 0, r = removable.size();
        int n = s.size(), m = p.size();

        while (l < r) {
            int mid = l + (r - l) / 2;
            string tmpS = s;

            for (int i = 0; i <= mid; i++) {
                tmpS[removable[i]] = '#';
            }

            if (isSubseq(tmpS, p)) {
                l = mid + 1;
            } else {
                r = mid;
            }
        }

        return l;
    }

private:
    bool isSubseq(const string& s, const string& p) {
        int i1 = 0, i2 = 0, n = s.size(), m = p.size();

        while (i1 < n && i2 < m) {
            if (s[i1] == p[i2]) {
                i2++;
            }
            i1++;
        }
        return i2 == m;
    }
};

class Solution {
    /**
     * @param {string} s
     * @param {string} p
     * @param {number[]} removable
     * @return {number}
     */
    maximumRemovals(s, p, removable) {
        let l = 0,
            r = removable.length;
        let n = s.length,
            m = p.length;

        const isSubseq = (tmpS) => {
            let i1 = 0,
                i2 = 0;
            while (i1 < n && i2 < m) {
                if (tmpS[i1] === p[i2]) {
                    i2++;
                }
                i1++;
            }
            return i2 === m;
        };

        while (l < r) {
            let mid = Math.floor(l + (r - l) / 2);
            let tmpS = s.split('');

            for (let i = 0; i <= mid; i++) {
                tmpS[removable[i]] = '#';
            }

            if (isSubseq(tmpS)) {
                l = mid + 1;
            } else {
                r = mid;
            }
        }

        return l;
    }
}

func maximumRemovals(s string, p string, removable []int) int {
    l, r := 0, len(removable)
    n, m := len(s), len(p)

    isSubseq := func(tmpS []byte) bool {
        i1, i2 := 0, 0
        for i1 < n && i2 < m {
            if tmpS[i1] == p[i2] {
                i2++
            }
            i1++
        }
        return i2 == m
    }

    for l < r {
        mid := l + (r-l)/2
        tmpS := []byte(s)

        for i := 0; i <= mid; i++ {
            tmpS[removable[i]] = '#'
        }

        if isSubseq(tmpS) {
            l = mid + 1
        } else {
            r = mid
        }
    }

    return l
}

class Solution {
    fun maximumRemovals(s: String, p: String, removable: IntArray): Int {
        var l = 0
        var r = removable.size
        val n = s.length
        val m = p.length

        fun isSubseq(tmpS: CharArray): Boolean {
            var i1 = 0
            var i2 = 0
            while (i1 < n && i2 < m) {
                if (tmpS[i1] == p[i2]) {
                    i2++
                }
                i1++
            }
            return i2 == m
        }

        while (l < r) {
            val mid = l + (r - l) / 2
            val tmpS = s.toCharArray()

            for (i in 0..mid) {
                tmpS[removable[i]] = '#'
            }

            if (isSubseq(tmpS)) {
                l = mid + 1
            } else {
                r = mid
            }
        }

        return l
    }
}

class Solution {
    func maximumRemovals(_ s: String, _ p: String, _ removable: [Int]) -> Int {
        var l = 0
        var r = removable.count
        var sArr = Array(s)
        let pArr = Array(p)
        let n = sArr.count
        let m = pArr.count

        func isSubseq(_ tmpS: [Character]) -> Bool {
            var i1 = 0
            var i2 = 0
            while i1 < n && i2 < m {
                if tmpS[i1] == pArr[i2] {
                    i2 += 1
                }
                i1 += 1
            }
            return i2 == m
        }

        while l < r {
            let mid = l + (r - l) / 2
            var tmpS = sArr

            for i in 0...mid {
                tmpS[removable[i]] = "#"
            }

            if isSubseq(tmpS) {
                l = mid + 1
            } else {
                r = mid
            }
        }

        return l
    }
}

Time & Space Complexity

Time complexity: $O((n + m) * \log k)$
Space complexity: $O(n)$

Where $n$ and $m$ are the lengths of the given strings $s$ and $p$ respectively. $k$ is the size of the array $removable$ .

Common Pitfalls

Off-by-One Errors in Binary Search Bounds

When setting up binary search, a common mistake is using incorrect bounds or midpoint calculations. The search space is 0 to len(removable), not 0 to len(removable) - 1, because we need to consider removing zero characters as a valid option. Additionally, confusing l < r vs l <= r loop conditions leads to either infinite loops or missing valid answers.

Modifying the Original String Instead of Using a Copy

When marking characters as removed (using '#' or similar), forgetting to create a fresh copy of the string for each binary search iteration causes state to persist incorrectly. The removed characters from a previous iteration pollute subsequent checks, leading to wrong results.

Incorrect Subsequence Check Logic

The subsequence check must skip removed indices while still correctly advancing through both strings. A common bug is incrementing the pattern pointer even when the current character is marked as removed, or failing to increment the source pointer when characters do not match. Both lead to incorrect subsequence determinations.