Number of Ways to Form a Target String Given a Dictionary
Time O(l * (w + n)) · Space O(n) · Official statement on LeetCode
Solutions
// Time: O(l * (w + n)), l is the length of a word, w is the number of words, n is the length of target
// Space: O(n)
// optimized from Solution2
class Solution {
public:
int numWays(vector<string>& words, string target) {
static const int MOD = 1e9 + 7;
vector<uint64_t> dp(size(target) + 1); // use uint64_t to avoid overflow
dp[0] = 1;
for (int i = 0; i < size(words[0]); ++i) {
vector<int> count(26);
for (const auto& w : words) {
++count[w[i] - 'a'];
}
for (int j = size(target) - 1; j >= 0; --j) {
dp[j + 1] += dp[j] * count[target[j] - 'a'] % MOD;
}
}
return dp.back() % MOD;
}
};
// Time: O(l * (w + n)), l is the length of a word, w is the number of words, n is the length of target
// Space: O(n)
class Solution2 {
public:
int numWays(vector<string>& words, string target) {
static const int MOD = 1e9 + 7;
// dp[i+1][j+1]: number of ways of target[0..j] using count[0..i].
vector<vector<uint64_t>> dp(2, vector<uint64_t>(size(target) + 1)); // use uint64_t to avoid overflow
dp[0][0] = dp[1][0] = 1;
for (int i = 0; i < size(words[0]); ++i) {
vector<int> count(26);
for (const auto& w : words) {
++count[w[i] - 'a'];
}
for (int j = 0; j < size(target); ++j) {
dp[(i + 1) % 2][j + 1] = dp[i % 2][j + 1] + dp[i % 2][j] * count[target[j] - 'a'] % MOD;
}
}
return dp[size(words[0]) % 2].back() % MOD;
}
};
Beginner Explanation
What is Number of Ways to Form a Target String Given a Dictionary?
Number of Ways to Form a Target String Given a Dictionary (LeetCode #1639) is a Hard problem that primarily trains dynamic programming.
How to think about it
- Restate the goal in your own words before coding.
- Work a tiny example by hand so the invariant becomes obvious.
- Identify the pattern — this problem aligns with dynamic programming.
- Only then translate the idea into code.
Why this problem matters
Hard problems force you to combine patterns and prove complexity carefully — interview gold.
AlgoForge explanations are original teaching notes. Always open the official problem statement on LeetCode for constraints and examples.
Interview Walkthrough
Interview approach for Number of Ways to Form a Target String Given a Dictionary
Opening (30–60 seconds)
- Clarify inputs/outputs and edge cases (empty input, single element, duplicates, overflow).
- State a brute force so the interviewer knows you can solve it naively.
- Propose the optimal direction tied to dynamic programming.
Core solution narrative
- Define the state you track (pointers, DP cell, set membership, stack top, etc.).
- Explain the transition when you process the next element.
- Call out time (O(l * (w + n))) and space (O(n)) before coding.
- Code cleanly; narrate variable names.
What interviewers listen for
- Correctness on edge cases
- Complexity honesty
- Ability to discuss trade-offs (e.g., hash map space vs. sort + two pointers)
Follow-up questions they may ask
- Can you solve it with less memory?
- What if the input stream is infinite / doesn't fit in RAM?
- How would tests look for adversarial inputs?
Optimized Approach
Optimized solution notes
The reference solutions on AlgoForge target O(l * (w + n)) time and O(n) space.
Pattern focus: dynamic programming
Use the pattern as a checklist:
- dynamic programming — confirm the invariant holds after each step
Multiple methods appear in the source solutions — compare them and explain when each is preferable.
Implementation tips
- Prefer readable names over micro-optimizations in interviews.
- Extract helpers only when they clarify (e.g., expand-around-center, DFS visit).
- After AC-level logic, re-scan for off-by-one and null checks.
Complexity Analysis
Complexity
| Measure | Bound |
|---|---|
| Time | O(l * (w + n)) |
| Space | O(n) |
How to justify this in an interview
- Time: count loops, map/set operations, and recursive branching; state average vs worst case if relevant.
- Space: include hash maps, recursion stack, and output allocation when the problem asks for it.
If your implementation differs from the reference, re-derive big-O from your code — never memorize a complexity you cannot defend.
Common Mistakes
Common mistakes on Number of Ways to Form a Target String Given a Dictionary
- Skipping edge cases — empty collections, single-element inputs, max constraints.
- Wrong invariant for dynamic programming — updating state too early or too late.
- Mutating input unexpectedly when the problem forbids it.
- Off-by-one in windows, ranges, or binary search bounds.
- Ignoring overflow / precision for integer arithmetic problems.
- Overengineering — jumping to an advanced structure when a simpler approach works.
Alternative Approaches
Alternatives
The source file includes more than one method. Compare:
- Primary optimized path — best complexity for typical interviews.
- Secondary approach — often brute force, sorting-based, or space-optimized variant.
Practice articulating when you would pick each (constraints, readability, follow-ups).
Edge Cases
Edge cases checklist
- Minimum input size
- Maximum input size / time limits
- Duplicates and already-sorted input
- Negative numbers / zeros (if applicable)
- Disconnected structures (graphs/trees)
- Single path vs branching recursion depth
Pattern Recognition
Spotting this pattern
Signal phrases that point to dynamic programming:
- Sorted input or ability to sort without changing the answer class
- Need for contiguous subarray / substring → consider sliding window
- Need for O(1) membership → hash set/map
- Optimal substructure + overlapping subproblems → DP
- Connectivity / components → graph DFS/BFS or Union-Find
Primary topics: dynamic programming.
Follow-up Interview Questions
Follow-ups
- How does the solution change if the input is a stream?
- Can you solve it in-place?
- What if duplicates must be handled differently?
- How would you parallelize the approach?
- Design tests that would break a buggy implementation.
Practice Recommendations
What to practice next
- Re-solve Number of Ways to Form a Target String Given a Dictionary in a second language (cpp, python).
- Drill 3–5 more problems tagged dynamic programming.
- Teach the solution out loud in under 5 minutes.
- Add this problem to your revision calendar in 3 days and 14 days.
Visualization
Study checklist
- Read the official problem statement on LeetCode
- Solve on paper / whiteboard first
- Implement the dynamic programming approach
- Verify edge cases from the checklist
- State time and space complexity aloud
- Compare with the AlgoForge reference solution
- Schedule a revision session
Revision notes
Number of Ways to Form a Target String Given a Dictionary (#1639) — Hard. Pattern: dynamic programming. Complexity: O(l * (w + n)) time / O(n) space. Re-derive the invariant before coding.
FAQs
What is the time complexity of Number of Ways to Form a Target String Given a Dictionary?+
The reference solutions aim for O(l * (w + n)) time and O(n) space. Always re-derive complexity from the code you write in the interview.
What pattern does Number of Ways to Form a Target String Given a Dictionary use?+
It primarily maps to dynamic programming, within the broader topic of dynamic programming.
Is Number of Ways to Form a Target String Given a Dictionary good for interviews?+
Yes — as a Hard problem it is a solid practice target. Pair it with related problems in the same pattern family for spaced repetition.
Where can I read the official statement?+
Open the official LeetCode page for constraints and examples: https://leetcode.com/problems/number-of-ways-to-form-a-target-string-given-a-dictionary/