daily-2026-03-28-2573-find-the-string-with-lcp

problems/2573-find-the-string-with-lcp/analysis.md

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+# 2573. Find the String with LCP
+
+[LeetCode Link](https://leetcode.com/problems/find-the-string-with-lcp/)
+
+Difficulty: Hard
+Topics: Array, String, Dynamic Programming, Greedy, Union-Find, Matrix
+Acceptance Rate: 41.0%
+
+## Hints
+
+### Hint 1
+
+Think about what the LCP matrix tells you about character equality. If `lcp[i][j] > 0`, what does that say about the characters at positions `i` and `j`? This relationship can help you group positions together.
+
+### Hint 2
+
+Use a greedy strategy to assign characters. Start with 'a' at the first unassigned position, and propagate: any position `j` where `lcp[i][j] > 0` must share the same character as position `i`. This is essentially a Union-Find / grouping problem. After building the string greedily, you need to verify it actually produces the given LCP matrix.
+
+### Hint 3
+
+The verification step is the critical insight. After constructing a candidate string, recompute the LCP matrix using dynamic programming from the bottom-right corner: if `word[i] == word[j]`, then `lcp[i][j] = lcp[i+1][j+1] + 1` (with boundary handling). If the recomputed matrix doesn't match the input, no valid string exists. This DP is the same recurrence that defines LCP for suffixes.
+
+## Approach
+
+The algorithm has two phases: **construction** and **verification**.
+
+**Phase 1 — Greedy Construction:**
+
+Iterate through positions `0` to `n-1`. For each unassigned position `i`, assign the smallest available character (starting from `'a'`). Then scan all positions `j > i`: if `lcp[i][j] > 0`, position `j` must have the same character as `i` (since a positive LCP means the suffixes starting at `i` and `j` share at least their first character). If `j` was already assigned a different character, or if we exhaust all 26 letters, return `""`.
+
+**Phase 2 — Verification:**
+
+The greedy phase only enforces a necessary condition (character equality from `lcp > 0`). It doesn't verify the exact LCP values. We recompute the full LCP matrix from the constructed string using the DP recurrence:
+
+- If `word[i] == word[j]`: `lcp[i][j] = lcp[i+1][j+1] + 1` (or `1` if at boundary)
+- If `word[i] != word[j]`: `lcp[i][j] = 0`
+
+Process from bottom-right to top-left. If the recomputed matrix matches the input exactly, return the string; otherwise return `""`.
+
+**Example walkthrough with `lcp = [[4,0,2,0],[0,3,0,1],[2,0,2,0],[0,1,0,1]]`:**
+
+- Position 0: unassigned → assign `'a'`. Check lcp[0][j]: lcp[0][1]=0, lcp[0][2]=2>0 → assign `'a'` to position 2. lcp[0][3]=0.
+- Position 1: unassigned → assign `'b'`. Check lcp[1][j]: lcp[1][2]=0, lcp[1][3]=1>0 → assign `'b'` to position 3.
+- Positions 2,3 already assigned. Result: `"abab"`.
+- Verify: recomputed LCP matches input. Return `"abab"`.
+
+## Complexity Analysis
+
+Time Complexity: O(n²) — both the greedy scan and the verification DP iterate over all pairs (i, j).
+
+Space Complexity: O(n²) — for the verification LCP matrix. The string itself is O(n).
+
+## Edge Cases
+
+- **Diagonal mismatch:** `lcp[i][i]` must equal `n - i` (a suffix's LCP with itself is its own length). If violated, return `""`.
+- **Asymmetry:** `lcp[i][j]` must equal `lcp[j][i]`. The verification step catches this.
+- **Value too large:** `lcp[i][j] > n - max(i,j)` is impossible. The verification catches this too.
+- **More than 26 groups:** If the matrix implies more than 26 distinct character groups, we run out of letters. Return `""`.
+- **n = 1:** Single character. `lcp` must be `[[1]]`, answer is `"a"`.

problems/2573-find-the-string-with-lcp/problem.md

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+---
+number: "2573"
+frontend_id: "2573"
+title: "Find the String with LCP"
+slug: "find-the-string-with-lcp"
+difficulty: "Hard"
+topics:
+  - "Array"
+  - "String"
+  - "Dynamic Programming"
+  - "Greedy"
+  - "Union-Find"
+  - "Matrix"
+acceptance_rate: 4100.8
+is_premium: false
+created_at: "2026-03-28T03:20:43.124847+00:00"
+fetched_at: "2026-03-28T03:20:43.124847+00:00"
+link: "https://leetcode.com/problems/find-the-string-with-lcp/"
+date: "2026-03-28"
+---
+
+# 2573. Find the String with LCP
+
+We define the `lcp` matrix of any **0-indexed** string `word` of `n` lowercase English letters as an `n x n` grid such that:
+
+  * `lcp[i][j]` is equal to the length of the **longest common prefix** between the substrings `word[i,n-1]` and `word[j,n-1]`.
+
+
+
+Given an `n x n` matrix `lcp`, return the alphabetically smallest string `word` that corresponds to `lcp`. If there is no such string, return an empty string.
+
+A string `a` is lexicographically smaller than a string `b` (of the same length) if in the first position where `a` and `b` differ, string `a` has a letter that appears earlier in the alphabet than the corresponding letter in `b`. For example, `"aabd"` is lexicographically smaller than `"aaca"` because the first position they differ is at the third letter, and `'b'` comes before `'c'`.
+
+
+
+**Example 1:**
+
+
+    **Input:** lcp = [[4,0,2,0],[0,3,0,1],[2,0,2,0],[0,1,0,1]]
+    **Output:** "abab"
+    **Explanation:** lcp corresponds to any 4 letter string with two alternating letters. The lexicographically smallest of them is "abab".
+
+
+**Example 2:**
+
+
+    **Input:** lcp = [[4,3,2,1],[3,3,2,1],[2,2,2,1],[1,1,1,1]]
+    **Output:** "aaaa"
+    **Explanation:** lcp corresponds to any 4 letter string with a single distinct letter. The lexicographically smallest of them is "aaaa". 
+
+
+**Example 3:**
+
+
+    **Input:** lcp = [[4,3,2,1],[3,3,2,1],[2,2,2,1],[1,1,1,3]]
+    **Output:** ""
+    **Explanation:** lcp[3][3] cannot be equal to 3 since word[3,...,3] consists of only a single letter; Thus, no answer exists.
+
+
+
+
+**Constraints:**
+
+  * `1 <= n == ``lcp.length == ``lcp[i].length` `<= 1000`
+  * `0 <= lcp[i][j] <= n`

problems/2573-find-the-string-with-lcp/solution_daily_20260328.go

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+package main
+
+// Greedy construction + DP verification.
+// 1. Assign characters greedily: for each unassigned position, use the next
+//    letter. Propagate: if lcp[i][j] > 0, positions i and j share a character.
+// 2. Verify by recomputing the LCP matrix via DP and comparing with input.
+func findTheString(lcp [][]int) string {
+	n := len(lcp)
+	word := make([]byte, n)
+
+	var c byte = 'a'
+	for i := 0; i < n; i++ {
+		if word[i] != 0 {
+			continue
+		}
+		if c > 'z' {
+			return ""
+		}
+		word[i] = c
+		for j := i + 1; j < n; j++ {
+			if lcp[i][j] > 0 {
+				if word[j] != 0 && word[j] != c {
+					return ""
+				}
+				word[j] = c
+			}
+		}
+		c++
+	}
+
+	// Verify: recompute LCP matrix from the constructed string and compare.
+	actual := make([][]int, n)
+	for i := range actual {
+		actual[i] = make([]int, n)
+	}
+
+	for i := n - 1; i >= 0; i-- {
+		for j := n - 1; j >= 0; j-- {
+			if word[i] == word[j] {
+				if i+1 < n && j+1 < n {
+					actual[i][j] = actual[i+1][j+1] + 1
+				} else {
+					actual[i][j] = 1
+				}
+			}
+			if actual[i][j] != lcp[i][j] {
+				return ""
+			}
+		}
+	}
+
+	return string(word)
+}

problems/2573-find-the-string-with-lcp/solution_daily_20260328_test.go

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+package main
+
+import "testing"
+
+func TestFindTheString(t *testing.T) {
+	tests := []struct {
+		name     string
+		lcp      [][]int
+		expected string
+	}{
+		{
+			name:     "example 1: alternating pattern abab",
+			lcp:      [][]int{{4, 0, 2, 0}, {0, 3, 0, 1}, {2, 0, 2, 0}, {0, 1, 0, 1}},
+			expected: "abab",
+		},
+		{
+			name:     "example 2: all same character aaaa",
+			lcp:      [][]int{{4, 3, 2, 1}, {3, 3, 2, 1}, {2, 2, 2, 1}, {1, 1, 1, 1}},
+			expected: "aaaa",
+		},
+		{
+			name:     "example 3: invalid lcp[3][3] = 3",
+			lcp:      [][]int{{4, 3, 2, 1}, {3, 3, 2, 1}, {2, 2, 2, 1}, {1, 1, 1, 3}},
+			expected: "",
+		},
+		{
+			name:     "edge case: single character",
+			lcp:      [][]int{{1}},
+			expected: "a",
+		},
+		{
+			name:     "edge case: invalid single element lcp[0][0] != 1",
+			lcp:      [][]int{{2}},
+			expected: "",
+		},
+		{
+			name:     "edge case: asymmetric matrix",
+			lcp:      [][]int{{2, 1}, {0, 1}},
+			expected: "",
+		},
+		{
+			name:     "edge case: two distinct characters",
+			lcp:      [][]int{{2, 0}, {0, 1}},
+			expected: "ab",
+		},
+	}
+
+	for _, tt := range tests {
+		t.Run(tt.name, func(t *testing.T) {
+			result := findTheString(tt.lcp)
+			if result != tt.expected {
+				t.Errorf("got %q, want %q", result, tt.expected)
+			}
+		})
+	}
+}