77** 1.1 Constraints & Complexity:**
88
99- ** Input Size:** n and k are both between 1 and 1000. The values can grow very large, so we need modulo 10^9 + 7.
10- - ** Time Complexity:** O(n * k) - we iterate k times, and for each iteration, we process n-1 elements.
10+ - ** Time Complexity:** O(n \ * k) - we iterate k times, and for each iteration, we process n-1 elements.
1111- ** Space Complexity:** O(n) - we maintain an array of size n.
1212- ** Edge Case:** When k=0, the array remains [ 1,1,...,1] , so the result is 1.
1313
@@ -17,7 +17,7 @@ The goal is to apply prefix sum operation k times. Each second, we compute the p
1717
1818** 1.3 Brute force vs. optimized strategy:**
1919
20- - ** Brute Force:** Simulate k seconds by computing prefix sums k times. This is O(n * k) time and O(n) space.
20+ - ** Brute Force:** Simulate k seconds by computing prefix sums k times. This is O(n \ * k) time and O(n) space.
2121- ** Optimized Strategy:** The same approach is already optimal. We can't avoid computing each prefix sum operation.
2222- ** Optimization:** Using modulo arithmetic ensures we don't overflow, and the direct simulation is the most straightforward approach.
2323
@@ -42,14 +42,14 @@ We apply prefix sum k=5 times.
4242
4343** 2.3 Trace Walkthrough:**
4444
45- | Second | Array State | Action |
46- | -------- | -------------| --------|
47- | 0 | [ 1, 1, 1, 1] | Initial state |
48- | 1 | [ 1, 2, 3, 4] | Prefix sum: arr[ 1] =1+1=2, arr[ 2] =1+2=3, arr[ 3] =1+2+3=4 |
49- | 2 | [ 1, 3, 6, 10] | Prefix sum: arr[ 1] =1+2=3, arr[ 2] =1+3+2=6, arr[ 3] =1+3+6=10 |
50- | 3 | [ 1, 4, 10, 20] | Prefix sum: arr[ 1] =1+3=4, arr[ 2] =1+4+3=10, arr[ 3] =1+4+10=20 |
51- | 4 | [ 1, 5, 15, 35] | Prefix sum: arr[ 1] =1+4=5, arr[ 2] =1+5+4=15, arr[ 3] =1+5+15=35 |
52- | 5 | [ 1, 6, 21, 56] | Prefix sum: arr[ 1] =1+5=6, arr[ 2] =1+6+5=21, arr[ 3] =1+6+21=56 |
45+ | Second | Array State | Action |
46+ | ------ | -------------- | ----------------------------------------------------------- |
47+ | 0 | [ 1, 1, 1, 1] | Initial state |
48+ | 1 | [ 1, 2, 3, 4] | Prefix sum: arr[ 1] =1+1=2, arr[ 2] =1+2=3, arr[ 3] =1+2+3=4 |
49+ | 2 | [ 1, 3, 6, 10] | Prefix sum: arr[ 1] =1+2=3, arr[ 2] =1+3+2=6, arr[ 3] =1+3+6=10 |
50+ | 3 | [ 1, 4, 10, 20] | Prefix sum: arr[ 1] =1+3=4, arr[ 2] =1+4+3=10, arr[ 3] =1+4+10=20 |
51+ | 4 | [ 1, 5, 15, 35] | Prefix sum: arr[ 1] =1+4=5, arr[ 2] =1+5+4=15, arr[ 3] =1+5+15=35 |
52+ | 5 | [ 1, 6, 21, 56] | Prefix sum: arr[ 1] =1+5=6, arr[ 2] =1+6+5=21, arr[ 3] =1+6+21=56 |
5353
5454** 2.4 Increment and Loop:**
5555
@@ -58,4 +58,3 @@ After k iterations, we have the final array state.
5858** 2.5 Return Result:**
5959
6060The value at index n-1 (index 3) after k=5 seconds is 56, which matches the example output.
61-
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