Sum is One

AtCoder

IDttpc2024_1_k

Time2000ms

Memory256MB

Difficulty—

You are given a sequence $A = (A_1, A_2, \dots, A_N)$ of length $N$ consisting of $0$s and $1$s. Define a simple undirected graph $G = (V, E)$ with $\frac{N (N - 1)}{2}$ vertices as follows: * For every pair of integers $(i, j)$ satisfying $1 \leq i < j \leq N$, $(i, j) \in V$. This vertex is called vertex $(i, j)$. * For every triple of integers $(i, j, k)$ satisfying $1 \leq i < j < k \leq N$ and $A_i + A_j + A_k = 1$, there is an edge connecting vertex $(i, j)$ and vertex $(j, k)$. * No other pairs of vertices have edges. Determine the number of connected components in $G$. You are given $T$ test cases. For each test case, compute the answer. ## Constraints * All input values are integers. * $1 \leq T \leq 10^5$ * $3 \leq N \leq 10^6$ * $A_i$ is $0$ or $1$ ($1 \leq i \leq N$) * The total sum of $N$ over all test cases in a single input is at most $10^6$. ## Input The input is given in the following format: $T$ $\text{case}_1$ $\text{case}_2$ $\vdots$ $\text{case}_T$ Here, $\text{case}_i$ represents the $i$\-th test case. Each test case is given in the following format: $N$ $A_1$ $A_2$ $\cdots$ $A_N$ ## Partial Score $30$ points will be awarded for passing the test set satisfying the following constraints: * $1 \leq T \leq 1000$ * $3 \leq N \leq 5000$ * The total sum of $N$ over all test cases in a single input is at most $5000$. [samples]

Samples

Input #1

4
5
1 0 0 1 0
5
1 1 1 1 1
12
0 0 1 1 1 0 0 0 1 0 1 0
20
0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1

Output #1

4
10
13
58

For the first test case, the connected components are as follows:

*   $\lbrace (1,2),(2,3),(2,4),(2,5),(3,4),(4,5) \rbrace$
*   $\lbrace (1,3),(3,5) \rbrace$
*   $\lbrace (1,4) \rbrace$
*   $\lbrace (1,5) \rbrace$

For the second test case, the graph has no edges, so the number of connected components is $10$.

API Response (JSON)

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    "name": "Sum is One",
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      "content": "You are given a sequence $A = (A_1, A_2, \\dots, A_N)$ of length $N$ consisting of $0$s and $1$s. Define a simple undirected graph $G = (V, E)$ with $\\frac{N (N - 1)}{2}$ vertices as follows: *   For ",
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "sync_url": null,
    "sign": "ttpc2024_1_k"
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      "statement_type": "Markdown",
      "content": "You are given a sequence $A = (A_1, A_2, \\dots, A_N)$ of length $N$ consisting of $0$s and $1$s. Define a simple undirected graph $G = (V, E)$ with $\\frac{N (N - 1)}{2}$ vertices as follows:\n\n*   For ...",
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