AllGraphCuts | Topcoder Archive

Problem Statement

All graphs in this problem statement are simple undirected graphs with nonnegative integer edge weights. Here, "simple" means that there are no self-loops and each pair of vertices is connected by at most one direct edge.

Given a connected graph G, a cut between vertices i and j is a set E of edges such that each path from i to j contains at least one edge from the set E. In other words, if you cut all edges in E, the vertices i and j will end in different connected components.

If E is a set of edges, the weight of E is the sum of weights of edges it contains. For any distinct vertices i and j, the min cut weight between i and j is the minimum weight of a cut between i and j. Additionally, for any vertex i we define that the min cut weight between i and i is 0.

You are given a int[] x with n*n elements. You would like to find a connected graph G with n vertices, labeled 0 through n-1. The graph G must have the following properties:

For each i and j, the min cut weight between vertices i and j must be exactly x[i*n+j].
G must have at most 1000 edges.
The weight of each edge must be between 0 and 10^5, inclusive.

For the given constraints it is guaranteed that whenever there is a graph that has the first property, there is a graph that has all three properties.

If there are no solutions, return the int[] {-1}. Otherwise, find any graph G that has the required properties and return a int[] with exactly m elements, where m is the number of edges in G. If G contains an edge (i,j) with weight w, the return value should contain the number (w*n*n + i*n + j).

Definition

Class:: AllGraphCuts
Method:: findGraph
Parameters:: int[]
Returns:: int[]
Method signature:: int[] findGraph(int[] x)
(be sure your method is public)

Constraints

n will be between 2 and 50, inclusive.
x will have exactly n*n elements.
Each element of x will be between 0 and 10^5, inclusive.

Examples

```
{0,1,
 1,0}
```
```
Returns: {6 }
```
In this case n = 2. The output can be decoded as a single edge between nodes 0 and 1 with weight 1 (i.e. 1*2*2 + 1*2 + 0 = 6).
```
{0,1,
 1,1}
```
```
Returns: {-1 }
```
Note that since we have x[1*2+1] != 0, there is no valid solution.
```
{0,2,2,
 2,0,2,
 2,2,0}
```
```
Returns: {10, 11, 14 }
```
This graph can be decoded as a triangle graph, where all the edge weights are 1.

{0,1,2,3,4,
 1,0,2,3,4,
 1,2,0,3,4,
 1,2,3,0,4,
 1,2,3,4,0}

Returns: {-1 }

```
{0,0,0,0}
```
```
Returns: {2 }
```
Note that some edge weights can be zero. The empty int[] {} is not a correct return value, since the graph G you return must be connected.

{
0, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3044, 3082,
2545, 0, 2348, 2545, 2545, 2545, 2013, 2545, 2545, 2440, 2495, 2262, 2342, 2545, 2545, 2545, 2545, 2545, 2545, 2545,
2348, 2348, 0, 2348, 2348, 2348, 2013, 2348, 2348, 2348, 2348, 2262, 2342, 2348, 2348, 2348, 2348, 2348, 2348, 2348,
2993, 2545, 2348, 0, 2606, 2623, 2013, 2993, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 2993, 2993,
2606, 2545, 2348, 2606, 0, 2606, 2013, 2606, 2606, 2440, 2495, 2262, 2342, 2606, 2606, 2606, 2606, 2606, 2606, 2606,
2623, 2545, 2348, 2623, 2606, 0, 2013, 2623, 2623, 2440, 2495, 2262, 2342, 2617, 2623, 2623, 2623, 2623, 2623, 2623,
2013, 2013, 2013, 2013, 2013, 2013, 0, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013,
3001, 2545, 2348, 2993, 2606, 2623, 2013, 0, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3001, 3001,
2626, 2545, 2348, 2626, 2606, 2623, 2013, 2626, 0, 2440, 2495, 2262, 2342, 2617, 2626, 2626, 2626, 2626, 2626, 2626,
2440, 2440, 2348, 2440, 2440, 2440, 2013, 2440, 2440, 0, 2440, 2262, 2342, 2440, 2440, 2440, 2440, 2440, 2440, 2440,
2495, 2495, 2348, 2495, 2495, 2495, 2013, 2495, 2495, 2440, 0, 2262, 2342, 2495, 2495, 2495, 2495, 2495, 2495, 2495,
2262, 2262, 2262, 2262, 2262, 2262, 2013, 2262, 2262, 2262, 2262, 0, 2262, 2262, 2262, 2262, 2262, 2262, 2262, 2262,
2342, 2342, 2342, 2342, 2342, 2342, 2013, 2342, 2342, 2342, 2342, 2262, 0, 2342, 2342, 2342, 2342, 2342, 2342, 2342,
2617, 2545, 2348, 2617, 2606, 2617, 2013, 2617, 2617, 2440, 2495, 2262, 2342, 0, 2617, 2617, 2617, 2617, 2617, 2617,
2773, 2545, 2348, 2773, 2606, 2623, 2013, 2773, 2626, 2440, 2495, 2262, 2342, 2617, 0, 2662, 2736, 2773, 2773, 2773,
2662, 2545, 2348, 2662, 2606, 2623, 2013, 2662, 2626, 2440, 2495, 2262, 2342, 2617, 2662, 0, 2662, 2662, 2662, 2662,
2736, 2545, 2348, 2736, 2606, 2623, 2013, 2736, 2626, 2440, 2495, 2262, 2342, 2617, 2736, 2662, 0, 2736, 2736, 2736,
2827, 2545, 2348, 2827, 2606, 2623, 2013, 2827, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 0, 2827, 2827,
3044, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 0, 3044,
3082, 2545, 2348, 2993, 2606, 2623, 2013, 3001, 2626, 2440, 2495, 2262, 2342, 2617, 2773, 2662, 2736, 2827, 3044, 0
}

Returns: {76801, 87602, 76403, 15604, 14005, 14406, 95607, 14408, 45609, 54010, 113211, 102812, 53613, 72414, 10015, 90416, 110417, 108818, 76819, 20022, 5223, 16824, 14025, 20426, 37227, 115628, 118829, 105230, 39631, 114032, 70833, 2434, 9235, 71636, 3637, 108438, 68439, 16443, 24444, 37245, 54046, 118447, 46048, 13249, 38450, 27651, 23652, 105253, 76054, 22455, 112056, 5657, 82458, 28859, 31264, 117665, 73666, 91667, 11268, 67269, 110870, 18471, 114872, 119673, 22074, 86075, 44876, 22477, 74478, 93679, 96085, 8486, 59687, 110088, 1689, 107290, 56091, 56092, 74893, 61294, 109295, 109296, 18097, 66498, 20899, 12506, 98907, 82908, 77309, 51710, 110111, 108112, 22513, 66514, 43315, 36516, 19717, 15318, 26519, 34127, 70128, 8929, 20930, 131, 81732, 80133, 29334, 68935, 53336, 81737, 36538, 57739, 35348, 96149, 25750, 19351, 15752, 4153, 80554, 107355, 96556, 70557, 66958, 48559, 33769, 81370, 55771, 15372, 61373, 72574, 11375, 2976, 66577, 108178, 57779, 590, 78991, 14992, 10593, 69794, 47395, 33796, 113397, 77398, 68999, 77811, 28212, 12613, 64614, 80215, 9816, 27017, 30618, 73819, 46632, 66633, 7834, 100635, 636, 7437, 4238, 76639, 87453, 16254, 10255, 31856, 50257, 18258, 3459, 73474, 69475, 63876, 41077, 3878, 28679, 13495, 56696, 109497, 101098, 116699, 60316, 107517, 12318, 98719, 51137, 91138, 81139, 108358, 119959, 106379 }

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