Pebbles

Time limit1sMemory limit128 MB

Summary
Place pebbles on an N by N board so no two touch even diagonally, maximizing the sum of covered cell values.
Level

Medium6 of 10

Topics
Dynamic programming, Bit manipulation, Brute force, Implementation
Solved
No attempts yet

Problem

You have an unlimited supply of pebbles to place on an N×NN \times N board, where 3≤N≤153 \le N \le 15. Every square holds a positive point value between 1010 and 9999 inclusive. For example, a 6×66 \times 6 board might look like this:

337426557954
675691724432
446422912961
613276505032
816556389636
387850929075

Place pebbles on the board subject to two rules:

  • At most one pebble may sit on any single square.
  • No two pebbles may occupy adjacent squares. Two squares are adjacent when they touch horizontally, vertically, or diagonally.

The board never wraps around, so squares at opposite ends of a row or column, and the two far corners, are not adjacent.

Your score is the sum of the point values of every square that holds a pebble. Maximize this score.

The input may contain several boards; report the best attainable score for each one.

Input

Each board is a block of NN lines, and every line lists NN space-separated point values (one per square). A blank line separates consecutive boards. Keep reading boards until the input ends.

Output

For each board, print a single integer on its own line: the maximum total score achievable by a valid pebble placement.

Examples1

  1. Example 1

    Input
    71 24 95 56 54
85 50 74 94 28
92 96 23 71 10
23 61 31 30 46
64 33 32 95 89

78 78 11 55 20 11
98 54 81 43 39 97
12 15 79 99 58 10
13 79 83 65 34 17
85 59 61 12 58 97
40 63 97 85 66 90

33 49 78 79 30 16 34 88 54 39 26
80 21 32 71 89 63 39 52 90 14 89
49 66 33 19 45 61 31 29 84 98 58
36 53 35 33 88 90 19 23 76 23 76
77 27 25 42 70 36 35 91 17 79 43
33 85 33 59 47 46 63 75 98 96 55
75 88 10 57 85 71 34 10 59 84 45
29 34 43 46 75 28 47 63 48 16 19
62 57 91 85 89 70 80 30 19 38 14
61 35 36 20 38 18 89 64 63 88 83
45 46 89 53 83 59 48 45 87 98 21

15 95 24 35 79 35 55 66 91 95 86 87
94 15 84 42 88 83 64 50 22 99 13 32
85 12 43 39 41 23 35 97 54 98 18 85
84 61 77 96 49 38 75 95 16 71 22 14
18 72 97 94 43 18 59 78 33 80 68 59
26 94 78 87 78 92 59 83 26 88 91 91
34 84 53 98 83 49 60 11 55 17 51 75
29 80 14 79 15 18 94 39 69 24 93 41
66 64 88 82 21 56 16 41 57 74 51 79
49 15 59 21 37 27 78 41 38 82 19 62
54 91 47 29 38 67 52 92 81 99 11 27
31 62 32 97 42 93 43 79 88 44 54 48
    Expected output
    572
683
2096
2755