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| Problem 116 - Unidirectional TSP, Explanations |
Algorithm
Process the matrix in the backward column order starting at the row before the last one. For each cell, add the smallest one of the three possible next number (right-up,right,right-down) to the current number. This way, the minimum of the first column is the numeric answer. To get the path, remember each parent choice and ouput the path starting at the lowest row first column cell with minimal score.
Trick
Don't forget the circular map.
In case of a score tie, the father is the smallest numbered one.
BEWARE of the so-called lexicographic order! It is based on the numbers, not on the strings. So, '1 2' > '1 1' and '1 10' > '1 2'.
There can be more than one minimum path starting location. Be sure to ouput the smallest one!
If you are in trouble with the formatted input, read my how to read input.
Additional Input
116.in
Additional Output
116.out
Problems that require minimum paths through some domain appear in many
different areas of computer science. For example, one of the
constraints in VLSI routing problems is minimizing wire length. The
Traveling Salesperson Problem (TSP) -- finding whether all the cities in
a salesperson's route can be visited exactly once with a specified limit
on travel time -- is one of the canonical examples of an NP-complete
problem; solutions appear to require an inordinate amount of time to
generate, but are simple to check.
This problem deals with finding a minimal path through a grid of
points while traveling only from left to right.
Given an 
matrix of integers, you are to write a
program that computes a path of minimal weight. A path starts anywhere
in column 1 (the first column) and consists of a sequence of steps
terminating in column n (the last column). A step consists of
traveling from column i to column i+1 in an adjacent (horizontal or
diagonal) row. The first and last rows (rows 1 and m) of a matrix
are considered adjacent, i.e., the matrix ``wraps'' so that it represents
a horizontal cylinder. Legal steps are illustrated below.
The weight of a path is the sum of the integers in
each of the n cells of the matrix that are visited.
For example, two slightly different 
matrices are shown below (the only difference is the numbers in the bottom
row).
The minimal path is illustrated for each matrix. Note that the path for
the matrix on the right takes advantage of the adjacency property of
the first and last rows.
The input consists of a sequence of matrix specifications. Each matrix
specification consists of the row and column dimensions in that order on
a line followed by 
integers where m is the row dimension
and n is the column dimension. The integers appear in the input in
row major order, i.e., the first n integers constitute the first row of
the matrix, the second n integers constitute the second row and so on.
The integers on a line will be separated from other integers by one or
more spaces. Note: integers are not restricted to being positive.
There will be one or more matrix specifications in an
input file. Input is terminated by end-of-file.
For each specification the number of rows will be between 1 and 10
inclusive; the number of columns will be between 1 and 100 inclusive.
No path's weight will exceed integer values representable
using 30 bits.
Two lines should be output for each matrix specification in the input
file, the first line represents a minimal-weight path, and the second
line is the cost of a minimal path. The path consists of a sequence of
n integers (separated by one or more spaces)
representing the rows that constitute the minimal path. If
there is more than one path of minimal weight the path that is
lexicographically smallest should be output.
5 6
3 4 1 2 8 6
6 1 8 2 7 4
5 9 3 9 9 5
8 4 1 3 2 6
3 7 2 8 6 4
5 6
3 4 1 2 8 6
6 1 8 2 7 4
5 9 3 9 9 5
8 4 1 3 2 6
3 7 2 1 2 3
2 2
9 10 9 10
1 2 3 4 4 5
16
1 2 1 5 4 5
11
1 1
19
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