Bridging signals

http://acm.pku.edu.cn/JudgeOnline/problem?id=1631

Description

'Oh no, they've done it again', cries the chief designer at the Waferland chip factory. Once more the routing designers have screwed up completely, making the signals on the chip connecting the ports of two functional blocks cross each other all over the place. At this late stage of the process, it is too expensive to redo the routing. Instead, the engineers have to bridge the signals, using the third dimension, so that no two signals cross. However, bridging is a complicated operation, and thus it is desirable to bridge as few signals as possible. The call for a computer program that finds the maximum number of signals which may be connected on the silicon surface without crossing each other, is imminent. Bearing in mind that there may be thousands of signal ports at the boundary of a functional block, the problem asks quite a lot of the programmer. Are you up to the task?

A typical situation is schematically depicted in figure 1. The ports of the two functional blocks are numbered from 1 to p, from top to bottom. The signal mapping is described by a permutation of the numbers 1 to p in the form of a list of p unique numbers in the range 1 to p, in which the i:th number specifies which port on the right side should be connected to the i:th port on the left side.Two signals cross if and only if the straight lines connecting the two ports of each pair do.

Input

On the first line of the input, there is a single positive integer n, telling the number of test scenarios to follow. Each test scenario begins with a line containing a single positive integer p < 40000, the number of ports on the two functional blocks. Then follow p lines, describing the signal mapping:On the i:th line is the port number of the block on the right side which should be connected to the i:th port of the block on the left side.

Output

For each test scenario, output one line containing the maximum number of signals which may be routed on the silicon surface without crossing each other.

Source

Northwestern Europe 2003


Solving

두 쌍의 포트들이 있다. 이 포트를 브릿지를 이용해서 연결을 하는데 왼쪽포트와 오른쪽포트의 쌍이 정해져 있고 1:1 매칭을 이룬다. 브릿지가 서로 교차하지 않게 연결을 하고자 할 때, 한 번에 얼마나 많은 포트를 연결할 수 있는가.

어찌보면 그래프 문제같기도 하고 이분매칭 같기도 하고 단번에 감이 잘안오는 문제이다.
문제를 풀면서 처음에는 O(n^2)의 알고리즘으로 풀었는데 n이 최대 40000이기 때문에 TLE로 고생을 하다 갑자기 생각난 Binary Seach를 통한 해법으로 겨우 풀었던 기억이 난다. 문제를 풀고나서 깨닳게 되었는데 이 문제는 Logest Incresing Sequence 문제였고 LIS는 O(n log n)으로 풀린다는 것을 나만 모르고 있었다 ㅠㅠ

아는 것이 힘이다.

문제가 LIS인 이유는 간단하다. i번째 브릿지가 A->B 라면 Ai=B라고 표현할 수 있다. 브릿지가  브릿지가 서로 교차하지 않고 연결을 하게 되면 선택된 Ai들을 모아보면 증가수열이다. 가장 많은 브릿지를 원하므로 결국엔 가장긴 증가수열을 구하는 문제가 된다.

#include <cstdlib> #include <cstdio> #include <vector> using namespace std; int main() { int T; scanf("%d", &T); while(T--) { int i, j, n; scanf("%d", &n); int link; int linkC = 0; int linkE = 0; int fi; vector<int> thres(n+1, n); //for (i = 0; i <= n; i++) thres[i] = n; for (i = 0; i < n; i++) { scanf("%d", &link); if (linkC == 0) { linkC = 1; linkE = link; thres[1] = link; } else { if (linkE < link) { linkC++; linkE = link; thres[linkC] = link; } else { int l = 1; int h = n-1; int t = (l+h)/2; while (t >= l) { if (thres[t] > link) { fi = t; h = t-1; t = (l+h)/2; } else { l = t+1; t = (l+h)/2; } } thres[fi] = link; if (fi == linkC) linkE = link; } } } printf("%d\n", linkC); } return 0; }